greater_tables_project/greater_tables/data/notes.csv

1		expr
2	0	$\sigma=0.075$
3	1	$(\s, 4-\s)$
4	2	$\le 1/(1-p)$
5	3	$x\mapsto \mathsf{E}(X_i\wedge x)$
6	4	$a(f)=\dfrac{gs_g}{1-f-fgs_g}$
7	5	$\int_0^1 \phi(p)dp=1$
8	6	$\ge 5000$
9	7	$E_2\not=0$
10	8	$\rho(-1_{A^c}) = \rho(-1_{B_l} - 1_{B_r}) = \rho(-1_{B_l}) + \rho(-1_{B_r})$
11	9	$X:\{\text{Explicit Events}\}\to\mathbb{R}$
12	10	$\tilde p<p$
13	11	$M(X_1, a)+M(X_2, a)=M(X_1+X_2, a)$
14	12	$\bar x + t\bar h$
15	13	$\rho(A_0) + k \mathsf E[N]$
16	14	$p-1=28$
17	15	$Z(u)=sum_i u_iX_i$
18	16	$X(\omega)=\omega$
19	17	$\bar S(a)$
20	18	$i=1,\dots,n_d$
21	19	$f_x(x_i, \hat x_i)$
22	20	$1-r_0$
23	21	$(rep.east) + (1.5, 1.5)$
24	22	$R_1(t)= \bar P^a_1(t)/(1-t)$
25	23	${1+1}*(1,.5)$
26	24	$g(S_t(a(t)))$
27	25	$ and derives $
28	26	$2^{-72}=1/4722366482869645213696=1/4.7\times 10^{21}$
29	27	$\mathsf{E}(X) = \displaystyle\int_0^\infty xf(x)dx = -xS(x)\vert_0^\infty + \displaystyle\int_0^\infty S(x)dx = \displaystyle\int_0^\infty S(x)dx$
30	28	$\mathbf{T_0}=(\mathsf{TVaR}_{p_j}(X_i))_{i,j}$
31	29	$\mu(\Omega)\not=1$
32	30	$D=\sum_{i\in I} D_i$
33	31	$Z$
34	32	$(X_i, a_i)$
35	33	$\hat\rho(A_{k_0}) \ge \rho(A_{k_0})$
36	34	$\rho(X_n)\uparrow 0$
37	35	$\rho_t(X) \ge \mathsf E[\rho_{t+1}(X)\mid \mathcal F_t]$
38	36	$A - \mathsf E[A] \succeq_2 A_0$
39	37	$GF$
40	38	$f\le 0$
41	39	$0<t<0.5$
42	40	$X=NF(\bar x)$
43	41	$q<\infty$
44	42	$\beta=1$
45	43	$|X|=X_++X_-$
46	44	$O(dt^2)$
47	45	$(F(x),x)=(1-S(x), x)$
48	46	$N_a$
49	47	$0\le N\le G$
50	48	$\rho(X)=E(gX)$
51	49	$mu\in\mathscr{P}[0,1]$
52	50	$11 million occurs a loss of $
53	51	$X\wedge 1:=\min(X,1)$
54	52	$=\iota=$
55	53	$0<b<1$
56	54	$\rho(0) = \rho(0+0)\le \rho(0)+\rho(0)$
57	55	$Z=\sigma(U)$
58	56	$X\le 0$
59	57	$p_i=i/(N+1)$
60	58	$\rho_m(X\wedge k)$
61	59	$w(s)$
62	60	$\mathsf{Pr}(X<0)=0$
63	61	$\not\Rightarrow$
64	62	$\rho(X \wedge a)$
65	63	$E=\tau=0$
66	64	$m=mg^{ak}/g^{ak}$
67	65	$[t_2,1]$
68	66	$y>0$
69	67	$v=(1+i)^{-1}$
70	68	$E_{\Bbb{Q}}[X] := E[Xg'(S(X))]$
71	69	$1-S(a)=F(a)=(\nu + \delta)F(a)$
72	70	$(Alice)+(0,-2.5)$
73	71	$\lambda / p$
74	72	$R_2=C_2$
75	73	$\tilde p=1-g(1-p)$
76	74	$R_i > C_i$
77	75	$-\log(1-\Phi(x))$
78	76	$g(s) = \dfrac{r_o+s(1+r_K)}{1+r_o+r_Ks}$
79	77	$\alpha_\epsilon=\alpha$
80	78	$\mathscr{P} =\{1+\lambda(\zeta-\mathsf{E}\zeta) \mid \zeta\ge 0, \|\zeta\|_q\le 1 \}$
81	79	$r_c\le r_i$
82	80	$(A.north east)+(0.1, -0.05)$
83	81	$\hat\rho(A_0)\ge \rho(A_0)$
84	82	$1=P(x) + Q(x)$
85	83	$G>c(x)$
86	84	$(X-a)^+=\max(X-a, 0)$
87	85	$(Alice)+(0,-3.5)$
88	86	$\phi\equiv 1$
89	87	$xy^4 / (x^2 + y^8)$
90	88	$\beta_i(t)/\alpha_i(t)$
91	89	$X=X(\bar x)=G\circ F(\bar x)=GF(\bar x)$
92	90	$X(\omega)=q(T(\omega))$
93	91	$\mathsf{E}(X_i/X ; X > a)$
94	92	$g=2\nu^4/(1-f)+3c+1$
95	93	$Y=c\in \mathbb R$
96	94	$R(x)=pd_i+(v-\nu^*)\sqrt{pq}$
97	95	$st=k$
98	96	$X\ge X+Y$
99	97	$L_{p,p+\delta}$
100	98	$s^*=1-p^*\le 1$
101	99	$\rho(X) = sup_Q \mathsf{E}_Q(X)$
102	100	$\bar P=\bar P_1+\bar P_2$
103	101	$1 for each $
104	102	$S_g = g\circ S$
105	103	$\pi(x)$
106	104	$\int S$
107	105	$\Delta\tilde p > \Delta p$
108	106	$(0,1)$
109	107	$\rho_{(g)}(X)=\int xg'(S(x))f(x)dx$
110	108	$\tpx=e^{-1}$
111	109	$\{ r_i \}$
112	110	$p \in [1,\infty]$
113	111	$\approx\sqrt{2Np}$
114	112	$\rho(X)=\int_0^1 q(p) \phi(p) dp$
115	113	$g_0$
116	114	$qq$
117	115	$q_{X+Y}=q_X+q_Y$
118	116	$(fun2.north west)+(-\spcer, \spcer)$
119	117	$p(1-\nu(p)-il(p))$
120	118	$0.725$
121	119	$D$
122	120	$=P=\mathrm{MV}(X\wedge a)$
123	121	$\Delta \tilde p\times T$
124	122	$L_0$
125	123	$\int_{1-p}^1 \phi(t)dt =\int_0^p \phi(1-t)dt=g(p)$
126	124	$x$
127	125	$\mathsf{E}[X_1g'(S(X))]$
128	126	$\mathsf{E}(X\wedge a)$
129	127	$0\le\beta<1$
130	128	$\rho_{t+1}(X)$
131	129	$X_i(X\wedge a)/X$
132	130	$P=\nu(\bar S + \iota a)$
133	131	$\rho_i(X_i)$
134	132	$\downarrow$
135	133	$\nabla_x f= \nabla_xq_\alpha -\nabla_x G$
136	134	$\eta\gg\zeta$
137	135	$v-\nu^*=\delta^*-d$
138	136	$\sup_n \| X_n \|< \infty$
139	137	$A=P+Q$
140	138	$B = g^{b} \pmod{p}$
141	139	$\alpha$
142	140	$X=C(\bar x)+N(\bar x)=$
143	141	$X_1$
144	142	$\mathrm{PQ}$
145	143	$v-\nu^*=(\iota^*-i)/v\nu^*$
146	144	$\mu_X\le\mu_X$
147	145	$\lambda X$
148	146	$g(x)\ge x$
149	147	$\rho(A_k)\le\hat\rho(A_0) + k\rho(N)=\hat\rho(A_k)$
150	148	$\rho_t$
151	149	$Z=\mathsf{E} Z$
152	150	$\beta$
153	151	$(A.north east)+(0.2, -0.05)$
154	152	$\tilde\rho_T=\rho_T$
155	153	$>$
156	154	$c_h>c=\mathsf{VaR}$
157	155	$a\ge c$
158	156	$F(x)=\mathsf{Pr}(X\le x)$
159	157	$X_i(x_i)$
160	158	$P + \rho_i(F_i) < \rho_i(X_i) \iff P < \rho_i(X_i) - \rho_i(F_i)$
161	159	$\tilde \rho$
162	160	$L^\infty(\Omega, \mathsf{P})$
163	161	$0\le Y\le 1$
164	162	$R(a)=\delta N(a)$
165	163	$\bar P$
166	164	$F_Y$
167	165	$(fun3a.south -| fun3a.south east)+(\smlspc,-\smlspc)$
168	166	$\sigma(1-t)=g'(t)$
169	167	$g'(1-p) dp$
170	168	$\mathsf{E}(X) = \int_0^1 q(p)dp$
171	169	$(3) \rightarrow (9 = 9) \rightarrow (27 = 4) \rightarrow (12 = 12) \rightarrow (36 = 13) \rightarrow (39 = 16) \rightarrow (48 = 2) \rightarrow (6 = 6) \rightarrow (18 = 18) \rightarrow (54 = 8) \rightarrow (24 = 1)$
172	170	$t=T_x<n$
173	171	$\le 89$
174	172	$\tilde F, \tilde S$
175	173	$u : (a, b) \to \mathbf R$
176	174	$\inf_t\ \{ t+(1-\alpha)^{-1}\mathsf{E}(Z-t)_+ \}$
177	175	$a=\mathsf{E}[X|A]$
178	176	$X_u$
179	177	$\sup$
180	178	$\mathsf{E}(L) = q(p)$
181	179	$p\delta -q\nu=p-\nu$
182	180	$(lee.east |- lee.south)+(0.375,-0.25)$
183	181	$(g^k, Pg^{ak})$
184	182	$X=x$
185	183	$\phi_Q=1-\phi_W$
186	184	$(X+Y-x-y)_+\le (X-x)_+ (Y-y)_+$
187	185	$g(S(a))$
188	186	$\int_0^\alpha$
189	187	$a-L$
190	188	$pd_i=F(x)d_i$
191	189	$g''(t)=-\phi'(1-t)\le 0$
192	190	$\lambda^Q$
193	191	$F_i = X_i(1 - (X\wedge a)/X)$
194	192	$\mathscr{P}=\{ dQ/dP\le 1/\alpha\}$
195	193	$178.7 billion of expenses. Commissions and brokerage accounted for 25.1 percent and claim adjustment services for 13.5 percent of the total. Taxes licenses and fees were 6.3 percent. However, their remaining expense items are broken out by expense category, such as employee salaries and benefits or advertising, rather than insurer value-add function. They also reported a cost of capital of 13 percent, applied to equity capital of $
196	194	$\zeta=\zeta(G)$
197	195	$ and the average thickness of the difference in support sets must be zero because the two support sets have the same measure $
198	196	$O(mn\times n^2)$
199	197	$F=(X-a)^+$
200	198	$mX$
201	199	$\rho(X)\ge 0$
202	200	$=dP(a)/da = g(S(a))$
203	201	$\mathsf{VaR}_p(X)$
204	202	$X=4$
205	203	$p=1-g^{-1}(1-\tilde p)$
206	204	$\bar x\mapsto \sum_i F_i(\bar x)$
207	205	$\mathsf{E}(W/X | X\ge x)$
208	206	$F_0$
209	207	$\zeta_t=0$
210	208	$\phi(s)=g'(s)=s^{1/\rho}/(s\rho)$
211	209	$EL_a =\mathsf{Pr}(Y>a)=1-\exp(-\lambda S(x))$
212	210	$YL$
213	211	$X\le 0\implies\rho(X)\le 0$
214	212	$u_1,\dots, u_n$
215	213	$t_2-\epsilon/2$
216	214	$F_t$
217	215	$p=F(\mathsf{E}(X))$
218	216	$1 \times 10^{24}$
219	217	$\nabla (\zeta NF) = \zeta\nabla NF$
220	218	$p=p_a$
221	219	$\iff$
222	220	$L,P,M,Q,a,LR,PQ,COC$
223	221	$\approx$
224	222	$\mathsf{E}(X_i\mid X)$
225	223	$\eta\gg \zeta:[0,1]\to\mathbb{R}$
226	224	$\phi:=\rho\circ F$
227	225	$i=0$
228	226	$\iota^*$
229	227	$\partial a/\partial x_1$
230	228	$\mathsf E[X_i]$
231	229	$\rho(Z)=\sup_{\zeta\in\mathcal{A}} \langle \zeta, Z \rangle$
232	230	$\Omega=[0,1]$
233	231	$s\in[0,1]$
234	232	$\bar\nu=1/(1+\bar\iota)$
235	233	$\rho(X+m)=\rho(X)-m$
236	234	$K = A^{k}=g^{ak} \pmod{p}$
237	235	$\rho E/(1-\tau) - rA$
238	236	$=E(X_i / X)$
239	237	$\mathscr{O}(\eta)$
240	238	$\mathbf{x}=(x_1,\dots,x_n)$
241	239	$t_1<t<t_2$
242	240	$K_i = \mathsf E[X_i \mid X \ge a] - \mathsf E[X_i]$
243	241	$\theta < 1$
244	242	$R_2(t)$
245	243	$q_p(\mathbf{x})=\mathsf{VaR}_p(X(\mathbf{x}))$
246	244	$(Bob) + (0,-2)$
247	245	$C < cx/a$
248	246	$g^{ks} = r^s$
249	247	$q=S(x)$
250	248	$1/\nu=1+\rho$
251	249	$\rho(X)=\mathsf{E}_Q(X)$
252	250	$R_2(t)\approx \mathsf{E}[X_2]$
253	251	$k=1,2,\dots,n-1$
254	252	$\rho_k$
255	253	$\mathsf{E}[e^sX]<\infty$
256	254	$\mathcal F_1 = \sigma(N)$
257	255	$B=\{ \omega\in\Omega \mid \zeta(\omega)>0 \}$
258	256	$1-g(S(x))=\tilde F(x)$
259	257	$a, b$
260	258	$\xtext$
261	259	$\bar h$
262	260	$g'(S(x))dF(x)$
263	261	$1=S(a) + \delta F(a) + \nu F(a)$
264	262	$\Omega=\mathbb{R}$
265	263	$\mathsf E[XY]\not=\mathsf E[X]\mathsf E[Y]$
266	264	$\sum \alpha_i=1$
267	265	$Z_p^\times$
268	266	$h_\epsilon$
269	267	$\rho(X) = \mathsf{E}(X) + \| (X-\mathsf{E} X)_+ \|_p$
270	268	$\mathbb{R}_+=[0\infty)$
271	269	$\delta_p+\nu_p=1$
272	270	$Z=g'(S(X))$
273	271	$\rho=0.6$
274	272	$\rho(L) = q(p)>q(p)$
275	273	$\mathsf{E}(X\mid X > a)$
276	274	$L^\infty$
277	275	$p(\nu_p-l_p)$
278	276	$\rho(B(s_u)) - \rho(B(s_l))$
279	277	$\rho(X)= (1+r_f)^{-1}\mathsf{E}_Q(X)$
280	278	$(1-{}_b\bar V)$
281	279	$a\theta^2=c$
282	280	$(1-\nu_p-il_p)/(\nu_p-l_p)=\iota_{1/2}$
283	281	$p=23$
284	282	$\nu=1/(1+\iota)=1-\delta$
285	283	$X=X_1+X_2+X_3$
286	284	$\rho(-k_i 1_{A_i}) \le c < 0$
287	285	$\bar a_x$
288	286	$a=1/c$
289	287	$\rho(-1_{A^c})=0$
290	288	$c=\bar A^{1}_{x:\lcroof{1}}/\bar a_{x:\lcroof{1}}$
291	289	$\mathcal F^G$
292	290	$\bar a_{x:\lcroof{1}}$
293	291	$g^ag^k=g^{a+k}$
294	292	$\pi_X(t)\le \pi_Y(t)$
295	293	$Y=\sum_i X_iY_i$
296	294	$(Alice)+(0,-3)$
297	295	$\beta_i(a)/\alpha_i(a) < 1$
298	296	$(3\times 6 + 2\times 2)/ 8 = 11/4$
299	297	$g\ge 0$
300	298	$X(u)$
301	299	$\displaystyle\int_0^1\phi(s)ds=\displaystyle\int_0^1\displaystyle\int_{1-s}^1\dfrac{\mu(dt)}{t}ds = \displaystyle\int_0^1\displaystyle\int_{1-t}^1ds\dfrac{\mu(dt)}{t}=\displaystyle\int_0^1\mu(dt)=1$
302	300	$\rho(A)>\hat\rho(A)$
303	301	$= \rho(B(s_l)) (1 - s) + \rho(B(s_u)) s$
304	302	$\int_0^1 μ(dt) = 1 - α < 1$
305	303	$P_idx_i$
306	304	$\omega_1,\omega_2\in\Omega$
307	305	$X\wedge a$
308	306	$C_2(0) = \mathsf{E}[X_2]$
309	307	$X(\mathbf{x})=\sum_i x_i X_i$
310	308	$\rho(X)=50=:r$
311	309	$|Z|$
312	310	$\rho(X)=\int_0^1 q(1-g^{-1}(1-t))dt$
313	311	$N(1-p)$
314	312	$1+2c(1-\mathsf{Pr}(Z>\mathsf{E} Z)$
315	313	$r$
316	314	$\bar P^a_i$
317	315	$E_2$
318	316	$m_j / r_j$
319	317	$\int_0^x (x-y)^{n-1}dG(y)$
320	318	$P =\{ Q \mid dQ/dP \le k \}$
321	319	$\pi = \mathsf E[PR]$
322	320	$A_{x+b}$
323	321	$\rho(-X_n)\downarrow 0$
324	322	$q(\epsilon)\approx q + \epsilon\mathsf{E}_q(X_i)$
325	323	$\mathsf{E}(Y(a))=\mathsf{E}(Y\wedge a)=\int_0^a S_Y(t)dt$
326	324	$g'(t)=1-r_0$
327	325	$\langle \zeta_{\bar x}, N(\bar x) \rangle$
328	326	$\sum_i h^i= 0$
329	327	$g(S(x))=1$
330	328	$(A.north east) + (-0.07mm,0)$
331	329	$E(XZ \mid \mathcal{G})=ZE(X \mid \mathcal{G})$
332	330	$(\nodespc/2, -\nodespc/2)$
