Question

$(X_{k})_{k }$ are IID, $P[X_{k}=1]=P[X_{k}=-1]=\frac{1}{2}$; $S_{n}=\sum_{k\leq n}X_{k}$ , $\tau=\inf\{n:S_{n}\in\{-a,b\}\}$. $E[\tau^{2}]=$?

score 46 · Answer 1 · 2023-12-13 12:53:53

46

Views

$(X_{k})_{k }$ are IID, $P[X_{k}=1]=P[X_{k}=-1]=\frac{1}{2}$; $S_{n}=\sum_{k\leq n}X_{k}$ , $\tau=\inf\{n:S_{n}\in\{-a,b\}\}$. $E[\tau^{2}]=$?

Published on 13 Dec 2023 - 12:53

score 71 · Answer 2 · 2026-03-26 12:36:38

71

Views

Karatzas & Shreve, Problem 2.22, Positivity of Stopping Time

Published on 26 Mar 2026 - 12:36

score 30 · Answer 3 · 2023-12-28 18:28:58

30

Views

Possible values for $\mathbb E[X_\tau]$ for supermartingales and submartingales.

Published on 28 Dec 2023 - 18:28

score 31 · Answer 4 · 2026-03-27 19:15:32

31

Views

Left tail bound for stopping time of a gaussian random walk

Published on 27 Mar 2026 - 19:15

score 63 · Answer 5 · 2026-03-27 21:34:36

63

Views

Expected value of stopped iid sequence.

Published on 27 Mar 2026 - 21:34

score 49 · Answer 6 · 2024-01-08 07:28:05

49

Views

Show that for $0<b<1<a$: $\mathbb P(\tau_a < \tau_b)=\frac{1-b^4}{a^4-b^4}$

Published on 08 Jan 2024 - 7:28

score 41 · Answer 7 · 2026-03-26 14:25:20

41

Views

Super-martingale at stopping time is a super-martingale

Published on 26 Mar 2026 - 14:25

score 143 · Answer 8 · 2024-01-11 11:15:27

143

Views

Calculate $\mathbb EX$, $VarX$ for $\tau=\inf \{t>0: \int_0^t e^{-(W_2^2+2W_s)}ds=2\}, X=\int_0^{\tau}e^{-(\frac{W_s^2}2+W_s)}dW_s$

Published on 11 Jan 2024 - 11:15

score 32 · Answer 9 · 2024-01-15 10:24:22

32

Views

Same Distribution of Stopping Time of Markov Chain Implied Same Spectral

Published on 15 Jan 2024 - 10:24

score 41 · Answer 10 · 2024-01-18 18:11:29

41

Views

Continuity of Optimal Stopping Time as a Function of the Starting Point

Published on 18 Jan 2024 - 18:11

score 38 · Answer 11 · 2024-01-26 02:29:46

38

Views

Proof Check: Expected flips of getting HT on a biased coin

Published on 26 Jan 2024 - 2:29

score 33 · Answer 12 · 2024-01-29 00:57:54

33

Views

Stopping time of a martingale $Y_n=e^{S_n}$

Published on 29 Jan 2024 - 0:57

score 53 · Answer 13 · 2024-02-01 19:56:03

53

Views

Stopping time and supremum of Brownian motion

Published on 01 Feb 2024 - 19:56

score 18 · Answer 14 · 2024-02-06 13:27:38

Is $\sigma_1 I_A + \sigma_2 I_{A^c}$ a stopping time for stopping times $\sigma_1,\sigma_2$ and $A\in\mathscr{F}_{n+1}$, $n\geq 0$?

score 72 · Answer 15 · 2026-03-26 14:19:49

72

Views

Definition of predictable stopping time

Published on 26 Mar 2026 - 14:19

List Question

$(X_{k})_{k }$ are IID, $P[X_{k}=1]=P[X_{k}=-1]=\frac{1}{2}$; $S_{n}=\sum_{k\leq n}X_{k}$ , $\tau=\inf\{n:S_{n}\in\{-a,b\}\}$. $E[\tau^{2}]=$?

Karatzas & Shreve, Problem 2.22, Positivity of Stopping Time

Possible values for $\mathbb E[X_\tau]$ for supermartingales and submartingales.

Left tail bound for stopping time of a gaussian random walk

Expected value of stopped iid sequence.

Show that for $0<b<1<a$: $\mathbb P(\tau_a < \tau_b)=\frac{1-b^4}{a^4-b^4}$

Super-martingale at stopping time is a super-martingale

Calculate $\mathbb EX$, $VarX$ for $\tau=\inf \{t>0: \int_0^t e^{-(W_2^2+2W_s)}ds=2\}, X=\int_0^{\tau}e^{-(\frac{W_s^2}2+W_s)}dW_s$

Same Distribution of Stopping Time of Markov Chain Implied Same Spectral

Continuity of Optimal Stopping Time as a Function of the Starting Point

Proof Check: Expected flips of getting HT on a biased coin

Stopping time of a martingale $Y_n=e^{S_n}$

Stopping time and supremum of Brownian motion

Is $\sigma_1 I_A + \sigma_2 I_{A^c}$ a stopping time for stopping times $\sigma_1,\sigma_2$ and $A\in\mathscr{F}_{n+1}$, $n\geq 0$?

Definition of predictable stopping time

Trending Questions

Popular # Hahtags

Popular Questions