Question

Show that $\mathbb{E}[X_t]=X_0e^{-ct}$ if $X_t=X_0e^{-ct}+\sigma e^{-ct}\int_0^te^{cs}dW_s$, $X_0\in\mathbb{R}$

score 69 · Answer 1 · 2026-04-04 18:22:24

69

Views

Show that $\mathbb{E}[X_t]=X_0e^{-ct}$ if $X_t=X_0e^{-ct}+\sigma e^{-ct}\int_0^te^{cs}dW_s$, $X_0\in\mathbb{R}$

Published on 04 Apr 2026 - 18:22

score 122 · Answer 2 · 2015-08-10 01:09:29

122

Views

Brownian motion exit time

Published on 10 Aug 2015 - 1:09

score 103 · Answer 3 · 2026-04-04 18:22:24

103

Views

Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$

Published on 04 Apr 2026 - 18:22

score 1.3k · Answer 4 · 2026-04-04 18:22:23

1.3k

Views

Show that $X_tY_t=X_0Y_0+\int_0^tX_sdYs+\int_0^tY_sdX_s+[X,Y](t)$, where $X_t,Y_t$ are Ito processes

Published on 04 Apr 2026 - 18:22

score 167 · Answer 5 · 2026-04-04 18:12:49

167

Views

Stopped Brownian motion proof

Published on 04 Apr 2026 - 18:12

score 2.1k · Answer 6 · 2015-08-11 00:28:05

2.1k

Views

Calculation with Ito processes, what is $ds \, dt$, $dW_t \, ds$ and $dW_s \, dW_t$?

Published on 11 Aug 2015 - 0:28

score 943 · Answer 7 · 2015-08-13 20:23:35

943

Views

What can you tell me about backward Brownian motion?

Published on 13 Aug 2015 - 20:23

score 159 · Answer 8 · 2026-03-29 07:34:42

159

Views

Independence of a hitting time and the underlying stochastic process

Published on 29 Mar 2026 - 7:34

score 53 · Answer 9 · 2026-03-29 07:22:26

53

Views

Solution to sde with specfic mean

Published on 29 Mar 2026 - 7:22

score 1.6k · Answer 10 · 2015-08-16 15:13:40

1.6k

Views

Reference book for Brownian Motion

Published on 16 Aug 2015 - 15:13

score 2.1k · Answer 11 · 2026-03-30 12:43:45

2.1k

Views

Show that Brownian motion on the unit circle is exponentially ergodic and has the uniform measure as its invariant distribution.

Published on 30 Mar 2026 - 12:43

score 408 · Answer 12 · 2015-08-19 08:14:35

408

Views

Can we have a Brownian motion under two different probability space?

Published on 19 Aug 2015 - 8:14

score 473 · Answer 13 · 2026-04-04 22:02:21

473

Views

Can someone, please, suggest some books for Stochastic Processes with exercises?

Published on 04 Apr 2026 - 22:02

score 946 · Answer 14 · 2026-04-05 00:24:33

946

Views

Expectation of exponential of integral of absolute value of Brownian motion

Published on 05 Apr 2026 - 0:24

score 1.3k · Answer 15 · 2015-08-20 22:06:32

1.3k

Views

What is the Skewness of a Geometric Brownian Motion?

Published on 20 Aug 2015 - 22:06

List Question

Show that $\mathbb{E}[X_t]=X_0e^{-ct}$ if $X_t=X_0e^{-ct}+\sigma e^{-ct}\int_0^te^{cs}dW_s$, $X_0\in\mathbb{R}$

Brownian motion exit time

Show that $\mathbb{E}\left(\int_0^1X_n(t)dW(t)-\int_0^1X(t)dW(t)\right)^2\to 0$

Show that $X_tY_t=X_0Y_0+\int_0^tX_sdYs+\int_0^tY_sdX_s+[X,Y](t)$, where $X_t,Y_t$ are Ito processes

Stopped Brownian motion proof

Calculation with Ito processes, what is $ds \, dt$, $dW_t \, ds$ and $dW_s \, dW_t$?

What can you tell me about backward Brownian motion?

Independence of a hitting time and the underlying stochastic process

Solution to sde with specfic mean

Reference book for Brownian Motion

Show that Brownian motion on the unit circle is exponentially ergodic and has the uniform measure as its invariant distribution.

Can we have a Brownian motion under two different probability space?

Can someone, please, suggest some books for Stochastic Processes with exercises?

Expectation of exponential of integral of absolute value of Brownian motion

What is the Skewness of a Geometric Brownian Motion?

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