333	331	$\{ v_i \}$
334	332	$\int_0^q = \int_0^{\mathsf{E}_q(X_2)} + \int_{\mathsf{E}_q(X_2)}^q$
335	333	$q(\epsilon)=q+\epsilon\mathsf{E}_q(X_1)$
336	334	$1-U$
337	335	$\log_{10}(N(m))) \propto -bm$
338	336	$\not=$
339	337	$[a, a+da]$
340	338	$1_Af_t(X)=1_Af_t(1_AX)$
341	339	$\mathbf{X}\times\mathbb{R}$
342	340	$X_i\ge 0$
343	341	$a>a(f)$
344	342	$p(a)=\nu S(a) + \delta = S(a) + \delta F(a) = 1-\nu F(a)$
345	343	$\sigma=0.125$
346	344	$D_n,D_n^*$
347	345	$X+Y$
348	346	$X_n \downarrow 0$
349	347	$\rho(X)=\int_0^1 q(s)g'(1-s)ds$
350	348	$\mathsf{E}(X) = \displaystyle\int_0^\infty xf(x)dx = \displaystyle\int_0^1 q(p)dp$
351	349	$\rho(X^{\oplus n}) \ge \rho(X^{\oplus n-1}) + \mathsf E[X] > \rho(X^{\oplus n-1})$
352	350	$\pi_\sigma(L)$
353	351	$X_1\wedge a$
354	352	$\rho_p$
355	353	$p=0,1$
356	354	$\hat \rho$
357	355	$X_p=^d Y_p$
358	356	$\mathbb{R}^2$
359	357	$B(b)\approx -b\mu_x$
360	358	$L(a)=$
361	359	$1 = m(x) + \nu F(x) = S(x)+\delta F(x) + \nu F(x)$
362	360	$I_i\in\{0,1\}$
363	361	$Y\ge X$
364	362	$X(\mathbf{1})$
365	363	$\bar P'(x)=P(x)$
366	364	$(34.05-23.81) / (100-34.05)=15.5$
367	365	$\rho_w$
368	366	$(-\x, 2)$
369	367	$a=F^{-1}(1-\delta)$
370	368	$\sigma=0.5, 1.0$
371	369	$Y\wedge a$
372	370	$[0,1]$
373	371	$\mathcal G$
374	372	$2/3$
375	373	$\bar Q'(x)=Q(x)$
376	374	$G(\bar x)$
377	375	$F(a)$
378	376	$p\delta_p$
379	377	$R_1(t)<\mathsf{E}[X_1]$
380	378	$5.14\times 10^{19}$
381	379	$\mathsf{TVaR}_p(X) \le r$
382	380	$0\le\alpha\le K$
383	381	$= 10^{1+6+12}=10^{19}$
384	382	$g^k$
385	383	$\phi(0)=0$
386	384	$g, g^2, \dots,g^{q-1}, g^q\equiv 1$
387	385	$||\cdot ||$
388	386	$g, g', g''$
389	387	$\Omega=\{1,2,3 \}$
390	388	$\sum x_iX_i$
391	389	$X_1=s$
392	390	$\mathbb{R}^n\to\mathbb{R}$
393	391	$X\wedge a\not\in \mathbf{X}$
394	392	$T_i\circ T$
395	393	$\delta+\nu=1$
396	394	$X(\cdot)$
397	395	$q+p\delta_p$
398	396	$q_Z(U)$
399	397	$1_A = 1 - 1_{A^c}$
400	398	$P=\rho(X\wedge a)$
401	399	$1-p \ge g^{-1}(1-p) \implies 1-g^{-1}(1-p) \ge p \implies q(1-g^{-1}(1-p))>q(p)$
402	400	$G=X_1+X_2$
403	401	$=\rho(B(\mathrm{current\ best\ estimate\ of\ } s)) = \rho(B(s))$
404	402	$\zeta, \zeta_t\ge 0$
405	403	$p=0.98,0.99$
406	404	$(3,6-4.724)$
407	405	$=\mathsf{E}(X_{i,2}(a))$
408	406	$\square\rho_i$
409	407	$602.6 billion and converted to net premium based on $
410	408	$(r,s)$
411	409	$F^{\times}_{23}$
412	410	$\mu_\sigma$
413	411	$E_{\Bbb{Q}}[Y]=E[Yg'(S(X))]$
414	412	$n\times 1$
415	413	$\phi(s)=\displaystyle\int_{1-s}^1\dfrac{\mu(dp)}{p}=\int_0^s\dfrac{\mu(dp)}{1-p}$
416	414	$(Alice)+(0,-1.75)$
417	415	$\rho(1_A) \le \rho(1)=1$
418	416	$1 \times 10^{20}$
419	417	$\rho(X)<\rho(Y)$
420	418	$1-U^2$
421	419	$G=N+C$
422	420	$\alpha_i$
423	421	$0<b\le 1$
424	422	$\mathsf{E}(X)=\mathsf{E}(X\wedge k) + \mathsf{E}(X-k)_+$
425	423	$g(s)=s^{0.75}$
426	424	$R_2(1) = \bar P^a_2(1)$
427	425	$\pi$
428	426	$yS(a)=y\times 1=y=$
429	427	$μ$
430	428	$\bar S(a) = \int_0^a S(x)dx$
431	429	$S<0$
432	430	$pl_p$
433	431	$\mathsf{Pr}(\cdot\mid N)$
434	432	$\rho(X)=\mathsf{TVaR}_p(X)$
435	433	$x^{a-1}e^{x/\theta}$
436	434	$Y=X\wedge a$
437	435	$\lambda(t)$
438	436	$δ$
439	437	$L^tf^*=m$
440	438	$D_i=\{\omega \mid X(\omega)=a_i\}\in\mathcal{F}$
441	439	$f(\cdot, \omega)$
442	440	$L_x^{x+dx}$
443	441	$\lambda=0.0725$
444	442	$1-\tilde p=g(S(x))=g(1-p)$
445	443	$\mathsf{E}(U(X))$
446	444	$V$
447	445	$1 \times 10^{13}$
448	446	$\sum_{i\in I}(D_i-N_I)$
449	447	$0 < t < 0.5$
450	448	$\bar F(a)=\int_0^a F(x)dx$
451	449	$\beta=\infty$
452	450	$\nabla\zeta$
453	451	$2n$
454	452	$\mathsf E[A] = \mathsf E[\mathsf E[X^{\oplus N}]] \le \mathsf E[\rho(X^{\oplus N})]$
455	453	$\beta_i(a)/\alpha_i(a) > 1$
456	454	$g'(s)=1$
457	455	$1-p=g(1-\hat p)$
458	456	$r_i=\rho(X_i)$
459	457	$\langle \zeta, G \rangle$
460	458	$g(s)=O(d)$
461	459	$g'(p)=\phi(1-p)$
462	460	$ be the compound of $
463	461	$\rho_t = \rho_t(-\rho_{t+1})$
464	462	$\circ$
465	463	$\mathsf{TVaR}_\beta\mathbin{\square}\mathsf{TVaR}_\gamma = \mathsf{TVaR}_\gamma$
466	464	$\rho=\rho_\phi$
467	465	$(rep.east) + (1.5, 0.5)$
468	466	$\mathsf{E}[h_\epsilon Y]\to\mathsf{E}[h Y]$
469	467	$\rho =$
470	468	$\phi(1-t)$
471	469	$A:=g^a \pmod{p}$
472	470	$a-L_0^a$
473	471	$1=v+d$
474	472	$\nu(p)=p$
475	473	$\rho(A+B)64.5>63.5=\rho(A)+\rho(B)$
476	474	$p<\infty$
477	475	$\alpha_i(X_u)= \text{E}[u_iX_i \mid X_u > F_u^{-1}(p)] = u_i \partial T/\partial u_i$
478	476	$\alpha_\epsilon-\alpha$
479	477	$F^{(2)}=[F^{(-2)}]^*$
480	478	$\mathrm{P}$
481	479	$q\in[1, \infty]$
482	480	$\uparrow$
483	481	$(0.5,1.5)$
484	482	$\omega\in\Omega$
485	483	$\phi(p)=1$
486	484	$0<a<q-1$
487	485	$^1$
488	486	$(D.south east)+(0.2, 0.05)$
489	487	$t=0,1,\dots, T$
490	488	$\int_0^\infty \phi(p)dp=1$
491	489	$1/t$
492	490	$X\ge 0$
493	491	$sgn(z)|z|^{1/(q-1)}/\|z\|_p^{q/p}$
494	492	$X(x+\epsilon)$
495	493	$L(a)$
496	494	$y$
497	495	$dS = \mathbf{n}dudv$
498	496	$c_x\approx\lambda$
499	497	$\delta=\rho/\nu$
500	498	$F=\Phi$
501	499	$g(s)=\sqrt{s}$
502	500	$\rho(A_k) \le \rho(A_0) + k \rho(N)$
503	501	$a\theta=1$
504	502	$g_{\min}(s):=\min_i (g_i(s)$
505	503	$g(s)=\displaystyle\int_{1-s}^1 \phi(t)dt = \displaystyle\int_0^s \phi(1-t)dt$
506	504	$\mathsf QV$
507	505	$\mathrm{LR}$
508	506	$d$
509	507	$x_1,1$
510	508	$g\in \nabla\rho(X)$
511	509	$S_X$
512	510	$v^b{}_bq_x(1-{}_b\bar V)$
513	511	$\tilde p=1-(1-p)^{1/\rho}$
514	512	$d=i/(1+i)=iv=1-v$
515	513	$\iota a$
516	514	$(X-a)^+ = \max(0,X-a)$
517	515	$C_1(t) < C_2(t)$
518	516	$x\times f(x)dx$
519	517	$-α(α-1)t^{α-1}$
520	518	$\rho(X^{\oplus n}) = n(v\mathsf E[X] + d\max(X))$
521	519	$(\langle X(\epsilon), \zeta_\epsilon \rangle - \langle X, \zeta \rangle)/\epsilon = \langle (X(\epsilon)-X)/\epsilon,\zeta \rangle = \mathsf{E}_Q(\nabla X)$
522	520	$>a$
523	521	$\mathsf{E}[g]\le 1$
524	522	$\mathsf{Pr}(I=1)=s$
525	523	$-norm less than $
526	524	$\bar M_i(a)$
527	525	$\mathbf{r}\ge 0$
528	526	$u_i\partial\pi / \partial u_i$
529	527	$\rho_m(X)=\rho_m(X\wedge k) + \rho_m((X-k)_+)$
530	528	$\bar P(x) = \bar S(x) + \bar R(x)$
531	529	$\mathsf{TVaR}_{p^*}$
532	530	$\rho(1_A) = \rho(1) = 1$
533	531	$s,t$
534	532	$ρ$
535	533	$[xf(x)] \times dx$
536	534	$p<1$
537	535	$c\in[0,1]$
538	536	$R(x)=pd+(\delta^*-d)\sqrt{pq}$
539	537	$g'(1-p)=\phi(p)$
540	538	$\nu=\nu_p$
541	539	$q=11$
542	540	$\bullet$
543	541	$\iff \mathcal A_{t+1}\subseteq \mathcal A_t$
544	542	$\sigma=0.5$
545	543	$n=1,2,\dots$
546	544	$\mathcal{A} = \{ X \mid \rho(X)\le 0 \}$
547	545	$age^2$
548	546	$\phi_{\bar x}(Z)=\langle Z,\zeta_{\bar x} \rangle$
549	547	$3.2 \times 10^{15}$
550	548	$P=L + \delta (a-L)$
551	549	$\mathsf{E}[X\cdot Z\circ T]=\mathsf{E}[X\cdot Z\circ T_B\circ T_A ]=\mathsf{E}[X \cdot Z\circ T_A]=\mathsf{E}[X\circ T_A^{-1}]=\mathsf{E}[X Z]$
552	550	$(X,Y)$
553	551	$\partial \zeta_{\bar x}/\partial x_i$
554	552	$\delta = \delta(p) = 1-\nu(p)$
555	553	$\lim_n \mathsf{E}_{\mathsf{Q}_n}(X)=\rho(X)$
556	554	$g^a\equiv n\pmod{p}$
557	555	$P(a) = S(a) + \delta F(a)$
558	556	$\hat\rho(A_k)$
559	557	$g'=0$
560	558	$X=X(I)$
561	559	$g(s)=\dfrac{r_{occ}+s(1+r_{use})}{1+r_{occ}+r_{use}s}$
562	560	$\mathsf{Q}_n\in\mathscr{P}$
563	561	$g^-1(p)$
564	562	$S=1-F$
565	563	${}^nS^{-1}_X(t)\le {}^nS^{-1}_Y(t)$
566	564	$\iota\alpha(X)=\iota a$
567	565	$\mathsf{E}[Y\mid X] = X$
568	566	$f^*_i$
569	567	$(fun3.north west)+(-\smlspc,\smlspc)$
570	568	$[0, 1]$
571	569	$(fun1a.south east)+(\smlspc,-\smlspc)$
572	570	$(rep.south) + (0.5, -1.0)$
573	571	$\rho^*=\rho(0.5)$
574	572	$g(S(x))\approx S(x)\approx 1$
575	573	$=(1-\alpha)\mathsf{TVaR}_\alpha(X)$
576	574	$B\cup B_t = (B\cap B_t) \cup C_t$
577	575	$X^n_t=1_{[1+T_n, \infty)}$
578	576	$(p-\nu-il)/(v-l)$
579	577	$G(x+th, \omega+d\omega) = c_k(x+th)$
580	578	$(k+1)\times 1$
581	579	$X_n\le 1$
582	580	$\langle NF(x), Th_i \rangle+\langle \partial NF/\partial x_i, \zeta_{GF(x)} \rangle$
583	581	$\partial a/ \partial x_i$
584	582	$\rho(I)\rho(X)=g(s)\rho(X)$
585	583	$\mathsf{Var}(\Pi)$
586	584	$\nu(p)=1/(1+\rho(p))$
587	585	$g:[0,1]\to[0,1]$
588	586	$g(S(a))-S(a)$
589	587	$\delta N(a)$
590	588	$n$
591	589	$H(x)$
592	590	$\rho(X)=\mathsf{E}_Q(X)=\mathsf{E}_Q(Y)+\mathsf{E}_Q(Z)$
593	591	$\rho(X)\le \rho(Y)$
594	592	$P_Q = \mathsf{P}[(X-a)V(a)]$
595	593	$q_j$
596	594	$(fun1.north west)+(-\medspc,\medspc)$
597	595	$\zeta NF$
598	596	$a = q_X(0.995)$
599	597	$S(x)=1=F(x)$
600	598	$d+v=1$
601	599	$K=g^k$
602	600	$b-a$
603	601	$(Bob)+(0,-2)$
604	602	$X^{\oplus N}$
605	603	$(a-X)^+:=\max(a-X, 0)$
606	604	$\rho_{1/2}$
607	605	$s,t \in[0,1]$
608	606	$10^{17}$
609	607	$M_0$
610	608	$\int_0^1 Z=1$
611	609	$\rho(X)=\mathsf{TVaR}_1=\esssup$
612	610	$\nu < 1$
613	611	$\pi : X\mapsto (X, \alpha(X))\mapsto E_g(X\wedge \alpha(X))$
614	612	$\rho_m(X) = \mathsf{E}(X) + (\rho_m(X)-\mathsf{E}(X))$
615	613	$(x-y)^n$
616	614	$u\in D_n=\{ u \mid u^{(k)} \ge 0, k=1,\dots,n-1, u^{(n-1)}\text{ nondecreasing} \}$
617	615	$\mathsf{E}(XZ \mid \mathcal{G})$
618	616	$\bar a_{\lcroof{n}}$
619	617	$(S(x) + \delta(F(x))F(x)) dx$
620	618	$m + ra = ks$
621	619	$Q$
622	620	$n-1$
623	621	$-1$
624	622	$(1-g(S(x)),x)$
625	623	$k_0>\ge 2$
626	624	$\Pi$
627	625	$\rho_i$
628	626	$\bar a_x = \bar a_{x:\lcroof{b}} + v^b{}_bp_x\bar a_{x+b}$
629	627	$v-l$
630	628	$\delta_p/\nu_p = \rho_p$
631	629	$\rho(-X+a)=\rho(-X) + a \le 0$
632	630	$r = g^k$
633	631	$(0,1) < 1$
634	632	$\mathcal F_1=\sigma(N)$
635	633	$dt$
636	634	$Z_1=Z\circ T_A$
637	635	$(fun4a.south -| fun3a.west)+(-\medspc,-\medspc)$
638	636	$dx=x_{i+1}-x_i$
639	637	$x=1.5, M=1.5,\sigma=0.75, K=6$
640	638	$ from policyholder as premium and capital $
641	639	$\hat X_i=\hat x_i$
642	640	$\nu(dx)$
643	641	$0.5$
644	642	$\liminf \rho(X_n) \ge \rho(X)$
645	643	$M(a)=\mathsf{E}(X\wedge a) + \delta N(a)$
646	644	$S(x) + \delta F(x)$
647	645	$(ckey1.north west)+(-\boundpad,\boundpad)$
648	646	$10 million I **must care more** about a loss of $
649	647	$1.5\times 10^{37}$
650	648	$Q\in \mathscr{P}$
651	649	$a,b$
652	650	$\zeta>0$
653	651	$\mathsf{E}(X_i \mid X \le a)$
654	652	$X=C+G$
655	653	$\rho(X)\le\liminf_{n\to\infty} \rho(X_n)$
656	654	$M_X(k)\le M_Y(k)$
657	655	$t=t_2$
658	656	$T_t$
659	657	$H:\mathcal X\to\mathbb R$
660	658	$\phi_i = \mathsf{E}(X_i)/\mathsf{E}(Y)$
661	659	$\mathsf{E}_Q(Y\mid X)\mathsf{E}(Z\mid X) = \mathsf{E}(YZ \mid X)$
662	660	$g(s) = t_{df}(t_{df}^{-1}(s)+\lambda)$
663	661	$\rho=\rho(p)$
664	662	$2*(1,1)$
665	663	$\lambda > 0$
666	664	$\rho=0.12$
667	665	${}_nE_x$
668	666	$\rho(T)\ge T$
669	667	$p\mathsf{E}[X\mid X<x_p]$
670	668	$\nu=\nu(p)$
671	669	$p-\nu$
672	670	$R>C$
673	671	$\delta=\log(1+i)$
674	672	$a=1$
675	673	$\approx (920+961)/2=940.5$
676	674	$(Alice) + (0,-2)$
