List questions from stochastic-processes

243

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Why aren't $B_s$ and $B_t$ independent for the one-dimensional standard Wiener process/Brownian motion?

Published on 25 Mar 2026 - 4:42

51

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$(X_{z+t}-X_{z})_{t\geq 0}$ satisfies "Strong Markov Property" where $X$ is càdlag process and $z$ discrete stopping time.

Published on 21 Mar 2026 - 9:39

2.9k

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Arrival Time in Poisson Process

Published on 24 Mar 2026 - 23:41

93

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Differentiation of local time in spatial variable

Published on 22 Mar 2026 - 6:42

52

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Asymptotic property concerning the last renewal time of a positive-drift random walk

Published on 25 Mar 2026 - 4:43

185

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Stochastic process’ mean-square derivative $X'(t)\equiv 0$ implies $X(t) \equiv X$?

Published on 25 Mar 2026 - 13:54

502

Views

Random graph process

Published on 25 Mar 2026 - 6:16

1.1k

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Arithmetic Brownian Motion

Published on 22 Mar 2026 - 13:27

182

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Limit involving a symmetric random walk

Published on 25 Mar 2026 - 3:01

59

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Expectation of stationary and ergodic process

Published on 04 Apr 2018 - 2:26

281

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Markov chain, define $\mathbb{P}_x$ as "conditioning on $X_0$"

Published on 04 Apr 2018 - 4:06

113

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Showing convergence of joint random variables

Published on 04 Apr 2018 - 4:20

48

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$f(X_{T})=f(X_{T^{-}})$ a.s. for all $f \in C_0$ implies $X_{T}=X_{T^{-}}$ a.s.

Published on 22 Mar 2026 - 12:37

33

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$\sum E_{x} |V_k - V_{k-1}|I\{\tau_C \ge k\} = \sum E_x[|E[(V_k - V_{k-1})|\mathscr{F}_{k-1}]I\{\tau_C \ge k\}|]$ for a submartingale Markov chain.

Published on 26 Mar 2026 - 2:47

88

Views

How to show that the quadratic variation of $p$ is $t$?

Published on 25 Mar 2026 - 3:03

List Question

Why aren't $B_s$ and $B_t$ independent for the one-dimensional standard Wiener process/Brownian motion?

$(X_{z+t}-X_{z})_{t\geq 0}$ satisfies "Strong Markov Property" where $X$ is càdlag process and $z$ discrete stopping time.

Arrival Time in Poisson Process

Differentiation of local time in spatial variable

Asymptotic property concerning the last renewal time of a positive-drift random walk

Stochastic process’ mean-square derivative $X'(t)\equiv 0$ implies $X(t) \equiv X$?

Random graph process

Arithmetic Brownian Motion

Limit involving a symmetric random walk

Expectation of stationary and ergodic process

Markov chain, define $\mathbb{P}_x$ as "conditioning on $X_0$"

Showing convergence of joint random variables

$f(X_{T})=f(X_{T^{-}})$ a.s. for all $f \in C_0$ implies $X_{T}=X_{T^{-}}$ a.s.

$\sum E_{x} |V_k - V_{k-1}|I\{\tau_C \ge k\} = \sum E_x[|E[(V_k - V_{k-1})|\mathscr{F}_{k-1}]I\{\tau_C \ge k\}|]$ for a submartingale Markov chain.

How to show that the quadratic variation of $p$ is $t$?

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