Question

$E[X|\mathcal{F}_{\max_{1 \leq k\leq d}R_k}]=\sum_{k=1}^d(-1)^{k-1}\sum_{1 \leq i_1<...<i_k \leq d}E[X|\mathcal{F}_{\min_{ 1 \leq p \leq k} R_{i_p}}]$

score 95 · Answer 1 · 2021-03-28 00:47:43

95

Views

$E[X|\mathcal{F}_{\max_{1 \leq k\leq d}R_k}]=\sum_{k=1}^d(-1)^{k-1}\sum_{1 \leq i_1<...<i_k \leq d}E[X|\mathcal{F}_{\min_{ 1 \leq p \leq k} R_{i_p}}]$

Published on 28 Mar 2021 - 0:47

score 103 · Answer 2 · 2021-03-28 05:17:22

103

Views

Example of event not in the $\sigma$-algebra of a stopping time.

Published on 28 Mar 2021 - 5:17

score 112 · Answer 3 · 2021-03-29 03:25:30

112

Views

Non-negative martingale $X_n \rightarrow 0$ a.s. prove that $P[X^* \geq x | \mathcal{F}_0]= 1 \wedge X_0 / x$

Published on 29 Mar 2021 - 3:25

score 113 · Answer 4 · 2021-03-29 06:11:31

113

Views

Prove a martingale is bounded

Published on 29 Mar 2021 - 6:11

score 29 · Answer 5 · 2021-03-30 10:48:40

29

Views

If $\tau:=\{t\ge t_0:f(t)\in B\}$, can we show that $f(\min(t,\tau))\not\in B$ for all $t$?

Published on 30 Mar 2021 - 10:48

score 102 · Answer 6 · 2021-03-31 09:12:07

102

Views

How can we prove $\left.\mathcal F_{σ}\right|_{\{\:σ\:\le\:\tau\:\}}=\left.\mathcal F_{σ\:\wedge\:\tau}\right|_{\{\:σ\:\le\:\tau\:\}}$?

Published on 31 Mar 2021 - 9:12

score 192 · Answer 7 · 2021-03-31 15:12:55

192

Views

Can we show $\text E\left[X\mid\mathcal F_\sigma\right]=\text E\left[X\mid\mathcal F_{\sigma\:\wedge\:\tau}\right]$ on $\{\sigma\le\tau\}$?

Published on 31 Mar 2021 - 15:12

score 54 · Answer 8 · 2021-04-02 18:05:21

54

Views

Markov Chain Probability of Coin Throw results

Published on 02 Apr 2021 - 18:05

score 25 · Answer 9 · 2021-04-03 11:04:13

25

Views

show $\max$ and $\min$ are stopping times.

Published on 03 Apr 2021 - 11:04

score 147 · Answer 10 · 2021-04-03 19:17:43

147

Views

Dynamics of probability of hitting a barrier

Published on 03 Apr 2021 - 19:17

score 139 · Answer 11 · 2021-04-08 08:29:19

139

Views

If $\tau_n,\tau$ are stopping times with $\tau=\inf_n\tau_n$, then $\text E[X\mid\mathcal F_{\tau_n}]\to\text E[X\mid\mathcal F_\tau]$

Published on 08 Apr 2021 - 8:29

score 392 · Answer 12 · 2021-04-12 03:23:56

392

Views

Exponential decay of tail probability for Brownian stopping time

Published on 12 Apr 2021 - 3:23

score 181 · Answer 13 · 2026-03-31 01:08:46

181

Views

Existence of a weak solution to an SDE

Published on 31 Mar 2026 - 1:08

score 130 · Answer 14 · 2021-04-17 09:35:49

130

Views

Is $(B_{\tau_b + (t - \tau_b)} - B_{\tau_b})_{t \ge 0}$ independent of $\mathcal{F}_{\tau_b}$ for a Brownian motion $(B_t)_{t\ge 0}$?

Published on 17 Apr 2021 - 9:35

score 80 · Answer 15 · 2021-04-18 01:42:29

80

Views

Continuous stopping time property

Published on 18 Apr 2021 - 1:42

List Question

$E[X|\mathcal{F}_{\max_{1 \leq k\leq d}R_k}]=\sum_{k=1}^d(-1)^{k-1}\sum_{1 \leq i_1<...<i_k \leq d}E[X|\mathcal{F}_{\min_{ 1 \leq p \leq k} R_{i_p}}]$

Example of event not in the $\sigma$-algebra of a stopping time.

Non-negative martingale $X_n \rightarrow 0$ a.s. prove that $P[X^* \geq x | \mathcal{F}_0]= 1 \wedge X_0 / x$

Prove a martingale is bounded

If $\tau:=\{t\ge t_0:f(t)\in B\}$, can we show that $f(\min(t,\tau))\not\in B$ for all $t$?

How can we prove $\left.\mathcal F_{σ}\right|_{\{\:σ\:\le\:\tau\:\}}=\left.\mathcal F_{σ\:\wedge\:\tau}\right|_{\{\:σ\:\le\:\tau\:\}}$?

Can we show $\text E\left[X\mid\mathcal F_\sigma\right]=\text E\left[X\mid\mathcal F_{\sigma\:\wedge\:\tau}\right]$ on $\{\sigma\le\tau\}$?

Markov Chain Probability of Coin Throw results

show $\max$ and $\min$ are stopping times.

Dynamics of probability of hitting a barrier

If $\tau_n,\tau$ are stopping times with $\tau=\inf_n\tau_n$, then $\text E[X\mid\mathcal F_{\tau_n}]\to\text E[X\mid\mathcal F_\tau]$

Exponential decay of tail probability for Brownian stopping time

Existence of a weak solution to an SDE

Is $(B_{\tau_b + (t - \tau_b)} - B_{\tau_b})_{t \ge 0}$ independent of $\mathcal{F}_{\tau_b}$ for a Brownian motion $(B_t)_{t\ge 0}$?

Continuous stopping time property

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