677	675	$v\mathsf E[X] + d\max(X)=\rho(X)$
678	676	$\rho_{(g)}=\max\{\mathsf{E}(ZX) \mid Z\in \mathcal{A}\}$
679	677	$G(x,\omega)=c_k(x)$
680	678	$P=\displaystyle\sum_i P_i$
681	679	$t \le g(t) = \displaystyle\frac{t}{1-p}$
682	680	$\lambda_{t}$
683	681	$P + \rho_i(F_i)$
684	682	$X\circ T=X$
685	683	$\sigma_\mu(\alpha) = \int_0^\alpha \frac{1}{1-p}\mu(dp)$
686	684	$X_i < cx/a$
687	685	$age$
688	686	$\zeta=\Omega$
689	687	$X = X_0 + M + A$
690	688	$l(p)= \nu(p)-\sqrt{(1-p)/p}$
691	689	$Y \Leftrightarrow \rho(X)\le \rho(Y)$
692	690	$\beta_i(t)<\alpha_i(t)$
693	691	$\sqrt{FS}\gg S$
694	692	$dx_i$
695	693	$\rho^*(\mu)=\infty$
696	694	$\mathsf{E}[hY]$
697	695	$U$
698	696	$\mathsf{TVaR}_{1}$
699	697	$\mathsf{E}(X_{i,2}(a))$
700	698	$p_a$
701	699	$4/3$
702	700	$\infty$
703	701	$12.318 / 260.81 = 4.7\%$
704	702	$1-\tilde p=g(S(x))$
705	703	$c_x-c_{\text{Nov 1}}$
706	704	$k-\rho_m(X)$
707	705	$P(X_1+X_2)=M(X_1+X_2, \psi(X_1+X_2))=$
708	706	$\rho(\cdot\mid\mathcal F_1)$
709	707	$h\in \nabla\rho(X)$
710	708	$\bar x$
711	709	$(v-\nu^*)\int_0^a \sqrt{F(x)S(x)}dx$
712	710	$Z\circ T_B=Z$
713	711	$\displaystyle\int_0^\infty xd(g\circ F)(x)$
714	712	$\rho:L_p\to\bar\mathbb{R}$
715	713	$x=z$
716	714	$A_k=A_0 + kN$
717	715	$ is the total return on invested assets and $
718	716	$b\approx 0.95$
719	717	$t=1-g(1)=0$
720	718	$\le_{\mathrm{cx}}$
721	719	$\nu(F(x))F(x) = \nu(p)p$
722	720	$q=1-p=S(x)$
723	721	$r_o,r_K$
724	722	$q(u_i)$
725	723	$\bar P^a(t)=\bar P^a_1(t)+\bar P^a_2(t)$
726	724	${}_b\bar V=1-\bar a_{x+b}/\bar a_x$
727	725	$\mathsf{E}(X) = \displaystyle\int_0^\infty xf(x)dx = -xS(x)\Big\vert_0^\infty + \displaystyle\int_0^\infty S(x)dx = \displaystyle\int_0^\infty S(x)dx$
728	726	$\theta(p)=q(1-g^{-1}(1-p))/q(p)$
729	727	$v-\nu^*$
730	728	$t\to 1$
731	729	$p(a) = \nu S(a) + \delta = \nu (S(a) + \rho)$
732	730	$1 \times 10^{19}$
733	731	$10^{20}$
734	732	$i=0.025$
735	733	$(1,1)$
736	734	$\nu$
737	735	$\rho_k\to\infty$
738	736	$\mathcal F_1=\sigma(I)$
739	737	$f(s) = -g''(1-s)(1-s)$
740	738	$r_O$
741	739	$50 of the amount allowed on each claim in the classes under subsections 3, 4, 4-B, 5 and 6 must be deducted from the claim and included in the class under subsection 8. Claims may not be cumulated by assignment to avoid application on the $
742	740	$k\mathsf B(s)$
743	741	$r_{qp}=\sqrt{pq}$
744	742	$\phi$
745	743	$\mathsf{MON}$
746	744	$g^{ak}=(g^a)^k$
747	745	$F_{\mathbf{x}}(t)=s$
748	746	$\rho(X)=\sum_i \mathsf{E}_\mathsf{Q}(X_i)$
749	747	$Y\le X=0$
750	748	$k=\mathsf E[X]$
751	749	$g\in\mathscr{P}$
752	750	$p(1-p)$
753	751	$x\not= y$
754	752	$\rho(X)= \sup_\zeta \langle \zeta, X \rangle$
755	753	$h_{i,\epsilon}$
756	754	$(X_1,\dots,X_n)$
757	755	$R$
758	756	$A=X_1 + \cdots + X_N$
759	757	$g\in\mathscr P$
760	758	$=F^{-1}(p)=$
761	759	$g^{-1}$
762	760	$q(1-g^{-1}(1-p))$
763	761	$Y=\log(X)$
764	762	$r = \nabla r$
765	763	$\bar S_i(\mathbf{x}; a) := \mathsf{E}[X_i(\mathbf{x}; a)]$
766	764	$N(m)$
767	765	$a_i = \mathsf E[X_i \mid X \ge a]$
768	766	$X\in L^\infty$
769	767	$-\int xdS=\int Sdx$
770	768	$p=\sigma^{-2}$
771	769	$X=\displaystyle\sum_i X_i$
772	770	$\partial \rho(X)$
773	771	$da > 0$
774	772	$s$
775	773	$a_1\not=a_2$
776	774	$H_k(X)=H_k(Y)$
777	775	$\theta(p)\equiv 1$
778	776	$\mathsf{E}_\mathsf{Q}(\cdot)$
779	777	$g(1-F(x))=1-\tilde p$
780	778	$\mu_t:=\lambda_t / \int_0^1\lambda_s \,ds$
781	779	$d\mathsf{Q}=g'(1-p)dp$
782	780	$\ge a$
783	781	$\mathcal{M}$
784	782	$k \in_{R} \{2,\dots,p-2\}$
785	783	$\int_0^x$
786	784	$F:\mathbb{R}^n\to \mathcal{X}^n$
787	785	$N$
788	786	$\bar M(a)$
789	787	$C_i=\partial \bar P^a/\partial x_i$
790	788	$\mathcal F_1\subseteq \mathcal F$
791	789	$2.6 \times 10^{12}$
792	790	$\| \sigma \|_p \le c$
793	791	$\|Z\| = \mathsf{E}(| Z|^p)^{1/p}$
794	792	$700 million. Enstar, which owns 9.1% of Watford’s common shares, at the same time agreed to abandon its quest to buy the insurer. In May 2020, activist investor Capital Returns Management LLC called for Watford to be sold or put into runoff, complaining about “consistently poor operating and stock performance” in comparison with its peers in the industry. When an initial offer of $
795	793	$t=0$
796	794	$0.125$
797	795	$=\mathsf{E}(X\mid X > a)$
798	796	$t\in[t, t+dt]$
799	797	$\rho_\phi(X)=\displaystyle\int_0^1 q(p)\phi(p)dp=\displaystyle\int_0^1 q(p)g'(1-p)dp=\displaystyle\int_0^\infty xg'(1-F(x))f(x)dx$
800	798	$\rho(X)=\mathsf{E}(q(U)\phi(U))=\mathsf{E}_Q(q(U))$
801	799	$-\log(1-\alpha)$
802	800	$ds=g'(1-t)dt$
803	801	$Z\ge 0$
804	802	$M_r$
805	803	$g^mA^r == r^s$
806	804	$b\mu_x v^b$
807	805	$\mathsf{E}_\mathsf{Q}(X_i) = \mathsf{E}_\mathsf{Q}(\mathsf{E}_\mathsf{Q}(X_i \mid X)) = \mathsf{E}_\mathsf{Q}(\mathsf{E}(X_i \mid X))$
808	806	$. Then $
809	807	$\iota$
810	808	$\epsilon >0$
811	809	$\hat\rho(X)<\rho(X)$
812	810	$\sum_i I_i=1$
813	811	$M(X_1, a_1)+M(X_1, a_2)=M(X_1, a_1+a_2)$
814	812	$=\mathsf{E}(\min(X,a))=\mathsf{E}(X\wedge a)$
815	813	$\pi'(s) = \displaystyle\frac{d}{ds}(g(s)g(k/s))$
816	814	$(A=g^a,a)$
817	815	$X \lt a$
818	816	$x=2, M=1.5,\sigma=0.75, K=6$
819	817	$h=H(A)$
820	818	$X\sim\text{Lognormal}(\text{mean}=5000, cv=3)$
821	819	$\rho(p)=\rho(F(x))$
822	820	$ of paying and $
823	821	$\mathsf{TVaR}(p)=(1-p)^{-1}\int_{p}^1 q(s)ds$
824	822	$p=\infty$
825	823	$x+dx$
826	824	$d\tilde p =g'(1-p)dp$
827	825	$X(x) = \sum_i x_iX_i$
828	826	$G>q_\alpha$
829	827	$M(a)=g(S(a)) - S(a)$
830	828	$x \times [f(x)dx]$
831	829	$S_Y(a)$
832	830	$\bar a_{x:\lcroof{n}}$
833	831	$\rho_\phi(X)=\displaystyle\int_0^1 q(p)\phi(p)dp=\displaystyle\int_0^\infty g(S(x))dx=\rho_{(g)}(X)$
834	832	$\delta\bar a_{x:\lcroof{n}}$
835	833	$\bar A^{1}_{x:\lcroof{1}}$
836	834	$\rho^*(\zeta-1)$
837	835	$(v-\nu^*)\sqrt{F(x)S(x)}$
838	836	$\theta=c=\nu^2$
839	837	$ because $
840	838	$\tilde F$
841	839	$(\partial \alpha/\partial x_i)q_X(\alpha)q_\zeta(1-\alpha)$
842	840	$A$
843	841	$-$
844	842	$\tilde W$
845	843	$\tilde p/p$
846	844	$\bar P_i(\mathbf{x},a):=\mathsf{E}_g[X_i(\mathbf{x}; a)]$
847	845	$(x)$
848	846	$\mathsf{TI}$
849	847	$t_1<\cdots<t_n$
850	848	$(g(S(a)) - S(a)) / (1 - g(S(a)))$
851	849	$\subseteq$
852	850	$S(a)$
853	851	$(rep.east) + (1.5, -1.75)$
854	852	$f(\square)\mapsto f(\square)-1$
855	853	$t/(1-t)$
856	854	$\hat\rho_{\mathcal F_1}$
857	855	$N\sim\text{Mixed Poisson}(\lambda=0.08 \times (\text{vehicles insured}), cv=0.075)$
858	856	$1/x$
859	857	$\rho(X) \le \rho(Y)$
860	858	$1-p=q$
861	859	$\mathsf{E}(X) = E(X_i \mid X\le a)F(a) + =E(X_i \mid X > a)S(a)$
862	860	$\text{E}(G^3)=g$
863	861	$X^{\oplus n}=X_1 + \cdots + X_n$
864	862	$Ann+V$
865	863	$Z\in L_1$
866	864	$F(t)=p$
867	865	$-k_i 1_{A_i}$
868	866	$31.5 million. Nine of Argonaut’s 11 top officers were fired, and Singleton began running the operations from headquarters in Los Angeles. Argonaut, one of the last large companies in the malpractice market, discontinued underwriting individual policies for the 20,000 physicians it covered. It continued to offer coverage to the 25 percent of the nation’s hospitals it covered, but at higher rates and covering fewer risks. In the meantime, the company collected $
869	867	$a=\inf$
870	868	$k_1 >0$
871	869	$X_n\downarrow 0$
872	870	$\rho=\text{AVaR}$
873	871	$R_1(t),R_2(t)$
874	872	$E_Q(N_i) = E_Q(\nabla \rho) + E_2$
875	873	$a\le X\le b$
876	874	$t<0.12$
877	875	$\text{E}(G^r)=\theta^r\Gamma(a+r)/\Gamma(a)$
878	876	$10 monthly premium and pay out as much as, say, $
879	877	$(fun4.north west)+(-\smlspc,\smlspc)$
880	878	$\mathsf E[A] \le \mathsf E[\rho(X^{\oplus N})] \le \rho(A)$
881	879	$a=\max X$
882	880	$s_l = f / (n+1)$
883	881	$M^{\tau_n}_t = M_{t \wedge \tau_n}$
884	882	$\rho(\cdot\mid \mathcal F_1)$
885	883	$0\le \alpha<1$
886	884	$n=2^2$
887	885	$H_g(X) \le H_g(Y)$
888	886	$\nu(p) = v-(v-\nu^*)\sqrt{(1-p)/p}$
889	887	$\mathsf{E}_Q$
890	888	$\nabla\partial\rho(Z)$
891	889	$\sigma=2.0,3.0$
892	890	$w \ge 0$
893	891	$Z=\frac{X-\mathsf{E}[X]}{\sigma(X)}$
894	892	$k= \mathsf{E}(X\wedge k) + (\rho_m(X\wedge k) - \mathsf{E}(X\wedge k)) + (k-\rho_m(X\wedge k))$
895	893	$=q(p)$
896	894	$\delta^2 p +\nu^2q-(p-\nu)^2=\delta^2 p -p\nu^2 -p^2+2p\nu =p(\delta^2 -\nu^2) -p^2+2p\nu =p(\delta -\nu) -p^2+2p\nu =p\delta -p^2 + p\nu = p-p^2$
897	895	$N\mid G$
898	896	$\mathsf{E}(L) = q(p)\delta$
899	897	$\rho(X)=\mathsf{E}[gX]$
900	898	$u_l>0$
901	899	$\alpha_i(t) = \mathsf{E}[X_i /X \mid X> t]\not=\mathsf{E}[X_i\mid X> t]/\mathsf{E}[X\mid X>t]$
902	900	$=Q=\mathrm{MV}(a-X)^+$
903	901	$\zeta=0$
904	902	$\mathsf{Var}(X_i)>0$
905	903	$\phi(0)$
906	904	$2^2\rightarrow 3^3-1=2\times 3^2 + 2\times 3 + 2 = 26$
907	905	$\hat\rho(Y)$
908	906	${}_tV$
909	907	$\tilde\rho$
910	908	$a=0$
911	909	$\square^\square-1$
912	910	$\rho_t(X) = \rho_t(-\rho_{t+1}(X))$
913	911	$\alpha_p = 1- (\| (X-\eta_{p,\alpha})_+\|_{p-1} / \| (X-\eta_{p,\alpha})_- \|_{p})^{p-1}$
914	912	$t$
915	913	$c\ge 1$
916	914	$g'(0)\le 1$
917	915	$\mathsf{E}(\theta)=1$
918	916	$\mathsf{TVaR}_{0.99}(X)=119.8=\mathsf{E}(W+Q\mid X\ge 100)=\mathsf{E}(W\mid X\ge 100) + \mathsf{E}(Q\mid X\ge 100)=19.8+100$
919	917	$\mathsf{E}(T)=74.25$
920	918	$X\wedge a:=\min(X,a)$
921	919	$P_Q$
922	920	$\bar\delta$
923	921	$\bar a_{40}=17.95$
924	922	$Y= IX$
925	923	$L^p$
926	924	$\mathsf E[A_0\mid N=n]=\mathsf E[X_0^{\oplus n}]=0$
927	925	$(asecret.east) + (0,-0.5)$
928	926	$\rho(X_n)\downarrow 0$
929	927	$\tilde p=\tilde p(p)$
930	928	$dp=$
931	929	$t=0.25$
932	930	$\zeta_\epsilon$
933	931	$s_s < s < s_f$
934	932	$\exp(n(e^\zeta-1))$
935	933	$M$
936	934	$0<p<1$
937	935	$10^6A_{75}=508676.91$
938	936	$R_i(t)>C_i(t)$
939	937	$\rho(X)=\sum_i \mathsf{E}_\mathbb{Q}(X_i)$
940	938	$se(\hat\beta)$
941	939	$1 - g(s)$
942	940	$j = 1, 2$
943	941	$\text{E}(G)=a\theta$
944	942	$\rho(-k_1 1_{A_1}) = k_1 \rho(-1_{A_1}) < c$
945	943	$H=G_0-F$
946	944	$-g''$
947	945	$\alpha=d$
948	946	$Y=h(Z)$
949	947	$\alpha(X)=a$
950	948	$(fun1a.south -| fun4a.south east)+(\smlspc,-\smlspc)$
951	949	$m=K^{-1}Km$
952	950	$\langle \cdot,\cdot\rangle:\mathcal{X}\times\mathcal{M}\to \mathbb{R}$
953	951	$p\in[1,\infty]$
954	952	$\mathsf{P}$
955	953	$q_Y$
956	954	$\bar P^a$
957	955	$\bar Q$
958	956	$\{X\le a\}$
959	957	$E_\mathsf{Q}(X_i) = E_\mathsf{Q}(E_\mathsf{Q}(X_i \mid X))$
960	958	$\phi:[0,1]\to [0,\infty)$
961	959	$q(p)=\mathsf{VaR}(p)$
962	960	$\rho(X)-a$
963	961	$m(p)$
964	962	$v^b{}_bq_x\bar a_{x+b} /\bar a_x=v^b{}_bq_x(1-{}_b\bar V)$
965	963	$\epsilon > 0$
966	964	$\mathsf{E}(Q/X | X\ge x)$
967	965	$v\mathsf E[X_i]$
968	966	$\tau>0$
969	967	$\Longleftrightarrow$
970	968	$\rho(X+Y)=\rho(X)+\rho(Y)$
971	969	$\lambda=0.1525$
972	970	$\mathsf E[X_i\mid X=x]$
973	971	$=a$
