List questions from stochastic-integrals

121

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How can we extend the Itō isometry for elementary processes to the class of square-integrable processes?

Published on 12 May 2017 - 18:00

67

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Solution to SDE $\mathrm{d} S(t) = \left( \beta S(t) + (1- \beta) S_0\right) \sigma \mathrm{d} W(t)$

Published on 29 Mar 2026 - 16:49

1.1k

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Density function of an (iid) stochastic process

Published on 15 May 2017 - 5:07

63

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If $τ_n$ are stopping times with $τ_n\uparrow T$ and $X_n$ are random variable, then $\sum_n1_{\left\{τ_{n-1}<t\right\}}X_n$ is convergent

Published on 02 Apr 2026 - 10:15

135

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Definition of elementary predictable processes

Published on 31 Mar 2026 - 5:39

38

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Show that $\sum_{n\in\mathbb N}X_n\left(W_t-W^{\tau_{n-1}}_t\right)$ is adapted whenever $W$ is adapted

Published on 31 Mar 2026 - 10:14

282

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$L^2$-bound for the stochastic integral

Published on 31 Mar 2026 - 5:39

1k

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Finding the stochastic differential equation satisfied by process Y

Published on 18 May 2017 - 13:41

402

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Application of Ito Isometry

Published on 18 May 2017 - 21:21

15.6k

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Stochastic Differential Equation solution for Geometric Brownian Motion

Published on 29 Mar 2026 - 18:30

131

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Show that the quadratic variation has a nondecreasing modification

Published on 23 Feb 2026 - 16:56

91

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Prove the equality $\mathbb{E}[(X_s^2-\mathbb{E}[X_s^2])(X_t^2-\mathbb{E}[X_t^2])]=2\mathbb{E}[X_sX_t]^2$

Published on 22 May 2017 - 22:01

135

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Stochastic integral under stopping

Published on 24 May 2017 - 14:40

154

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If the SDE of an Ito process is identically zero, are the drift and volatility both identically zero?

Published on 25 May 2017 - 14:14

5.8k

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Conversion between solution to Stratonovich SDE and Itô SDE

Published on 29 Mar 2026 - 18:30

List Question

How can we extend the Itō isometry for elementary processes to the class of square-integrable processes?

Solution to SDE $\mathrm{d} S(t) = \left( \beta S(t) + (1- \beta) S_0\right) \sigma \mathrm{d} W(t)$

Density function of an (iid) stochastic process

If $τ_n$ are stopping times with $τ_n\uparrow T$ and $X_n$ are random variable, then $\sum_n1_{\left\{τ_{n-1}<t\right\}}X_n$ is convergent

Definition of elementary predictable processes

Show that $\sum_{n\in\mathbb N}X_n\left(W_t-W^{\tau_{n-1}}_t\right)$ is adapted whenever $W$ is adapted

$L^2$-bound for the stochastic integral

Finding the stochastic differential equation satisfied by process Y

Application of Ito Isometry

Stochastic Differential Equation solution for Geometric Brownian Motion

Show that the quadratic variation has a nondecreasing modification

Prove the equality $\mathbb{E}[(X_s^2-\mathbb{E}[X_s^2])(X_t^2-\mathbb{E}[X_t^2])]=2\mathbb{E}[X_sX_t]^2$

Stochastic integral under stopping

If the SDE of an Ito process is identically zero, are the drift and volatility both identically zero?

Conversion between solution to Stratonovich SDE and Itô SDE

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