974	972	$P_{x+b}-P_x > 0$
975	973	$0<\alpha<2$
976	974	$p(\delta_p-il_p)$
977	975	$1 - \mathsf{Pr}(Z>\mathsf{E} Z)$
978	976	$[a,b]$
979	977	$(valu\x.south east)+(\boundpad,-\boundpad)$
980	978	$\rho(X) = \sup_{\zeta\in A} \langle \zeta, X \rangle$
981	979	$P(a) = \nu S(a) + \delta = \nu (S(a) + \rho)$
982	980	$(X,a_2)$
983	981	$\mathsf{E}_\mathsf{Q}(Y\mid X)\mathsf{E}(Z\mid X) = \mathsf{E}(YZ \mid X)$
984	982	$r=0.045$
985	983	$a$
986	984	$F(x):=\mathsf{Pr}(X\le x)$
987	985	$C_{1,\cdot}$
988	986	$\mathsf{E}_Q(\cdot)$
989	987	$g(s)=(s/1-p)^\alpha\wedge 1$
990	988	$\omega$
991	989	$ = a bond with probability $
992	990	$p=0.1$
993	991	$26 \rightarrow 2\times 4^2 + 2\times 4 + 1=41 \rightarrow 60 \rightarrow 83 \rightarrow 109\rightarrow\dots$
994	992	$x=3$
995	993	$p\delta_p/p\nu_p=\iota_p$
996	994	$t=0.5$
997	995	$c\ge 1/2$
998	996	$\mathbb{R}^n$
999	997	$\phi(t) = g'(1-t)$
1000	998	$k<k_0$
1001	999	$\rho(X) = \inf\{ \alpha \mid X+\alpha \in \mathcal{A} \}$
1002	1000	$a\theta=1-s$
1003	1001	$g'(t)<1$
1004	1002	$0 \ge \rho(Y-X) \ge \rho(Y) - \rho(X)$
1005	1003	$B<C<A$
1006	1004	$\square$
1007	1005	$0 < \mu < \lambda$
1008	1006	$F_n^{-1}(1)=\frac{1}{(n-1)!}\mathsf{E}[\min(X_1,\dots, X_{n-1}]$
1009	1007	$=\mathsf{E}(X_i/X \mid X \le a)$
1010	1008	$X(\mathbf{x})(\omega)=q_\omega(\mathbf{x})$
1011	1009	$p>0.5$
1012	1010	$X+\epsilon Y$
1013	1011	$1-\Phi(x)=\Phi(-x)$
1014	1012	$>q(p)$
1015	1013	$k\ge n$
1016	1014	$\alpha(X_u) = \text{E}[X\mid X > F_u^{-1}(p)]$
1017	1015	$E(u(X)) \le E(u(Y))$
1018	1016	$p_n=\mathsf{Pr}(N=n)$
1019	1017	$\zeta-\zeta_\epsilon$
1020	1018	$\mathsf{E}_Q(X) =\mathsf{E}(\theta X /\mathsf{E}(\theta))$
1021	1019	$g(s)g(t)=O(d^2)< g(s)$
1022	1020	$Y\le a$
1023	1021	$\zeta\in\partial(X)$
1024	1022	$\rho(T)$
1025	1023	$13809$
1026	1024	$n+2$
1027	1025	$P(a) = L(a) + \iota (a-P(a)) = \nu L(a) + \delta a$
1028	1026	$x_1$
1029	1027	$\sum_j \mathsf{TVaR}_{p_j}(X)m_j$
1030	1028	$\mathscr{O}(\zeta)=\{\zeta T \mid T\in MPT\}$
1031	1029	$\rho(1_A) = 1$
1032	1030	$g'(x)=0$
1033	1031	$\{X>a\}$
1034	1032	$\alpha(\cdot)$
1035	1033	$h(t)=\int_0^t F_Z^{-1}(1-u)\,du$
1036	1034	$g''(p)=-\phi'(1-p)\le 0$
1037	1035	$x = 0$
1038	1036	$(\bar a_x - \bar a_{\lcroof{b}})/\bar a_x$
1039	1037	$t=0=1$
1040	1038	$P=\rho_{PH}(X)$
1041	1039	$\mathbf{x}'$
1042	1040	$\mathrm{L}$
1043	1041	$\mathsf{E}(X) = \mathsf{E}(X\mid X \le a)F(a) + \mathsf{E}(X\mid X > a)S(a)$
1044	1042	$(1-t)/t$
1045	1043	$c=1.124$
1046	1044	$(Alice) + (0,-4)$
1047	1045	$\mathsf{cov}(h^i, Y(\mathbf{X})) = \mathsf{E}_P[h^iY(X)]$
1048	1046	$0<\alpha_1<\alpha_2<1$
1049	1047	$\rho(X,a)=\int_0^a S(x) + \delta(F(x))F(x)dx$
1050	1048	$l(p)= \nu-\sqrt{p(1-p)}$
1051	1049	$p-1=22$
1052	1050	$q_{\cdot}(\mathbf{x})$
1053	1051	$\cdot$
1054	1052	$\nabla\rho(X)=\{h\}$
1055	1053	$i=0,\dots,n-1$
1056	1054	$ is time cheap. Indeed, the condition implies the denominator is $
1057	1055	$g(\sqrt{st})^2$
1058	1056	$g(s)g(t)-g(st)$
1059	1057	$0.475$
1060	1058	$(ckey2.north west)+(-\boundpad,\boundpad)$
1061	1059	$\rho_t(X) = \displaystyle{1}{\beta} \log \mathsf E[e^{-\beta X}\mid \mathscr F_t]$
1062	1060	$\bar A_{x+b}$
1063	1061	$\rho(X+\epsilon Y)-\rho(X)$
1064	1062	$\prec_3$
1065	1063	$\rho(T)=76.11$
1066	1064	$\bar R(a)$
1067	1065	$4.7\times 10^{21} / 10^{19} = 470 \text{\,seconds} \approx 8\text{mins}$
1068	1066	$X^{\oplus n}$
1069	1067	$\sup \{ \mathsf{E}(LZ) \mid Z \preceq \sigma \}$
1070	1068	$(Alice)+(0,-2)$
1071	1069	$g(s) = \max(g_m, g^0(s))$
1072	1070	$g'(1)=\alpha < 1$
1073	1071	$X_-:=\max(-X,0)$
1074	1072	$g'(1-s)$
1075	1073	$X\in \mathcal X$
1076	1074	$\mathsf{E}_Q(N_i) =$
1077	1075	$\iff P +\rho_i(F_i) < \rho_i(X_i) \iff P < \rho_i(X_i) - \rho_i(F_i)$
1078	1076	$g'(S(x))=dQ/dP$
1079	1077	$G=\sum_i N_i(x_i) + C_i(x_i)$
1080	1078	$F_X$
1081	1079	$5 \times 10^9$
1082	1080	$1-\tilde p$
1083	1081	$\mathsf{E}_Q=\mathsf{E}$
1084	1082	$1- \nu F(x)$
1085	1083	$\delta_p=1-\nu_p=\rho_p\nu_p$
1086	1084	$X^{\oplus n} -\mathsf E[X] = X^{\oplus n-1} + (X'-\mathsf E[X])$
1087	1085	$\mathsf{E}[Y]=1$
1088	1086	$\langle \mu,Y \rangle - \langle \mu,X \rangle = \langle \mu, Y-X \rangle \ge 0$
1089	1087	$q(p)=c$
1090	1088	$\mu$
1091	1089	$\mathsf{E}(X_ig'(S))$
1092	1090	$x=0$
1093	1091	$p\delta_p/p\nu_p=\rho_p$
1094	1092	$g'(t)=αt^{α-1}$
1095	1093	$s_u = (f+1) / (n+1)$
1096	1094	$1-t=g^{-1}(1-s)$
1097	1095	$\rho(X)=\mathsf{E}_\mathsf{Q}[X]$
1098	1096	$X=q_X(U)$
1099	1097	$A=\sum_n 1_{N=n}X^{\oplus n}$
1100	1098	$i\in I$
1101	1099	$L_p$
1102	1100	$\mathsf{CoTVaR}(X_i)$
1103	1101	$g(st) = \displaystyle\frac{st}{1-p} < \displaystyle\frac{s}{1-p}= g(s)g(t)$
1104	1102	$\rho(X)=\lim_n \rho(X_n)$
1105	1103	$\sigma=0.45$
1106	1104	$\tilde F(x)=\mathsf{Pr}(\tilde X-\lambda\le x-\lambda)=\Phi(x-\lambda)$
1107	1105	$x_iX_i$
1108	1106	$(0,0)$
1109	1107	$\alpha_i(t)$
1110	1108	$q_X$
1111	1109	$g(1)=1$
1112	1110	$g'(1-s)=\phi(s)$
1113	1111	$\mathcal{M}\subset\mathscr{P}[0,1]$
1114	1112	$34.05$
1115	1113	$\mathsf{Pr}(X>a)>1-\alpha$
1116	1114	$k>m$
1117	1115	$m(x) = \nu S(x) + \delta = \nu (S(a) + \rho)$
1118	1116	$\sqrt{FS}$
1119	1117	$P_{x+b}-P_x$
1120	1118	$c_k$
1121	1119	$(X, a)$
1122	1120	$\mathsf{E}(X_i / X)$
1123	1121	$ is a measure on $
1124	1122	$k_i(a) = \phi_i(a) k(a)$
1125	1123	$\rho(G(\bar x))=\langle \zeta_{\bar x}, G(\bar x) \rangle$
1126	1124	$(a-X)^+$
1127	1125	$\langle \zeta, G \rangle=\int q_G q_\zeta$
1128	1126	$g(p)\ge p$
1129	1127	$\rho(m) = \rho(0) - m$
1130	1128	$\mathsf{cov}(X_1, N | G = const_j) f_G(const_j)$
1131	1129	$f'_\omega (\bar x, h)$
1132	1130	$g^a$
1133	1131	$\mathsf{VaR}$
1134	1132	$\bar P_{x+b}$
1135	1133	$L_0^{a-Y}$
1136	1134	$\sigma=0.25$
1137	1135	$(\rho)$
1138	1136	$\bar P^a(t):=\bar P^a(1-t, t)$
1139	1137	$\mathsf{E}(X)=\int S(x)dx$
1140	1138	$a=q_p(\mathbf{x})$
1141	1139	$a(x)$
1142	1140	$u^{iv}\le 0$
1143	1141	$\mathsf{E}(X \mid X\ge q_{1/k}(X))$
1144	1142	$\bar a_x = (1-\bar A_x)/\delta$
1145	1143	$(g^{k})^a = K$
1146	1144	$s=1$
1147	1145	$X=Y+Z$
1148	1146	$\le$
1149	1147	$(-\x*0.75, -2)$
1150	1148	$\mathsf{E}(YZ\mid X)=Z\mathsf{E}(Y\mid X)$
1151	1149	$X_+:=\max(X,0)$
1152	1150	$N_i=N_i(x_i)$
1153	1151	$50) of the amount allowed on each claim in the classes under subsections (3) to (7), inclusive, of this section, shall be deducted from the claim and included in the class under subsection (9) of this section. Claims may not be cumulated by assignment to avoid application of the fifty dollars ($
1154	1152	$\bar G'(a)=\frac{d\bar G}{da}=G(a)$
1155	1153	$\nu(p)<1$
1156	1154	$\mathsf{E}_q(X_1)$
1157	1155	$\mathsf{E}(L)$
1158	1156	$X_c$
1159	1157	$s_u$
1160	1158	$T_{x,\delta}$
1161	1159	$\phi'(s)=\mu(ds)/(1-s)\ge 0$
1162	1160	$SD(G')=\nu$
1163	1161	$x+t$
1164	1162	$x=0.5, M=1.5,\sigma=0.75, K=6$
1165	1163	$g'(1)$
1166	1164	$\nu(p) F(x)$
1167	1165	$c_l<c=\mathsf{VaR}$
1168	1166	$E_\mathsf{Q}(X_i \mid X)$
1169	1167	$d(1-d)=v(1-v)=dv$
1170	1168	$\hat\rho(A)<\rho(A)$
1171	1169	$Q=A-P$
1172	1170	$c<0$
1173	1171	$Z=\sum_i b_i1_{E_i}$
1174	1172	$A\in\mathcal{G}$
1175	1173	$X=W+Q$
1176	1174	$Z=Z(\mathbf{X})$
1177	1175	$0 \le 0$
1178	1176	$\tau=0.5$
1179	1177	$\lambda$
1180	1178	$C = cx/a$
1181	1179	$=\dfrac{1}{1-p}\displaystyle\int_{p}^1 q(p)dp$
1182	1180	$\mathsf{SA}$
1183	1181	$\beta_i(t)/\alpha_i(t)<g(S(t))/S(t)$
1184	1182	$Z\preceq \sigma$
1185	1183	$p(a) = 1 - \nu F(a)$
1186	1184	$C^{D+E}$
1187	1185	$\beta((a-X)^+)$
1188	1186	$xf(x)$
1189	1187	$\rho(X)\ge -\rho(-X)$
1190	1188	$l(p)= \nu(p)-\sqrt{p(1-p)}$
1191	1189	$d=1/(1+r)$
1192	1190	$\mathrm{MV}$
1193	1191	$v+l$
1194	1192	$2^{256}=115792089237316195423570985008687907853269984665640564039457584007913129639936=1.2\times 10^{77}$
1195	1193	$kS = m + Ra$
1196	1194	$\hat\rho(X)\ge \rho(X)$
1197	1195	$A_{k_0}$
1198	1196	$\sum_i a_i1_{D_i}$
1199	1197	$\rho(X\wedge a)$
1200	1198	$E(X_i/X \mid X)$
1201	1199	$697.6 billion in 2016, $
1202	1200	$\mathsf{E}(Z \mid \mathcal{G})=Z$
1203	1201	$E(G')=1-f$
1204	1202	$\zeta\in \mathcal{Z}*$
1205	1203	$O(n)$
1206	1204	$1_A$
1207	1205	$X(x)=x$
1208	1206	$p\in [1, \infty]$
1209	1207	$iota^*$
1210	1208	$A=\mathsf E[X]N + A_0\succeq \mathsf E[X]N$
1211	1209	$\mu_x = A+Bc^x$
1212	1210	$dQ/dp=\phi(p)$
1213	1211	$F_Z^{-1}(U)\in\mathscr{P}$
1214	1212	$A=\rho_{\mathsf{TVaR}}(X)$
1215	1213	$ for all $
1216	1214	$C_2(0)>\mathsf{E}[X_2]$
1217	1215	$M(0)=1$
1218	1216	$2\nu$
1219	1217	$c_k-G\le 0$
1220	1218	$\forall X\in L^p$
1221	1219	$B_t$
1222	1220	$\nu_p$
1223	1221	$Q(a) = (L-a)V(a) = (L-a)^+$
1224	1222	$p\nu_p$
1225	1223	$L_{\sigma_1}\subset L_{\sigma_2}$
1226	1224	$g(x)=x$
1227	1225	$(p,q(p))$
1228	1226	$S(x)dx$
1229	1227	$\nabla p$
1230	1228	$Z_1$
1231	1229	$\mathsf{E}[X_i(1) \mid X(\mathbf{x}) = q_p(\mathbf{x}) ]$
1232	1230	$g(S(x))=q(\tilde p)\phi(\tilde p)$
1233	1231	$r_f$
1234	1232	$\bar P_{40}=6908.82$
1235	1233	$\phi(p)$
1236	1234	$D_i-N_i > 0$
1237	1235	$A=0.00022$
1238	1236	$(X,a_1)$
1239	1237	$\rho(X)=\int g(S(x))dx$
1240	1238	$X_i(\mathbf{x}; a)$
1241	1239	$0.5<t<1$
1242	1240	$g<q$
1243	1241	$ν$
1244	1242	$\bar\iota$
1245	1243	$0.318 / 260.81 = 0.13\%$
1246	1244	$0.4-x^2/4.6-\log(x)$
1247	1245	$X\ge Y\implies \rho(X) \ge \rho(Y)$
1248	1246	$[\alpha,1)$
1249	1247	$x_i=q(u_i)=F^{-1}(u_i)$
1250	1248	$\epsilon(\mathsf{E}_q(X_1)-t)$
1251	1249	$\rho_\sigma$
1252	1250	$(rep.east) + (1.5, -0.5)$
1253	1251	$l_c\le l_i$
1254	1252	$\mathsf{TVaR}_1=\esssup$
1255	1253	$R_2(t) > R_2(0)$
1256	1254	$\sigma_\mu(\alpha) = \displaystyle\int_0^\alpha\dfrac{1}{1-u}\mu(du)$
1257	1255	$\rho(X+Y)\le\rho(X) + \rho(Y)$
1258	1256	$X \prec_n Y$
1259	1257	$\phi_i(a)\mathsf{E}(Y\wedge a) = \mathsf{E}(X_i(a))$
1260	1258	$\rho(X+x)=\rho(X)-x$
1261	1259	$F:\mathbb{R}^n \to \mathcal{X}$
1262	1260	$S>0$
1263	1261	$G = C + \sum_i N_i$
1264	1262	$\sqrt{2Np}=19$
1265	1263	$(fun1a.south -| fun3a.south east)+(\smlspc,-\smlspc)$
1266	1264	$g'$
1267	1265	$Y-X\le 0$
1268	1266	$\rho(X-a)=\rho(X)-a$
1269	1267	$\mathsf E[F_i]$
1270	1268	$750,000,000). The deposit shall be made subject to the approval of the commissioner under those rules and regulations that he or she shall promulgate. The deposit shall be maintained at a deposit value specified by the commissioner, but in any event no less than one hundred thousand dollars ($
1271	1269	$\mathbb{Q}$
1272	1270	$f(s) = \alpha(1-\alpha)(1-s)^{\alpha-1}$
1273	1271	$\nu \in\mathscr{P}[0,1]$
1274	1272	$a\ll \sum_i a_i$
1275	1273	$g^{-1}(x)\le s$
1276	1274	$\mathsf{TVaR}_p(X)=\frac{1}{1-p}\int_p^1 F_X^{-1}(t)dt$
1277	1275	$A_1$
1278	1276	$g_n$
1279	1277	$\bar R$
1280	1278	$\mathsf{E}_\mathsf{P}$
1281	1279	$1/(1-\alpha)$
1282	1280	$u'''>0$
1283	1281	$Z_a$
1284	1282	$t = 1$
1285	1283	$id\times\tau$
1286	1284	$[0.37, 0.55]$
1287	1285	$B(1/2)$
1288	1286	$n=2,3$
1289	1287	$m(p)=q+p\delta_p$
1290	1288	$\rho(-X)$
1291	1289	$X=X_c + X_n$
1292	1290	$\sigma=0.15$
1293	1291	$\rho(\cdot)$
1294	1292	$[a,a+da]$
1295	1293	$(s,t)$
1296	1294	$g'(0)>1$
1297	1295	$\le 1$
1298	1296	$q=1-p$
1299	1297	$\rho(X)\ge -\rho(-X)\ge a$
1300	1298	$(\mathsf{E}_q(X_1)-s)/\mathsf{E}_q(X_1)$
1301	1299	$200 of losses otherwise payable to any claimant under this subsection. All claims under life insurance policies and annuity contracts, whether for death proceeds, annuity proceeds or investment values, must be treated as loss claims. Claims may not be cumulated by assignment to avoid application of the $
1302	1300	$X_p =F_X^{-1}(p + (1-p)U_X$
1303	1301	$X_i(\alpha)$
1304	1302	$=\mathsf{E}(X_i/X \mid X > a)$
1305	1303	$N=1$
1306	1304	$a\wedge b:=\min(a,b)$
1307	1305	$t_2-\epsilon$
1308	1306	$X_1,X_2$
1309	1307	$q(1)$
1310	1308	$\theta<1$
1311	1309	$\sum_i X_i(a) = X\wedge a$
1312	1310	$X(T(s))=q(s)$
1313	1311	$\tpx=\exp(-\int_0^t \mu_{x+s}ds)$
1314	1312	$H$
1315	1313	$g^{kS}=R^S$
1316	1314	$a\mapsto n=g^a\pmod{p}$
1317	1315	$(x, g(S(x)))$
1318	1316	$0 \le \rho(0) = \rho(X-X) \le \rho(X) + \rho(-X)$
1319	1317	$\bar a_{\lcroof{b}}=(1-v^b)/\delta$
1320	1318	$CV=\nu=\sqrt{a}\theta$
1321	1319	$\psi$
1322	1320	$3.2 \times 10^{18}$
1323	1321	$a_i=\rho_i(\tilde X_i)$
1324	1322	$\rho(X-\rho(X))=\rho(X)-\rho(X)=0$
1325	1323	$v$
1326	1324	$\lambda_{x+t}=\lambda\mu_{x+t}$
1327	1325	$\rho(X + \rho(X))=0$
1328	1326	$\lambda=(1-\alpha_p)^{-1}$
1329	1327	$\backslash$
1330	1328	$\delta=\iota\nu$
1331	1329	$\mathsf{E}[X_2]$
1332	1330	$\rho(xX)=x\rho(X)$
1333	1331	$R_1(t) = \bar P^a_1(t)/(1-t)$
1334	1332	$g^{ak}=(g^k)^a$
1335	1333	$f(0)=0$
1336	1334	$(fun5.north east)+(\medspc,\medspc)$
1337	1335	$p = 1-g^{-1}(1-\bar p)$
1338	1336	$1-p$
1339	1337	$C_1$
1340	1338	$x<\mathsf{VaR}_p(X)$
1341	1339	$μ = δ_α$
1342	1340	$P_c, P_n$
1343	1341	$g(s) =$
1344	1342	$\rho_\phi$
1345	1343	$\rho_\min(L_i)=\rho_i(L_i)$
1346	1344	$\mathsf{E}(X_i \mid X=x)$
1347	1345	$g(s)=s^{2/3}$
1348	1346	$\epsilon(\mathsf{E}_q(X_1)-s)$
1349	1347	$\sigma\in L_q$
1350	1348	$a\ge \psi(X)$
1351	1349	$l_p=\nu_p-\nu_{1/2}\sqrt{\bar p}$
1352	1350	$(N,m)$
1353	1351	$s=0$
1354	1352	$x^∗$
1355	1353	$C_t$
1356	1354	$\mathsf{E}(X_i\mid X=x)$
1357	1355	$i=1$
1358	1356	$\tau_n$
1359	1357	$200 of losses otherwise payable to any claimant under this subsection other than the federal government. All claims under life insurance and annuity policies, whether for death proceeds, annuity proceeds or investment values, shall be treated as loss claims. Claims may not be cumulated by assignment to avoid application of the $
1360	1358	$G=f+G'$
1361	1359	$-\partial g(S(x))/\partial x$
1362	1360	$\mathcal X^\perp$
1363	1361	$\mathsf{E}_P[h_0]=\mathsf{E}_P[h_{i,\epsilon}]=1$
1364	1362	$EL_a =\mathsf{Pr}(Y>a) = \mathsf{Pr}(\max(X_1, \dots, X_N)>a)=\mathsf{Pr}(\text{one or more events $
1365	1363	$\mathsf{E}_\mathbb{Q}$
1366	1364	$\rho(0) = 0$
1367	1365	$xf_i(x)$
1368	1366	$\delta \ge 0$
1369	1367	$Z'=ZT$
1370	1368	$X \preceq_{sl} Y$
1371	1369	$q(p)=F^{-1}(p)=\mathsf{VaR}_p(X)$
1372	1370	$A=X_1 + \cdots X_N$
1373	1371	$x\mapsto |x|$
1374	1372	${}^1S^{-1}=S^{-1}$
1375	1373	$m$
1376	1374	$f$
1377	1375	$g(s)=1$
1378	1376	$\mathsf{E}[X_1]=\mathsf{E}[X_2]$
1379	1377	$1-EL$
1380	1378	$100$
1381	1379	$C_k$
1382	1380	$COC = (P-L) / Q$
1383	1381	$\mathsf{E}_Q(X \mid \mathcal{G})\mathsf{E}(Z \mid \mathcal{G}) = E(XZ \mid \mathcal{G})$
1384	1382	$c=\sup_{0\le\alpha<1} \dfrac{\int_\alpha^1 \sigma_2}{\int_\alpha^1 \sigma_1}$
1385	1383	$\mathcal{M}_{X,r_X}=\{m \in\mathcal{M} \mid \rho_m(X) = r_X \}$
1386	1384	$\mathsf{E}_\mathbb{Q}(X_i) = \mathsf{E}_\mathbb{Q}(\mathsf{E}_\mathbb{Q}(X_i \mid X)) = \mathsf{E}_\mathbb{Q}(\mathsf{E}(X_i \mid X))$
1387	1385	$ "the standard way to obtain the $
1388	1386	$\rho(X)=\mathsf{E}[hX]$
1389	1387	$R(a)$
1390	1388	$f(x, \cdot)\in L_p(\Omega, \mathcal{F}, \mathcal{P})$
1391	1389	$\pi'(\sqrt k)=0$
1392	1390	$\rho_{m'}(Y) < 89$
1393	1391	$i>0$
1394	1392	$(L^t)^+$
1395	1393	$P(x) = \sum_i P_i(x)$
1396	1394	$\dots$
1397	1395	$X=X_+-X_-$
1398	1396	$\mathsf{Var}(\pi)=\bar p/(\nu_p-l_p)^2$
1399	1397	$q_X(p)$
1400	1398	$a=a(f)$
1401	1399	$(1-\alpha)^{-1} \min_c c(1-\alpha) + \mathsf{E}(X-c)_+$
1402	1400	$d=i/(1+i)$
1403	1401	$\nu(p)$
1404	1402	$(rep.south) + (0.5, -2.70)$
1405	1403	$\mathsf{Pr}(Z>\mathsf{E}(Z))$
1406	1404	$r=50$
1407	1405	$\inf_\eta \{ \eta + \phi(X_\eta) \}$
1408	1406	$X+tY$
1409	1407	$p_1, \dots, p_N$
1410	1408	$\text{Var}(G)=a\theta^2$
1411	1409	$r=3$
1412	1410	$Var(G) = a\theta^2$
1413	1411	$\delta F$
1414	1412	$P(X) = M(X, \psi(X))$
1415	1413	$a\ge 0$
1416	1414	$X(p)=F^{-1}(p)$
1417	1415	$K = (A)^{b} = g^{ab}$
1418	1416	$YN$
1419	1417	$\bar P_{75}=53123.19$
1420	1418	$x\to\infty$
1421	1419	$m_1 / r_1 > m_2 / r_2$
1422	1420	$0.1$
1423	1421	$\Delta \tilde p< \Delta p$
1424	1422	$l_p=0$
1425	1423	$X_i(u_i)$
1426	1424	$k>0$
1427	1425	$\mathsf{E}(L) = F^{-1}(p) dp$
1428	1426	$X_i(a)=(X\wedge a)X_i/X$
1429	1427	$\rho_t(X)$
1430	1428	$1-l-(\nu-l)=\delta$
1431	1429	$Q_\epsilon \to Q$
1432	1430	$
1433	1431	$\rho(0X)=\rho(0)=0\rho(X)=0$
1434	1432	$r_X=\mathsf{TVaR}_p(X)$
1435	1433	$. If the insurer has a single insured there is no notion of default: the insured has purchased a policy covering losses up to a limit $
1436	1434	$R_1(t)$
1437	1435	$X=q=F^{-1}$
1438	1436	$Q(a)$
1439	1437	$q_2(t)=t^2$
1440	1438	$\mathcal{A}$
1441	1439	$F:\mathbb{R}^n\to\mathcal{X}^n$
1442	1440	$\eta\ge$
1443	1441	$\subset [\essinf X ,\esssup X]$
1444	1442	$a'=\mathsf{E}[X|A^c]$
1445	1443	$1,2,3,\dots$
1446	1444	$g\circ S$
1447	1445	$2\square^2 + 2\square + 2$
1448	1446	$L_p dp$
1449	1447	$A_k=X_{k,1} + \cdots + X_{k, N}$
1450	1448	$X_n\uparrow 0$
1451	1449	$\mathsf{Pr}(\mathsf B(s)=1)=s$
1452	1450	$C_1(t)=C_2(t)=\bar P^a(t)$
1453	1451	$Q(a) = 1 - P(a) = 1 - g(S(a))$
1454	1452	$\rho(X)=\sup\{ \mathsf{E}(XZ) \mid Z\ge 0, \mathsf{E}(Z)=1, \mathsf{E}(Z\log(Z))\le\log(1/(1-\alpha)) \}$
1455	1453	$C_i$
1456	1454	$\bar Q(a)$
1457	1455	$\bar P_i$
1458	1456	$\mathsf{E}(X_i \mid G=q)=:\mathsf{E}_q(X_i)$
1459	1457	$\mathsf{E}[XZ] = \mathsf{cov}(X,Z) \le \sigma(X)\sigma(Z)\le \sigma(X)$
1460	1458	$\nu=1/(1+\rho)$
1461	1459	$\mathscr{P}=\{ (1-p)^{-1}1_A \mid P(A)\le 1-p \}$
1462	1460	$\phi(x)=-\int_x^1 (s-x)^{n-1}d\tau(s)$
1463	1461	$m(x)=S(x)+d_iF(x)+(v-\nu^*)\sqrt{F(x)S(x)}$
1464	1462	$\partial B$
1465	1463	$\mathsf B(s)$
1466	1464	$t^*$
1467	1465	$X,Y,X+Y$
1468	1466	$a=(X\wedge a) + (a-X)^+$
1469	1467	$(rep.south) + (0.5, -1.85)$
1470	1468	$ for $
1471	1469	$L_a^{a+y}$
1472	1470	$(\sqrt{st}, \sqrt{st})$
1473	1471	$\sum_{n\ge 0} 1_{N>n} X_n$
1474	1472	$X\wedge a =\min(X,a)$
1475	1473	$\mathsf{TVaR}_p(X)$
1476	1474	$L_{p,\delta}(\omega)=\begin{cases} q(p) & \omega\in (p,p+\delta] \\ 0 & \omega\not\in (p, p+\delta]\end{cases}$
1477	1475	$\mathbb{R}\times \mathbb{R}$
1478	1476	$\beta_i(t)/\alpha_i(t)> 1 > g(S(t)) / S(t)$
1479	1477	$\ge 5000 / \text{Probability}$
1480	1478	$\rho(A_k)\ge \mathsf{E}[A_k] = k\mathsf{E}[N]$
1481	1479	$1 \times 10^{15}$
1482	1480	$q\phi$
1483	1481	$R_i=\alpha p_i + \beta r_{qp,i} + \gamma\, \text{controls}_i$
1484	1482	$CV(G) = SD(G') = \nu$
1485	1483	$+$
1486	1484	$\eta=(1-\alpha)^{-1}1_A$
1487	1485	$E(X^k)=E(Y^k)$
1488	1486	$2 \times 10^{14}$
1489	1487	$a=a(\mathbf{x})$
1490	1488	$a=a(x)$
1491	1489	$g\in D_n^*=\{ g \mid (-1)^{k+1} g^{(k)} \ge 0, k=1,\dots,n-1, (-1)^n g^{(n-1)}\text{ nonincreasing} \}$
1492	1490	$\log(1-\Phi(x))$
1493	1491	$S(x_1)-S(x_2)\approx f(x_1)(x_2-x_1)$
1494	1492	$\zeta_t\to\zeta$
1495	1493	$R_1(t)<R_1(0)$
1496	1494	$1.25 \times 10^{14}$
1497	1495	$\mathsf{E}[XZ_1]$
1498	1496	$C_i(t) = \partial \bar P^a/\partial x_i$
1499	1497	$1-w$
1500	1498	$\delta(\sqrt{st},\sqrt{st})\ge 0$
1501	1499	$S_{\tilde X}$
1502	1500	$2\square^2 + 11$
1503	1501	$g(1-p)=1- \tilde p$
1504	1502	$\mathsf{E}(X_i \mid X \ge a)$
1505	1503	$Q\in\mathscr{P}$
1506	1504	$(x^{-1}-x^{-3})\phi(x)$
1507	1505	$g(s) = s^{b}$
1508	1506	$ is average invested assets, equal to $
1509	1507	$0=p_0 < p_1 < p_2 < p_3=1$
1510	1508	$1 -p = g(1-\hat p)$
1511	1509	$\mathcal F_1=\sigma(I_1,\dots,I_n)$
1512	1510	$g=1$
1513	1511	$\mu-\nu$
1514	1512	$F(x)=p$
1515	1513	$Q=a-P$
1516	1514	$R_2(t) > C_2(t)$
1517	1515	$ is $
1518	1516	$\mathcal A_\rho= \{ X\mid \rho(X)\le 0 \}$
1519	1517	$X \prec_n^* Y$
1520	1518	$\nu F(a)$
1521	1519	$\mathsf{E}(L)=\int_0^\infty S(x)dx$
1522	1520	$K_Q=19.473$
1523	1521	$X=X_i + \hat X_i$
1524	1522	$500/year HO insurance then I don't really notice it compared to upkeep, mortgage, property tax etc. It is just a sunk cost. But if I pay $
1525	1523	$ and investor equity $
1526	1524	$(x-a)_+^\alpha$
1527	1525	$r_{pq}$
1528	1526	$\mathsf{E}$
1529	1527	$c\le a$
1530	1528	$g(s)g(k/s)$
1531	1529	$\phi(1-p)=g'(p)$
1532	1530	$k= \mathsf{E}(X\wedge k) + (\rho_m(X) - \mathsf{E}(X\wedge k)) + (k-\rho_m(X))$
1533	1531	$\langle \zeta_{\bar x}, N_i \rangle$
1534	1532	$Z>\mathsf{E} Z$
1535	1533	$\int_0^1 dp$
1536	1534	$\Bbb{Q}$
1537	1535	$T_A$
1538	1536	$E_\mathsf{Q}(X_i\mid X)=E(X_i\mid X)$
1539	1537	$\beta=0$
1540	1538	$O(dt)$
1541	1539	$V=m(L(1+e)P+rS) + (eL+\rho S)$
1542	1540	$0<a\le 99$
1543	1541	$g(s) = \min(1, a+bs)$
1544	1542	$\sigma=2$
1545	1543	$t\in(0,1)$
1546	1544	$p>1$
1547	1545	$\displaystyle\int_0^1 \text{AVaR}_\alpha(X)d\alpha$
1548	1546	$\rho_m$
1549	1547	$b_i$
1550	1548	$\mu_{x+t}$
1551	1549	${}_tp_x=\mathsf{Pr}(T_x > t) =\mathsf{Pr}(T_0 > x+t \mid T_0 > x)$
1552	1550	$\mathsf{P}(B)=0$
1553	1551	$m_j=m([p_{j-1},p_j])$
1554	1552	$(0,\dots,0,r_0,\dots, r_k)$
1555	1553	$\| X_n \|_\infty \le 1$
1556	1554	$dF=-d(g\circ S)=$
1557	1555	$\rho(X+tY)=\langle \zeta_t, X+tY \rangle$
1558	1556	$\pi'(k)=...$
1559	1557	$g:\text{thin layer risk}\mapsto\text{price}$
1560	1558	$(x-\mu_x)^+$
1561	1559	$(\mathsf{E}_q(X_1)(1-\epsilon\mathsf{E}_q(X_2)/q), \mathsf{E}_q(X_2)(1+\epsilon \mathsf{E}_q(X_1)/q))$
1562	1560	$5 \times 10^{14}$
1563	1561	$\rho(Z)=\int_0^1\eta(\tau)\mathsf{VaR}_\tau(Z)d\tau$
1564	1562	$ xx billion, of which California workers compensation deposits account for $
1565	1563	$-\int xd(g\circ S)=\int g(S(x))dx$
1566	1564	$2$
1567	1565	$(p,q(1-g^{-1}(1-p)))$
1568	1566	$S(a)da$
1569	1567	$\partial Y/\partial x_i$
1570	1568	$\sum_i F_i=F$
1571	1569	$\mathsf{E}(X) + c\mathsf{E}(| X-\mathsf{E}(X) |^p)^{1/p}$
1572	1570	$\mathcal X^\perp = \{X\in\mathcal X\mid \exists U\text{ uniform[0,1] rv independent of } X\}$
1573	1571	$\alpha(X)$
1574	1572	$\bar A^{1}_{x:\lcroof{n}}$
1575	1573	$\mathsf{TVaR}_{p_2}(X)\ge r$
1576	1574	$\mathsf{TVaR}_p(X)=$
1577	1575	$g(s)=s^{1/4}$
1578	1576	$\rho(X+tY)\ge \rho(X) + \langle \zeta, tY \rangle$
1579	1577	$X_n\to X$
1580	1578	$\rho(X - b)=\rho(X)-b\le 0$
1581	1579	$t=2$
1582	1580	$Q\in \partial\rho(X)$
1583	1581	$g=\mathsf{E}(G^3)=\nu^3 skew(G')+3c+1$
1584	1582	$375-185=190 > 0$
1585	1583	$C_1(t) < \bar P^a(1, 0)$
1586	1584	$i=1,2$
1587	1585	$\partial\rho(Z)$
1588	1586	$\rho(L) = q(1-g{-1}(1-p))\delta > \mathsf{E}(L)$
1589	1587	$\rho(p)$
1590	1588	$1-\delta\bar a_{x:\lcroof{n}}-\bar A_{x:\lcroof{n}}=0$
1591	1589	$\theta=(1-f)/a$
1592	1590	$\mathsf{Var}(B(p))=p(1-p)$
1593	1591	$p\in[0,1]$
1594	1592	$\mathsf{COH}+\mathsf{FAT}$
1595	1593	$=E(X_i \mid X \ge a)$
1596	1594	$\zeta$
1597	1595	$\mathcal{M}_{X,r}=\mathsf{var}nothing$
1598	1596	$\rho(X\mid \mathcal F_1) =\mathsf E[X g'\mathsf{Pr}(X>x\mid \mathcal F_1) ]$
1599	1597	$\alpha=d_i$
1600	1598	$\{ \zeta>0 \} = \{ G>c(x) \}$
1601	1599	$(v-\nu^*)\sqrt{FS}$
1602	1600	$\mathsf{TVaR}_{p=1}=\esssup$
1603	1601	$F_i = X_i(1 - (X\wedge a) / X)$
1604	1602	$t>0.25$
1605	1603	$X^∗_i = (X − x^∗)I_{A^∗_i} + x^∗ / n$
1606	1604	$H_k=H_{g_k}$
1607	1605	$\lambda\mu_t$
1608	1606	$(Bob) + (0,-4)$
1609	1607	$1 assets: $
1610	1608	$\sum_i P_i(a)=P(a)$
1611	1609	$\rho GF$
1612	1610	$\rho=0.5, x=1.5, M=1.5,\sigma=0.75, K=8$
1613	1611	$q_Z$
1614	1612	$\langle \mu,tX \rangle - \rho(tX) =t(\langle \mu,X \rangle - \rho(X))$
1615	1613	$^{*}$
1616	1614	$\hat p$
1617	1615	$\delta(F(x))=\delta$
1618	1616	$L_x^{x+dx}=L_0^{x+dx} - L_0^x$
1619	1617	$M(a)$
1620	1618	$\alpha < 1$
1621	1619	$a-X\le 0$
1622	1620	$>0$
1623	1621	$\tilde \rho(X)=\mathsf{E}(X) + \inf_t \rho(X-t)$
1624	1622	$Y\circ T=g(X\circ T)$
1625	1623	$\mathsf{E}[X_1]$
1626	1624	$\rho(X)=-U(X)$
1627	1625	$-\epsilon(\mathsf{E}_q(X_2)-s)$
1628	1626	$E_2=0$
1629	1627	$\mu_{x+t}=-\dfrac{d}{dt}\log({}_tp_x)$
1630	1628	$a\mapsto g^a \pmod{p}$
1631	1629	$(fun1a.south -| fun5a.east)+(\smlspc,-\smlspc)$
1632	1630	$10^{16}$
1633	1631	$X=X(x_i)=\sum_i X_i(x_i)$
1634	1632	$t \le 1-p$
1635	1633	$\rho(X+c)=\rho(X) + c$
1636	1634	$h\in\mathscr P$
1637	1635	$il$
1638	1636	$697.6 billion underlying Table \ref{tab-equity-what-if} this implies $
1639	1637	$q=S(a)$
1640	1638	$\rho(0)=0$
1641	1639	$Q_\epsilon$
1642	1640	$k_i=\mathsf{E}_Q(X_i)$
1643	1641	$\rho(X)\ge\rho(X+Y)\ge \rho(X)+\mathsf{E}[gY]$
1644	1642	$\rho(A)\le \rho(N)\rho(X)$
1645	1643	$k>\max(N)\max(|X|)$
1646	1644	$\bar P^a(1,0)<\bar P^a(0,1)$
1647	1645	$st \le 1-p < s$
1648	1646	$X-\sum f_i(X)$
1649	1647	$\bar P_x = (1/\bar a_x)-\delta$
1650	1648	$\beta=v-\nu^*$
1651	1649	$\mathscr{F}$
1652	1650	$310 billion in premiums annually in California. Since 2011 the California Department of Insurance received more than 1,000,000 calls from consumers and helped recover over $
1653	1651	$d_i=iv=i/(1+i)$
1654	1652	$\sigma=0.35$
1655	1653	$t=0.37$
1656	1654	$R_2(t)<C_2(t)$
1657	1655	$X>a$
1658	1656	$X(t)$
1659	1657	$(4-\s, \s)$
1660	1658	$1 excess attachment $
1661	1659	$f(\alpha):=\mathsf{E}[X^\alpha-Y^\alpha]$
1662	1660	$t=1-g(0)=1$
1663	1661	$x=0.1, M=1.5,\sigma=0.75, K=6$
1664	1662	$\partial\rho(X)=\{\zeta\}$
1665	1663	$t>t_2$
1666	1664	$x\ge 0$
1667	1665	$Q(a)=\nu N(a)$
1668	1666	$(3+2)/2=5/2$
1669	1667	$\displaystyle\int_0^\infty xg'(1-F(x))f(x)dx = -xg(S(x))\vert_0^\infty + \displaystyle\int_0^\infty g(S(x))dx=\displaystyle\int_0^\infty g(S(x))dx$
1670	1668	$(K=g^k, mg^{ak})$
1671	1669	$kN$
1672	1670	$\mathsf{E}_Q(X \mid \mathcal{G}) = E(X \mid \mathcal{G})$
1673	1671	$F(x)$
1674	1672	$[l_c, r_c)$
1675	1673	$\mathsf{Var}(B(p)/p\nu_p)=p(1-p)/(p\nu_p)^2$
1676	1674	$F(a)=p$
1677	1675	$\mathsf{E}[x_iX_i\mid X(\mathbf{x}) \le a]F_{\mathbf{x}};a) = \mathsf{E}[x_iX_i 1_{X(\mathbf{x}) \le a}]$
1678	1676	$(-1)^nf^{(n)}(x)<0$
1679	1677	$h^i = \lim_{\epsilon\downarrow 0}(h_{i,\epsilon}-h_0)/\epsilon$
1680	1678	$Z_1=q_Z(U)$
1681	1679	$[1,2]$
1682	1680	$\approx 10^{-40}$
1683	1681	$\hat\rho(A_k) =\rho(\rho((X+k)^{\oplus N})) = \rho(\rho(X^{\oplus N})+kN)= \hat\rho(A_0) + k\rho(N)$
1684	1682	$\tau=0.156$
1685	1683	$\mathsf{E}_\mathsf{Q}(X)$
1686	1684	$f_G$
1687	1685	$424) for the initial filing of each letter of credit utilized pursuant to subdivision (a). In addition, the commissioner shall require payment, in advance, of a fee of two hundred eighty-three dollars ($
1688	1686	$\displaystyle\int_0^\infty xdF(x)$
1689	1687	$(4.5-\s, \s)$
1690	1688	$g(s) = t_{df}(\Phi^{-1}(s)+\lambda)$
1691	1689	$B-p(\nu(p) + il(p))$
1692	1690	$R, S$
1693	1691	$a = b$
1694	1692	$\nabla \zeta=0$
1695	1693	$X\sim\text{Lognormal}(\mu=19.9, \sigma=2.36)$
1696	1694	$\sqrt{F(x)S(x)}$
1697	1695	$\rho(X)=35/9$
1698	1696	$X(p)$
1699	1697	$\langle X(\epsilon),\zeta_\epsilon \rangle-\langle X,\zeta \rangle=\langle X(\epsilon)-X,\zeta \rangle$
1700	1698	$\rho_{t+1}(X)=\rho_{t+1}(Y)\implies \rho_{t}(X)=\rho_{t}(Y)$
1701	1699	$\bar P_x:=\bar A_x / \bar a_x$
1702	1700	$p=0.5$
1703	1701	$(\Omega, \mathcal{F}, \mathbb{P})$
1704	1702	$l\ge 1$
1705	1703	$X(\omega)=$
1706	1704	$g(st) = 1= g(s)g(t)$
1707	1705	$\int_x^\infty$
1708	1706	$p=F(a)=1-q$
1709	1707	$\bar S$
1710	1708	$(ckey\x.north west)+(-\boundpad,\boundpad)$
1711	1709	$\rho_{t+1}(X) = \rho_{t+1}(Y) \implies \rho_{t}(X) = \rho_{t}(Y)$
1712	1710	$\rho_\phi=\mathsf{E}$
1713	1711	$\rho(X)=\int_\Omega X(\omega)\theta(\omega)dP(\omega)$
1714	1712	$B(0.5)$
1715	1713	$U\subset\Bbb{R}^n$
1716	1714	$a(x) = \sum_i x_i a_i = \sum_i x_i v_i a$
1717	1715	$\phi(p)dp$
1718	1716	$\gamma$
1719	1717	$p\in (0, 1)$
1720	1718	$ since $
1721	1719	$p\mapsto q(\hat p)=q(1-g^{-1}(1-p))$
1722	1720	$S =$
1723	1721	$p(x)$
1724	1722	$H(x)=y$
1725	1723	$x\mapsto \mathsf E[f(X_2)\mid X_1=x]$
1726	1724	$B(b)>0$
1727	1725	$\mathsf E[X^{\oplus n}]\le\rho(X^{\oplus n})$
1728	1726	$g(st) = \displaystyle\frac{st}{1-p} < 1 = g(s)g(t)$
1729	1727	$\pi_X(t_{2j-1})\le \pi_Y(t_{2j-1})$
1730	1728	$ϕ$
1731	1729	$i=1,\dots, n_r$
1732	1730	$\mathsf PV$
1733	1731	$\le 1/(1-\alpha)$
1734	1732	$A<B<C$
1735	1733	$B_l$
1736	1734	$t<1<0.5<t_2$
1737	1735	$(Bob) + (0,-2.5)$
1738	1736	$\rho(B(s_l))$
1739	1737	$g(s)=e^\alpha p/(e^\alpha p + (1-p))$
1740	1738	$\mathcal T(X)=\hat\rho(X) - \rho(X)$
1741	1739	$g, p, A=g^a, m$
1742	1740	$t=n\wedge T_x$
1743	1741	$\pi_g(X)=\int_a^{\alpha(X)} g(S(t))dt$
1744	1742	$\zeta\in\mathcal{A}$
1745	1743	$\delta$
1746	1744	$p=10^{-6}$
1747	1745	$\mathsf{E}(X_i\wedge x)$
1748	1746	$w_1, w_2$
1749	1747	$X + \epsilon Y$
1750	1748	$\zeta\ge 0$
1751	1749	$X_i-F_i$
1752	1750	$A=\partial \rho(0)$
1753	1751	$C_1(t)=C_2(t)$
1754	1752	$X\tilde N(0,\sigma^2)$
1755	1753	$dF=-dS=$
1756	1754	$\rho(A)=4.875 > \hat\rho(A)=4.8125$
1757	1755	$\nabla_y f=-\nabla_y G$
1758	1756	$\| f^*-f\|_2$
1759	1757	$\iota(0.5)=\iota^*$
1760	1758	$\rho_{t+1}(X) \ge \rho_{t+1}(Y) \implies \rho_{t}(X) \ge \rho_{t}(Y)$
1761	1759	$\rho(X_1\mid \mathcal F_1)\le \rho(X_2\mid \mathcal F_1)$
1762	1760	$\mathsf{E}(X|X\ge a)$
1763	1761	$ and $
1764	1762	$L_0^a$
1765	1763	$\rho(X)=\int_0^1 \mathsf{TVaR}_p(X)m(dp)$
1766	1764	$g(S(x))\to d$
1767	1765	$0.1525$
1768	1766	$l$
1769	1767	$U=X$
1770	1768	$\rho_m(X)=r$
1771	1769	$=1.75$
1772	1770	$\rho(X) = \max \{ \rho_\phi(X) \mid \phi\in A \}$
1773	1771	$\zeta\in\mathscr{P}$
1774	1772	$\rho$
1775	1773	$Z_i$
1776	1774	$x=q(p)$
1777	1775	$\rho(-1_{A^c}) = c < 0$
1778	1776	$\delta(p)=1-\nu(p)=d+(\delta^*-d)\sqrt{(1-p)/p}$
1779	1777	$\mathsf{E}[Z_1]=1$
1780	1778	$X_t=1_{[1,\infty)}$
1781	1779	$N\sim\text{Poisson}(1.74)$
1782	1780	$M(a)=\mathsf{E}(X\wedge a)+dN(a)+(\delta^*-d)\displaystyle\int_0^a \sqrt{F(x)S(x)}dx$
1783	1781	$c=\mathsf{VaR}$
1784	1782	$L^\infty(a, b)$
1785	1783	$dp$
1786	1784	$\tilde p=\tilde F(F^{-1}(p))=1-\tilde S(F^{-1}(p))=1-g(S(F^{-1}(p)))=1-g(1-F(F^{-1}(p)))=1-g(1-p)$
1787	1785	$D_i-N_i$
1788	1786	$1-t=g(1-s)$
1789	1787	$\dfrac{d}{dx}g(S(x))=-g'(S(x))f(x)$
1790	1788	$\mathsf{Pr}(Y\le a)=\exp(-\lambda (1-F(x)))=\exp(\lambda (\int_0^x f(s)ds -1))$
1791	1789	$0.06333 / 247.798 = 0.026\%$
1792	1790	$X_n$
1793	1791	$dx$
1794	1792	$_1$
1795	1793	$S_i$
1796	1794	$\mathsf{E}(X_i/X)$
1797	1795	$g(p)$
1798	1796	$g(s)=\displaystyle\frac{s}{1-p}\wedge 1$
1799	1797	$\mathsf{E}(X\wedge a)=\int_0^a S(x)dx$
1800	1798	$\mathscr P$
1801	1799	$})$
1802	1800	$\bar\delta=\bar\iota\bar\nu$
1803	1801	$g'(1-p)$
1804	1802	$k$
1805	1803	$J$
1806	1804	$\hat\rho(A)\ge \rho(A)$
1807	1805	$t=b$
1808	1806	$x=4, M=1.5,\sigma=0.75, K=6$
1809	1807	$\delta=\rho\nu$
1810	1808	$E(X_i \mid X=a)$
1811	1809	$c\ge \mathsf{E}[cg]$
1812	1810	$ϕ(1-t)=g'(t)$
1813	1811	$\rho:\mathcal{X}\to \mathbb{R}$
1814	1812	$q_{Z_k}$
1815	1813	$\rho=\rho_\gamma$
1816	1814	$T^{-1}$
1817	1815	$X(p)=q(p)$
1818	1816	$\\leftrightarrow$
1819	1817	$F(x_1)=1-S(x_1)=p$
1820	1818	$V(c)=0$
1821	1819	$\bar P_1$
1822	1820	$X_i$
1823	1821	$\mathsf{E}(X)=\sum_i x_i$
1824	1822	$a>1$
1825	1823	$(\delta^*-d)\sqrt{FS}$
1826	1824	$\mathsf{ABOVE}$
1827	1825	$C_i(t^*)=R_i(t^*)$
1828	1826	$T_n$
1829	1827	$\text{E}(G)=M_G'(0)=1$
1830	1828	$pl(p)$
1831	1829	$P(A)=1-\alpha$
1832	1830	$\mathsf{E}(L) = F^{-1}(p)dp$
1833	1831	$\rho(X) = \sup_{\mu\in \mathcal{A}} \langle \mu, X \rangle$
1834	1832	$\bar P(x+dx) - \bar P(x)$
1835	1833	$a=98,99,\dots,104$
1836	1834	$F^-1$
1837	1835	$E_\mathsf{Q}(X_i)= E_\mathsf{Q}(E(X_i \mid X))$
1838	1836	$\hat\rho_N$
1839	1837	$a,b=\pm 1/n$
1840	1838	$N\times r$
1841	1839	$U(x)$
1842	1840	$p=0.99$
1843	1841	$g(t) = \mathsf E[u(X-\pi(R+tQ) +R+tQ)]$
1844	1842	$\mathbf{X}=(X_1,\dots,X_n)$
1845	1843	$\rho_m(Y)$
1846	1844	$2\square^2 + 2\square - 1$
1847	1845	$\bar P^a(t)$
1848	1846	$q(\hat p)$
1849	1847	$g(0)=0,\ g(1)=1$
1850	1848	$\Leftrightarrow$
1851	1849	$\delta_p/\nu_p = \iota_p$
1852	1850	$100\cdot (1-g(s))$
1853	1851	$\delta=\iota/(1+\iota)$
1854	1852	$\bar X\ge 0$
1855	1853	$1-g(s)$
1856	1854	$X,Y$
1857	1855	$(g)$
1858	1856	$\mathscr{P} = \{P\}$
1859	1857	$\displaystyle\int_0^\infty xg'(S(x))f(x)dx$
1860	1858	$P'$
1861	1859	$\displaystyle\int_0^\infty xf(x)dx$
1862	1860	$Y\le 0$
1863	1861	$0\le\beta\le \gamma\le 1$
1864	1862	$\tilde S(x)=g(S(x))$
1865	1863	$\rho_{t+1}(X)\le\rho_{t+1}(Y)$
1866	1864	$N=365$
1867	1865	$b\le 1$
1868	1866	$g^a=g^{\log_g(n)}=n$
1869	1867	$(2,-\x*0.75)$
1870	1868	$r_X$
1871	1869	$\min_{\eta\in \mathbb{R}} \eta + \alpha \mathsf{E}(X-\eta)_+ -\beta\mathsf{E}(X-\eta)_-$
1872	1870	$\bar P_x$
1873	1871	$T_s(p) = \mathsf{TVaR}_p(s)$
1874	1872	$\bar A_{x:\lcroof{n}} = \bar A^{1}_{x:\lcroof{n}} + e^{-\delta n}{}_np_x$
1875	1873	$\partial \rho(X)=argmax_{\zeta\in A} \langle \zeta, X \rangle$
1876	1874	$=\mathsf{E}(X \mid X\le a)$
1877	1875	$p_i(a)=\phi_i(a)p(a)$
1878	1876	$\mathsf{E}(X_i(a))$
1879	1877	$Y$
1880	1878	$f_x(x_i, \hat x_i) = f(x_i, \hat x_i) / f_X(x)$
1881	1879	$\mathbf{x}=(1-t, t)$
1882	1880	$\mu\in \mathscr{P}$
1883	1881	$0 \le f'(z) \le 1$
1884	1882	$p=0. $
1885	1883	$\bar Z = F(\bar x)$
1886	1884	$[0,1]\to [0,1]\times [0,1]$
1887	1885	$2.592 \times 10^{16}$
1888	1886	$u_i$
1889	1887	$\zeta_t$
1890	1888	$\rho = AVaR$
1891	1889	$X(u)=X_1(u_1) + X_2(u_2)$
1892	1890	$E2$
1893	1891	$g'(0)$
1894	1892	$ at $
1895	1893	$1/(1+r_f)$
1896	1894	$\le a$
1897	1895	$f(x)dx = dp$
1898	1896	$\mathsf{E}(X)=$
1899	1897	$X_3$
1900	1898	$g'(S(x))$
1901	1899	$(Alice) + (0,-3.75)$
1902	1900	$x=q(1-g^{-1}(1-\tilde p))$
1903	1901	$d=iv=i/(1+i)$
1904	1902	$m =$
1905	1903	$\tau_\sigma(\alpha) = \int_\alpha^1 \sigma$
1906	1904	$\rho(-1_{B_l}) \le \rho(-1_{B_r})$
1907	1905	$(g(s)-s)/(1-g(s))$
1908	1906	$p\in [0,1]$
1909	1907	$\rho_{(g)}$
1910	1908	$X^{\oplus 2}$
1911	1909	$(\Omega, \mathcal{F}, \mathsf{P})$
1912	1910	$[l_i, r_i)$
1913	1911	$(1-X)^+$
1914	1912	$A=\sum_i I_iX_i$
1915	1913	$\sup\{ \mathsf{E}[Y\sigma(U)] \mid U\text{\ uniform} \}$
1916	1914	$X>F_u^{-1}(p)$
1917	1915	$R_2(t)= \bar P^a_2(t)/t$
1918	1916	$d\tilde p/dp = g'(1-p)=\tilde f(F^{-1}(p))/f(F^{-1}(p))$
1919	1917	$\rho(X) = \mathsf{E}(X) + c\mathsf{E}( |X-\mathsf{E}(X)|^p)^{1/p}$
1920	1918	$30,000 per accident up to $
1921	1919	$\sigma(X)$
1922	1920	$A^c\supset A_1\supset A_2\supset \dots$
1923	1921	$C > cx/a$
1924	1922	$\omega < 1/n$
1925	1923	$\phi_W(a)=\mathsf{E}(W/Y \mid Y>a)$
1926	1924	$\mathsf{E}(X_i/X \mid X > a)$
1927	1925	$q_L(\tau_\sigma^{-1}(U)$
1928	1926	$4.7\times 10^{21} / 10^{19} \approx 8\text{mins}$
1929	1927	$\mathsf{E}(\min(X_i,a))=\mathsf{E}(X_i\wedge a)$
1930	1928	$v=1/(1+i)$
1931	1929	$\tau_\sigma(p)=\int_0^p\sigma(u)du$
1932	1930	$50 of the amount allowed on each claim in the classes under paragraphs II, V, and VI except claims of the guaranty associations as defined in RSA 404-B, 404-H, 404-D, and 408-B shall be deducted from the claim. Claims may not be cumulated by assignment to avoid application of the $
1933	1931	$(p-\nu)/\nu$
1934	1932	$50.00) of the amount allowed on each property, casualty or fidelity claim in the classes under Subsections B through F of this section, shall be deducted from the claim and included in the class under Subsection I of this section. Claims may not be cumulated by assignment to avoid application of the fifty dollar ($
1935	1933	$r=0.038$
1936	1934	$X_1(x_1), \dots, X_n(x_n)$
1937	1935	$ into aggregate premiums $
1938	1936	$u_0,u_1,\dots,u_k$
1939	1937	$S(1-t,t;x)$
1940	1938	$\mathsf{E}[gY]\le 0$
1941	1939	$\mathsf{TVaR}_0(\cdot)=\mathsf{E}[\cdot]$
1942	1940	$\mathsf{E}(X-c_l)_+$
1943	1941	$P(a)=\mathsf{E}(Y\wedge a)+\rho K(a)$
1944	1942	$\iota(p)$
1945	1943	${}_b\bar V$
1946	1944	$X_i=q(p_i)$
1947	1945	$x_1<x_2<x_3<\dots$
1948	1946	$R = g^k \pmod{p}$
1949	1947	$Y=-\rho_{t+1}(X)$
1950	1948	$\tilde X$
1951	1949	$\tilde S(x):=g(S(x))=F(x)=e^{-\mu x/\rho}$
1952	1950	$G$
1953	1951	$1{X>q}$
1954	1952	$Z=d\mathbb{Q}/d\mathbb{P}$
1955	1953	$Z^* = \sum_i \alpha_i Z\circ T_i$
1956	1954	$X(t):=X(\mathbf{x})=(1-t)X_1 + tX_2$
1957	1955	$(ccc.south |- mcc.south)+(0,-0.5)$
1958	1956	$\sum t_i=\infty$
1959	1957	$(fun1a.south -| fun2a.east)+(\smlspc,-\smlspc)$
1960	1958	$1_D$
1961	1959	$\rho(X)=\mathsf{E}_\mathsf{Q}(X)$
1962	1960	$T_{700,100}$
1963	1961	$< 1$
1964	1962	$t=q-s$
1965	1963	$0$
1966	1964	$M_X(k)=M_Y(k)$
1967	1965	$\{ X=a \}$
1968	1966	$a = M(a)+Q(a)= \mathsf{E}(X\wedge a) + \delta N(a) + \nu N(a)$
1969	1967	$[p, p+dp]$
1970	1968	$(v-\nu^*)\sqrt{pq}=$
1971	1969	$(X−x^∗)I_{B_i}$
1972	1970	$r=g^k$
1973	1971	$n=g^a\pmod{p} \mapsto a=\log_g(a)$
1974	1972	$10^{13}$
1975	1973	$\gamma = 2/\sqrt(a) = 2\nu$
1976	1974	$\sigma=1$
1977	1975	$0\le \tau\le 1$
1978	1976	$(fun2.north west)+(-\smlspc,\smlspc)$
1979	1977	$\rho_g$
1980	1978	$\mathsf{E}(X) = E(X_i \mid X\le a)F(a) + E(X_i \mid X > a)S(a)$
1981	1979	$\alpha>1$
1982	1980	$b \in_{R} \{2,\dots,p-2\}$
1983	1981	$N\times 1$
1984	1982	$g$
1985	1983	$(Bob)+(0,-2.5)$
1986	1984	$\alpha=\text{E}[X \mid X > F_u^{-1}(p)]$
1987	1985	$(B.north east) + (-0.07mm,0)$
1988	1986	$\mathsf{E}[Y]$
1989	1987	$\mathsf{E}[X^k]=\mathsf{E}[Y^k]$
1990	1988	$\mathsf E[X]\rho(N) \le \rho(A)$
1991	1989	$\sigma$
1992	1990	$C_2$
1993	1991	$S(a)=1-p$
1994	1992	$\nu=\nu(F(a))=\nu(p)$
1995	1993	$\tau_\sigma(p)=\int_0^p \sigma$
1996	1994	$100\cdot g(s)$
1997	1995	$\phi(1)\le 1$
1998	1996	$\mathsf{E}(X)=0$
1999	1997	$\mathsf{E}(X_i\mid X=x)f_X(x)/x$
2000	1998	$\mathbf{T}^+\mathbf{r}$
2001	1999	$\mathsf{E}[Y\tilde W] = n^{-1}\sum_T \mathsf{E}[Y \cdot W\circ T] = n^{-1}\sum \mathsf{E}[Y\circ T^{-1} \cdot W] = \mathsf{E}[YW]$
2002	2000	$\mathsf{Q}_1$
2003	2001	$D_i$
2004	2002	$(Bob) + (0,-1)$
2005	2003	$-1\le X_n\le 0$
2006	2004	$[F(x)](\cdot)$
2007	2005	$g_k(s) = 1-(1-s)^k$
2008	2006	$10^{15}$
2009	2007	$P_i(a)=\phi_i(a)P(a)$
2010	2008	$F^{-1}(1-g^{-1}(1-p))$
2011	2009	$(Alice) + (0,-1)$
2012	2010	$\mathsf{Pr}(X>a)=S(a)$
2013	2011	$b\le a$
2014	2012	$\tilde\rho(X) = \mathsf{E}_Q(Y(\mathbf{X}))=\mathsf{E}_Q(Y)$
2015	2013	$\mathsf{E}_\mathbb{Q}(Z \mid X)=\mathsf{E}(Z \mid X)$
2016	2014	$\bar P_i(a)$
2017	2015	$L_\infty$
2018	2016	$k=1,\dots,K$
2019	2017	$\delta(p) F(x)=dF(x) + (\delta^*-d)\sqrt{FS}$
2020	2018	$k<\sup X$
2021	2019	$t=0,1$
2022	2020	$M_i\not=C_i$
2023	2021	$S(x)\to 0$
2024	2022	$\mathsf{E}[X_2 Z_1] = \mathsf{E}[X_2]\mathsf{E}[Z_1] =\mathsf{E}[X_2]$
2025	2023	$P(a)= S(a) + \bar\delta F(a)$
2026	2024	$\rho(L) = F^{-1}(p)g'(1-p)dp$
2027	2025	$\rho_t(X)=\rho_t(-\rho_{t+1}(X))$
2028	2026	$\bar p=1$
2029	2027	$\mathsf{E}(L_\sigma)= \int_0^1 q_L(s)\sigma(s)ds =:\pi_\sigma(L)$
2030	2028	$\rho(A)\le\rho(A_0) +\mathsf E[X]\rho(N)$
2031	2029	$\ge\mathsf{VaR}_p$
2032	2030	$T_x$
2033	2031	$\mathbf{m}=(m_j)$
2034	2032	$0\lt p \lt 1$
2035	2033	$B^2$
2036	2034	$\mathsf{E}(Q|X\ge a)$
2037	2035	$X=0$
2038	2036	$e^* \in E^*$
2039	2037	$-Y\ge 0$
2040	2038	$F^{-1}(U)$
2041	2039	$\kappa_i(x)$
2042	2040	$C<B<A$
2043	2041	$e =$
2044	2042	$\rho^*(\mu) = \sup_{X\in\mathcal{X}} \{ \langle \mu,X \rangle - \rho(X) \}$
2045	2043	$g(s)=\displaystyle\int_{1-s}^1 \phi(p)dp = \displaystyle\int_0^s \phi(1-p)dp = \min(s/(1-p), 1)$
2046	2044	$p=F(a)$
2047	2045	$5 \times 10^{10}$
2048	2046	$l(p)=\nu(1-2\sqrt{p(1-p)}$
2049	2047	$t=t_1$
2050	2048	$i=0.02, 0.04$
2051	2049	$\int \zeta dP=1$
2052	2050	$\mapsto$
2053	2051	$\rho_g(\cdot)$
2054	2052	$\Longrightarrow$
2055	2053	$p+q=1$
2056	2054	$\mathsf{Q}\in \mathscr{P}$
2057	2055	$\{G = q_j\}$
2058	2056	$\rho(A_k)$
2059	2057	$\zeta\in\partial \rho(X)$
2060	2058	$G\le c(x)$
2061	2059	$1-g(S(a))$
2062	2060	$0=p_0<p_1<\cdots<p_n=1$
2063	2061	$2/\sqrt{a}= 2\nu/(1-f)$
2064	2062	$N=\sum_{i\in I} N_i + N_a$
2065	2063	$a=0.02, b=1.310$
2066	2064	$(fun2a.south west)+(0,-2*\spcer)$
2067	2065	$(fun2a.south -| fun4a.east)+(\spcer, -\spcer)$
2068	2066	$P = \mathsf{E}(X) + \iota K$
2069	2067	$b>0$
2070	2068	$\mathsf E[Q\mid \mathcal F_1]$
2071	2069	$n=1,2,3,\dots$
2072	2070	$p(a) = S(a) + \rho k(a)$
2073	2071	$n\ge 1$
2074	2072	$\rho(X) = \sup_{\zeta\in\mathcal{A}} \langle \zeta,X \rangle$
2075	2073	$O(mn\times n\log(n))$
2076	2074	$x\mapsto (f(x), g(x))$
2077	2075	$\sigma=2.70$
2078	2076	$w$
2079	2077	$\Phi$
2080	2078	$a<a(f)$
2081	2079	$X\wedge a=\max(X,a)$
2082	2080	$(C.north east)+(1.3, 0)$
2083	2081	$\mathsf{E}(W \mid X\ge a)$
2084	2082	$X>q(\alpha)$
2085	2083	$1-\tilde p=g(1-p)=g(S(x))$
2086	2084	$0\le a-L_0^a\le a$
2087	2085	$F^{\times}_{359}$
2088	2086	$S\not=xf$
2089	2087	$q(\hat p)=q(1-g^{-1}(1-p))$
2090	2088	$a \le b$
2091	2089	$\sum_i \phi_i(a) = 1$
2092	2090	$N=N(\bar x)$
2093	2091	$C_{2,\cdot}$
2094	2092	$T_0$
2095	2093	$r_0$
2096	2094	$1,2,\dots, m$
2097	2095	$dQ/dP$
2098	2096	$n \ll p$
2099	2097	$1-2c\mathsf{Pr}(Z>\mathsf{E} Z)$
2100	2098	$1 \times 10^{16}$
2101	2099	$f_X$
2102	2100	$(\mathsf{E}(X_i)-\mathsf{E}(X\wedge a))/\mathsf{E}(X_i)$
2103	2101	$p(a)$
2104	2102	$c=$
2105	2103	$dv$
2106	2104	$\mu_{t+1}=\mu_t$
2107	2105	$\epsilon$
2108	2106	$X'$
2109	2107	$\rho(A_k) \le \rho(A_0) + k\rho(N)$
2110	2108	$Y_n$
2111	2109	$\delta(s)=g(s)g(k/s)-g(k)$
2112	2110	$Y>a$
2113	2111	$R(x)$
2114	2112	$X_u=X=u_1X_1 + u_2X_2$
2115	2113	$=\int_0^c S(x)dx = \int_0^c xf(x)dx + cS(c)$
2116	2114	$Q \sim P$
2117	2115	$=L/(1+r)$
2118	2116	$[x,x+dx)$
2119	2117	$, $
2120	2118	$f(x)<\infty$
2121	2119	$1-S(a)=F(a)$
2122	2120	$=\dfrac{g(s)-s}{1-s}$
2123	2121	$\langle \zeta_{\bar x}, X_i \rangle$
2124	2122	$i$
2125	2123	$\lambda S(a)$
2126	2124	$a \ge a'$
2127	2125	$g'(S(X))$
2128	2126	$\bar P = \bar S + \bar R$
2129	2127	$a<1$
2130	2128	$p+dp$
2131	2129	$L_1$
2132	2130	$1-\hat p=g^{-1}(1-p)$
2133	2131	$\mathsf{E}_q(X_2)$
2134	2132	$\mathsf{E}(X_i(a)) = E(X_i \mid X\le a)F(a) + aE(X_i/X \mid X> a)S(a)$
2135	2133	$\eta_{p,\alpha_1}(X) < \eta_{p,\alpha_2}(X)$
2136	2134	$μ = t ν$
2137	2135	$1-S(x)=F(x)$
2138	2136	$\delta=1-\nu=\rho\nu$
2139	2137	$\bar A_{x:\lcroof{n}}$
2140	2138	$\mathscr{O}(\zeta)$
2141	2139	$X=\mathsf E[Y\mid X]$
2142	2140	$\rho_{t+1}(-\rho_{t+1}(X))=\rho_{t+1}(X)$
2143	2141	$\rho(n^{-1}\sum X\circ T) = n^{-1}\sum \rho(X\circ T)$
2144	2142	$=\int_0^\infty xf(x)dx = \int_0^\infty S(x)dx = \int_0^1 q(p)dp$
2145	2143	$\notiff$
2146	2144	$\hat\rho$
2147	2145	$\lambda=0.045$
2148	2146	$[x, x+dx)$
2149	2147	$C$
2150	2148	$\mathsf{E}(B)=p$
2151	2149	$O(mn\log(n))$
2152	2150	$\mathcal F^{NS}$
2153	2151	$P(\alpha(X))$
2154	2152	$F(a)/\nu F(a)=1/\nu=1+\rho$
2155	2153	$da$
2156	2154	$(\partial P_i / \partial x_i)dx_i$
2157	2155	$\tilde p=g(p)$
2158	2156	$\min(X,a)=X \wedge a$
2159	2157	$1=S(x)+F(x)$
2160	2158	$(valu2.south east)+(\boundpad,-\boundpad)$
2161	2159	$\theta > 1$
2162	2160	$[0,1]\to[0,1]$
2163	2161	$\lambda_t=\lambda \mu_t$
2164	2162	$\ge 5$
2165	2163	$A = \{ \zeta \mid \|\zeta\|_q\le c, \zeta\ge 0 \}$
2166	2164	$U(X)<U(X1_{A^c} + \mathsf E[X\mid A]1-A)$
2167	2165	$X ∈ L^p$
2168	2166	$\bar S'(x)=S(x)$
2169	2167	$[0,1)$
2170	2168	$\mathbf{x}$
2171	2169	$2 \times 10^{19}$
2172	2170	$j$
2173	2171	$1 layer covering losses at or above the $
2174	2172	$Z\not=0$
2175	2173	$g(st)= \displaystyle\frac{st}{1-p} \le \displaystyle\frac{s}{1-p}\displaystyle\frac{t}{1-p}=g(s)g(t)$
2176	2174	$a-Y$
2177	2175	$(A.south east) + (-0.07mm,0)$
2178	2176	$\mathsf{E}(X\mid X > a) = (\mathsf{E}(X)-\mathsf{E}(X\mid X \le a)F(a))/S(a)$
2179	2177	$D(x)$
2180	2178	$\mathsf{E}(X_i(a)) = \mathsf{E}(X_i \mid X \le a)F(a) + a\mathsf{E}(X_i/X \mid X > a)S(a)$
2181	2179	$nG$
2182	2180	$y\ge x$
2183	2181	$d=iv$
2184	2182	$\mathsf E[T_s T_t] \ge \mathsf E[T_s] \mathsf E[T_t]=g(s)g(t)$
2185	2183	$\rho(X) = \mathsf{E}[gX]$
2186	2184	$(Bob)+(0,-3.5)$
2187	2185	$\mathsf{Pr}(X=\mathsf{E}(X))=0$
2188	2186	$u\mapsto \mathsf{E}[X_i/u\mid X(t)=u]$
2189	2187	$X_2$
2190	2188	$\displaystyle\int g(S_X) = \sup\{ E_Q(X) \mid Q(A)\le g(P(A)), \forall A\in \mathcal{F}) \}$
2191	2189	$(LL^t)^{-1}L^t$
2192	2190	$g\leftrightarrow \rho$
2193	2191	$g(s)$
2194	2192	$a=P+Q$
2195	2193	$n=2$
2196	2194	$Z=d\mathsf{Q}/d\mathsf{P}$
2197	2195	$n=3$
2198	2196	$W$
2199	2197	$g(t)=O(d)$
2200	2198	$\sqrt{F(x)S(x)}\approx \sqrt{S(x)}$
2201	2199	$\ge 50,000$
2202	2200	$g=3$
2203	2201	$10^{19}$
2204	2202	$L_\infty\subset L_p \subset L_\sigma\subset L_1$
2205	2203	$^{**}$
2206	2204	$s\mapsto g(s)$
2207	2205	$X=\sum_{i=1}^n X_i$
2208	2206	$\tilde F^{-1}(\tilde p)=F^{-1}(p)$
2209	2207	$[p, d+dp]$
2210	2208	$\rho_g(X)=\int xg'(S(x))f(x)dx$
2211	2209	$\mathsf{E}_q(X_1)/q$
2212	2210	$\delta(s,t)\ge 0$
2213	2211	$\delta F(x)$
2214	2212	$\lambda=0.045, 0.0625, 0.085, 0.125,$
2215	2213	$\bar \zeta$
2216	2214	$\Delta p\times T$
2217	2215	$1+2c(Z-\tau)$
2218	2216	$s_l$
2219	2217	$\mathbf{T}^+$
2220	2218	$\alpha\in [0,1]$
2221	2219	$\mathsf{E}_\mathsf{Q}[Y \mid X] = \mathsf{E}[Y \mid X]$
2222	2220	$\epsilon\mathsf{E}_q(X_1)$
2223	2221	$0\le (-X_n) \le 1$
2224	2222	$\rho(X+tY)-\rho(X) = \langle \zeta_t, X+tY \rangle -\rho(X) \le \langle \zeta_t, X+tY \rangle - \langle \zeta_t, X \rangle = \langle \zeta_t, tY \rangle$
2225	2223	$=18\times 4 = 72$
2226	2224	$1/N$
2227	2225	$(\delta^*-d)\int_0^a \sqrt{F(x)S(x)}dx$
2228	2226	$t_1$
2229	2227	$\rho(1_A)=1$
2230	2228	$g(s)=s$
2231	2229	$(x+b)$
2232	2230	$\mathsf{E}(U(Z))=\mathsf{E}(U(Z) \mid A) = \mathsf{E}(U(X))p + \mathsf{E}(U(Y))(1-p)$
2233	2231	$\phi_i(a)=\mathsf{E}(X_i/Y \mid Y>a)$
2234	2232	$s\le 1-p < t$
2235	2233	$B(b)$
2236	2234	$\rho(X_n)=1$
2237	2235	$\rho(X)=\max_{Q\in\mathsf{Q}} \mathsf{E}_Q(X)$
2238	2236	$x\in\mathbb{R}^n$
2239	2237	$1 \times 10^{23}$
2240	2238	$N=4$
2241	2239	$H(X) > -H(-Y)$
2242	2240	$1=\nu+\delta$
2243	2241	$t=0.55$
2244	2242	$t = 0$
2245	2243	$=E(X_i \mid X=a)$
2246	2244	$\Delta p$
2247	2245	$p+q=1=\nu+\delta$
2248	2246	$(\delta^*-d)\sqrt{pq}=$
2249	2247	$X_i, Y$
2250	2248	$A=X+Y$
2251	2249	$\rho(X)<\infty$
2252	2250	$b^2 \mu_x /2$
2253	2251	$\displaystyle\int_0^\infty S(x)dx$
2254	2252	$\Phi_i(y)=\mathsf{E}(X_i \mid Y = y)$
2255	2253	$-g''(t)=α(α-1)t^{α-2}$
2256	2254	$g^mA^R=g^m(g^a)^R=g^{m+Ra}$
2257	2255	$l_p>0$
2258	2256	$\sigma(X_1)$
2259	2257	$\mathsf{E}_\mathsf{Q}(Y \mid X) = \mathsf{E}(Y \mid X)$
2260	2258	$B$
2261	2259	$f=1$
2262	2260	$p(1-p)/p^2(\nu_p-l_p)^2$
2263	2261	$\rho(X)=r$
2264	2262	$Z(t\mathbf{X})=tZ(\mathbf{X})$
2265	2263	$\mu_x = -d\log(\tpx)/dt = \lim_{t\downarrow 0} {}_tq_x/t$
2266	2264	$g(s)=s^{1/\rho}$
2267	2265	$(k+1)\times n$
2268	2266	$f_{\hat i}$
2269	2267	$2^{20}$
2270	2268	$\beta_i/\alpha_i$
2271	2269	$EL$
2272	2270	$B_i$
2273	2271	$\phi F$
2274	2272	$du$
2275	2273	$a,0\le a\le\infty$
2276	2274	$g^{ak}$
2277	2275	$[\alpha_\epsilon,1]$
2278	2276	$t\ge 0$
2279	2277	$\mathsf{E}[g]\ge 1$
2280	2278	$p=1-g^{-1}(1-p)$
2281	2279	$[t-dt, t]$
2282	2280	$s_l < s < s_u$
2283	2281	$x = F^{-1}(1-g^{-1}(1-\tilde p))$
2284	2282	$p-\nu-il$
2285	2283	$50 of the amount allowed on each claim in the classes under subsections 2 to 6 shall be deducted from the claim and included in the class under subsection 8. Claims shall not be cumulated by assignment to avoid application on the $
2286	2284	$(s_{i}, g(s_{i}))$
2287	2285	$d\nu=d\mu/\alpha$
2288	2286	$\text{E}[X_i \mid X]$
2289	2287	$x\mapsto 1/x$
2290	2288	$\mathcal P$
2291	2289	$[a,a+1)$
2292	2290	$\not =$
2293	2291	$a_{i-1} < a_i < a_{i+1}$
2294	2292	$\rho_g(X)=$
2295	2293	$V(X)>0$
2296	2294	$\rho(X)=\int_0^\infty g(S(x))dx$
2297	2295	${}^1S=S$
2298	2296	$^{2}$
2299	2297	$i=0.02$
2300	2298	$q(\epsilon)/(1+\epsilon)\approx (q+\epsilon\mathsf{E}_q(X_1) )(1-\epsilon)=q-\epsilon(q-\mathsf{E}_q(X_1))=q-\epsilon E_q(X_2)$
2301	2299	$Z_k \succeq_2 (Z_k\mid N)$
2302	2300	$\mathsf E[(a-X)^+]$
2303	2301	$= \rho(B(s_l)) (1 -g(s)) + \rho(B(s_u)) g(s)$
2304	2302	ho=0.5, x=3, M=1.5,\sigma=0.85, K=8$
2305	2303	$X(t)=X(\mathbf{x})=(1-t)X_1 + tX_2$
2306	2304	$a=\infty$
2307	2305	$H(x)\not=H(y)$
2308	2306	$\mathbf{x}=(x_1,x_2)$
2309	2307	$Q(x) = \nu(F(x))F(x)$
2310	2308	$\mathcal{Z}=\{ Z\in L^\infty\mid \mathsf{E}[Z]=0, \mathsf{E}[Z^2]\le 1 \}$
2311	2309	$B=2.7\times 10^{-6}$
2312	2310	$(lee.east |- lee.north)+(0.25,0.25)$
2313	2311	$Y=g(X)$
2314	2312	$p\delta(p)/p\nu(p)=\iota(p)$
2315	2313	$M(a)=\mathsf{E}(X\wedge a)+d_iN(a)+(v-\nu^*)\displaystyle\int_0^a \sqrt{F(x)S(x)}dx$
2316	2314	$\bar\delta(a)$
2317	2315	$\displaystyle\int_0^1 \mathsf{AVaR}(p)\mu(dp) = \displaystyle\int_0^1 \dfrac{1}{1-p}\displaystyle\int_{p}^1 q(s)ds \mu(dp) =\displaystyle\int_0^1\displaystyle\int_0^s \dfrac{\mu(dp)}{1-p}q(s)ds=\displaystyle\int_0^1\displaystyle\int_{1-s}^1 \dfrac{\mu(dp)}{p}q(s)ds=\displaystyle\int_0^1\phi(s)q(s)ds$
2318	2316	$ϕ(s) = α^{-1}1_{[1-α, 1)}(s)$
2319	2317	$\rho_m(X)$
2320	2318	$X-b\le 0$
2321	2319	$\theta$
2322	2320	$(\s,4.5-\s)$
2323	2321	$\rho(2X)= \rho(X+X)=\rho(X)+\rho(X)=2\rho(X)$
2324	2322	$g(S)\not=q\phi$
2325	2323	$[0, t_1]$
2326	2324	$t\to 0$
2327	2325	$g'(t)dt < dt$
2328	2326	$R,S$
2329	2327	$X\circ T$
2330	2328	$s = f/n$
2331	2329	$h_0$
2332	2330	$X=a$
2333	2331	$p=F(x)$
2334	2332	$r\times 1$
2335	2333	$D-N = \sum_{i\in I} (D_i-N_i) - N_a$
2336	2334	$\bar\delta,\bar\nu$
2337	2335	$0.0625$
2338	2336	$\mathsf{TVaR}_p=\dfrac{1}{1-p}\displaystyle\int_p^1 F^{-1}(p)dp$
2339	2337	$\ll$
2340	2338	$s>0$
2341	2339	$E_i$
2342	2340	$O(\delta^2)$
2343	2341	$(a,b)$
2344	2342	$n=\square^\square$
2345	2343	$m(x)=S(x)+\delta(p)F(x)=S(x)+dF(x)+(\delta^*-d)\sqrt{F(x)S(x)}$
2346	2344	$\zeta\in\mathscr{O}(\eta)$
2347	2345	$f(x,y)=q_\alpha(x) - G(x,y)$
2348	2346	$f(X)$
2349	2347	$\rho(X)\le \liminf \rho(X_n)$
2350	2348	$\pi(X)=\int_a^{\alpha(X)} g(S(t))dt$
2351	2349	$X$
2352	2350	$\rho(X) = \mathsf{E}(X) + c\| X-\mathsf{E}(X) \|_p$
2353	2351	$2\square^2 + 2\square$
2354	2352	$[0.2, 0.85]$
2355	2353	$v_i = a_i/a$
2356	2354	$a+da$
2357	2355	$Q=(P+P')/2$
2358	2356	$μ = w_1 δ_{α_1} + w_2 δ_{α_2}$
2359	2357	$G_0$
2360	2358	$(\bar P_{x+b} - \bar P_x)\bar a_{x+b}=\bar A_{x+b}-\bar P_x \bar a_{x+b}=: {}_b\bar V$
2361	2359	$L(a)=\mathsf{E}(X\wedge a)$
2362	2360	$Y_i=\partial Y/\partial x_i$
2363	2361	$\alpha=1$
2364	2362	$B(b)\approx -b\mu_xv^b \approx {-}_bq_xv^b = -A^{\, 1}_{x:\lcroof{b}}$
2365	2363	$X\circ\tau$
2366	2364	$(X^∗_1, \dots, X^∗_n)$
2367	2365	$\phi_{\bar x}$
2368	2366	$dF(X)$
2369	2367	$-1_{B_r}$
2370	2368	$p(1-\nu(p))=p\delta(p)$
2371	2369	$g(s)=a^\alpha$
2372	2370	$u=(u_1, u_2)$
2373	2371	$\lambda^Q_t = \lambda^Q\mu_t$
2374	2372	$\rho(1)=1$
2375	2373	$u''<0$
2376	2374	$X(\mathbf{x})$
2377	2375	$\langle X_i, \zeta \rangle$
2378	2376	$\mathsf{E}(X \mid X\le a)$
2379	2377	$D_\lambda$
2380	2378	$g(0)=r_0$
2381	2379	$p(1-p)/(\nu-l)^2=0.5(1-0.5)=0.25$
2382	2380	$i=1,\dots,n$
2383	2381	$\displaystyle\int_0^\infty xf(x)dx = \displaystyle\int_0^\infty S(x)dx$
2384	2382	$\epsilon\to 0$
2385	2383	$\bar p$
2386	2384	$A^k=(g^a)^k$
2387	2385	$g:[0,1]\to [0,1]$
2388	2386	$16\times 4=64$
2389	2387	$g(s)=d+vs$
2390	2388	$\omega\mapsto q(\omega)=F^{-1}(\omega)$
2391	2389	$\mathbf{x}=\mathbf{1}$
2392	2390	$\nu^*$
2393	2391	$q(p)+y$
2394	2392	$\mathsf{E}(X)=\int_0^\infty xf(x)dx = \int_0^\infty S(x)dx$
2395	2393	$c_k-G=\gamma_k$
2396	2394	$Z_1=Z\circ T$
2397	2395	$p\not=0.5$
2398	2396	${}_tq_x=1-\tpx$
2399	2397	$L_2(\Omega)$
2400	2398	$n:=\nabla_yG/\|\nabla_y G\|$
2401	2399	$\{X = a\}$
2402	2400	$\phi(s)\ge 0$
2403	2401	$g(s) = \Phi(\Phi^{-1}(s)+\lambda)$
2404	2402	$\mathbf{T}$
2405	2403	$\partial\bar P/ \partial a$
2406	2404	$X\not\equiv 0$
2407	2405	$\mathsf{E}_\mathsf{Q}(X_i \mid X=x)=\mathsf{E}(X_i \mid X=x)$
2408	2406	$k\ge 0$
2409	2407	$a(\mathbf{x}) =\mathsf{VaR}_p(X(\mathbf{x}))= q_p(\mathbf{x})$
2410	2408	$u''(z+t)$
2411	2409	$\rho(X) = \mathsf{E}(X) + V(X)$
2412	2410	$F^{-1}(1-s)$
2413	2411	$\rho_i(X_i) - \rho_i(F_i)$
2414	2412	$\mathsf{E}_Q(Y)=\tilde \rho(X)$
2415	2413	$R_i<C_i$
2416	2414	$\delta_p=1-\nu_p$
2417	2415	$\mathsf{E}(X\wedge k)$
2418	2416	$\pm 1$
2419	2417	$\sigma=0.225$
2420	2418	$t\ge 0.5$
2421	2419	$(a,A)$
2422	2420	$t> t^*$
2423	2421	$X_i=x_i$
2424	2422	$(fun5.north west)+(-\smlspc,\smlspc)$
2425	2423	$\tpx \mu_{x+t}$
2426	2424	$(s_{i+1}, g(s_{i+1}))$
2427	2425	$r_P < r$
2428	2426	$T_B$
2429	2427	$X\in \mathcal A_{t,t+1} + \mathcal A_{t+1}\iff -\rho_{t+1}(X)\in\mathcal A_{t+1}$
2430	2428	$(1+\theta)\rho$
2431	2429	$\rho(A_k) \le \rho(A_0) + k \rho(N) \le \hat\rho(A_0) + k\rho(N)=\hat\rho(A_k)$
2432	2430	$EL=\mathsf E[X\wedge a]$
2433	2431	$dt=g'(1-s)ds=\phi(s)ds$
2434	2432	$A^∗_i$
2435	2433	$I=[0,1]$
2436	2434	$\bar R'(x)=R(x)$
2437	2435	$X=X(\mathbf{x})$
2438	2436	$Y_n=-X_n$
2439	2437	$2^{256}\approx 10^{77}$
2440	2438	$P$
2441	2439	$\mathsf{Pr}(X\le a)=F(a)$
2442	2440	$g(t)=1$
2443	2441	$Y=\max(X_1, \dots, X_N)$
2444	2442	$α$
2445	2443	$p=0$
2446	2444	$0\le\lambda \le 1$
2447	2445	$\phi(x)/x$
2448	2446	$=P + r(P+S)$
2449	2447	$\nabla g'$
2450	2448	$(f)$
2451	2449	$\iota^*=0.125$
2452	2450	$R_2 > C_2$
2453	2451	$\delta\bar a_{x:\lcroof{n}}+\bar A_{x:\lcroof{n}}=1$
2454	2452	$(g^k, Km)$
2455	2453	$p=100043$
2456	2454	$6 \times 10^7$
2457	2455	$[0,1]\to \mathbb{R}$
2458	2456	$b$
2459	2457	$2\square^2 + \square + 5$
2460	2458	$99<a\le 100$
2461	2459	$\mathcal F_1$
2462	2460	$A^c$
2463	2461	$\eta_{p,\alpha}$
2464	2462	$\bar \iota$
2465	2463	$\rho(G) = \mathsf{E}_Q(G)$
2466	2464	$=\mathsf{E}(X-c)_-=\int_0^c (c-x)f(x)dx$
2467	2465	$500,000 per claimant except that workers' compensation claims are paid in full; $
2468	2466	$s\in[k,1]$
2469	2467	$\eta$
2470	2468	$\Omega$
2471	2469	$=1$
2472	2470	$\sup_{\mu\in M} \int CTEd\mu$
2473	2471	$\liminf \rho(-k_i 1_{A_i}) \ge \rho(0)=0$
2474	2472	$E_Q(X_i(a)\mid X)$
2475	2473	$Z'$
2476	2474	$B_{\cdot}$
2477	2475	$\rho_t(X)\le \rho_t(Y)$
2478	2476	$B(s)$
2479	2477	$Q_0, Q_{i,\epsilon}$
2480	2478	$N(\bar x)=N(F(\bar x))$
2481	2479	$\int_0^s q_Z(1-t)dt\le g(s)$
2482	2480	$N(a)=\int_0^a F(x)dx=a-\mathsf{E}(X\wedge a)$
2483	2481	$\mathsf{Var}(\pi)$
2484	2482	$\mathsf{FAT}'$
2485	2483	$a\ge 1$
2486	2484	$\bar S(a)=\mathsf{E}(X\wedge a)$
2487	2485	$\mathsf E[Y_i\mid S]$
2488	2486	$da = p(a)da + (1-p(a))da$
2489	2487	$1 \times 10^{10}$
2490	2488	$(1+\epsilon)x_1$
2491	2489	$j=1,\dots r$
2492	2490	$\tilde Z_1:=\mathsf{E}[Z_1\mid X] = \tilde Z$
2493	2491	$X^{\oplus 1}$
2494	2492	$f_t$
2495	2493	$X=c$
2496	2494	$P(a)+K(a)=a$
2497	2495	$Z\circ T$
2498	2496	$\bar P(a)$
2499	2497	$\epsilon x_1$
2500	2498	$a-\bar P(a)$

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