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Mathematics Stack Exchange

Questions tagged [stochastic-integrals]

This tag is used for questions about stochastic integrals – especially for calculations. For questions related to more theoretical aspects of stochastic integrals such as their constructions, (stochastic-analysis) may be more appropriate.

A stochastic integral is an integral of stochastic processes with respect to stochastic processes. This may include Ito's integrals, but also variants such as Stratonovich integrals.

2333 questions
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Expectation of geometric brownian motion

I was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t + \mu B_t)$$ I wanted to calculate the expectation…
Stijn
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ito vs Stratonovich

I need to sum up the advantages of ito and stratonovich. I often heard, that the Stratonovich integral lacks the important property of the Itō integral, which does not "look into the future". Can you explain me why Stratonovich looks into the…
user112260
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Applying Ito To Geometric Brownian Motion

I'm trying to understand the example problem on the Wikipedia page for Ito's Lemma and need it dumbed down a little bit. $$dS = S(\sigma dB + \mu dt)$$ $$ f(S) = log(S) $$ Given Ito's lemma, below: $$ df(t, X_t) = (\frac{\partial f}{\partial t} +…
Soo
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Trying to integrate a stochastic RV, $\int_0^t sZ_s \, ds$

I'm not taking an official class (actuarial exams), some fellow "students" created a question (forum discussion), considering the integral in title. This is my attempt at a solution with no real justifications \begin{align} \int_0^tsZ_s \, ds & =…
Brady Trainor
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stochastic differential equation

$X_t$ is a weak solution to the SDE with $dX_t = ( −\alpha X_t + \gamma)dt + \beta dB_t$ , $\forall t \geq 0$ $X_0 = x_0$. $\;\;\;\alpha$, $\beta$, and $\gamma$ constants, and $Bt$ is a Brownian motion. I need to find the PDE for the transition…
lisa
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In Itô's lemma, if you wish to take expectations, when can you ignore the stochastic integral term?

Fix $d,k \in \mathbb{N}$. Let $\,b\colon \mathbb{R}^d \to \mathbb{R}^d\,$ and $\,\sigma\colon \mathbb{R}^d \to \mathbb{R}^{d \times k}\,$ be locally Lipschitz functions such that the Itô SDE $$dX_t=b(X_t)dt+\sigma(X_t)dW_t $$ has global existence…
Julian Newman
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1 answer

Stochastic Integral

I've just learned about stochastic integral and only know how to evaluate $\int\limits^{t}_{0} W(s)\mathrm{d}W(s)$. Could anyone give me some instruction on how to evaluate the following integrals? $$\mathbb{Var}\left(\int_{1}^2 (W_t)^2\mathrm{d}W_t…
BVFanZ
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How to calculate this easy stochastic integral?

I have a relatively simple homework for stochastic calculus that I recently started to learn. I cannot seem to calculate the following integral: $$ \int_0^t s dW_s $$ In principle, it should be solved by guessing some primitive function $f(W_t,t)$,…
in_finiti
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Doleans-Dade exponential formula

How do I apply the Doleans-Dade exponential formula for the following Lévy stochastic differential equation: $$dZ_t=Z_t\left(\theta_1(t)dW_t^{(1)} +\theta_2(t)dW_t^{(2)}+\int_\mathbb R \theta(s,x)\mu(ds,st)-v(ds,dt)\right),$$ where $W_t^{(1)}$ and…
Vaolter
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Expectation of Ito integral, part 2, and Fubini theorem

I previously asked a question (Expectation of Ito integral). I have additional questions on the same subject. Let's say that we have an Ito process such as $$ X(t)=X(0) + \int_0^t a ds + \int_0^t b X(s) dW(s) $$ where a and b are constants and W(t)…
user154506
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1 answer

Expectation of Ito integral

The expectation of an Itô stochastic integral is zero $$ E[\int_0^t X(s)dB(s)\,]=0 $$ if $$ \int_0^t E[X^2(s)]ds\,<\infty $$ It is sometimes possible to check this condition directly if the Itô integrand is simple enough but how would you do it if…
user154506
3
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1 answer

stochastic integrals and inequalities (boundedness)

We have $X(t)=[X_1(t)\ X_2(t)\ X_3(t)\ \dots\ X_n(t)]$ and $Y(t)=[Y_1(t)\ Y_2(t)\ Y_3(t)\ \dots\ Y_n(t)]$ are two stochastic process such that: $$\sup E[Y_1^2] \leq K,$$ on $[t_0, T]$ with $K$ a positive integer. Is that true $$E\int^T_{t_0}…
fidel
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is integral with respect to finite variation process of finite variation?

is integral with respect to finite variation process of finite variation? $\int_{[0,t]}X_sdA_s$, where X is $\mathcal{B}\times F$-measurable. If no in general, under what conditions? Question 2: if I have integral w.r. to local martingale. What is…
c-walk
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What is the solution of this stochastic integration?

This is the integration $$\int_0^t {\tau\ dW(\tau)}$$ where $W(t)$ is Wiener Process. I've check using wolfram mathematica that the solution is $$\frac{t}{\sqrt{3}} W(t)$$ But, I completely don't know why. I'm new in SDE. The clue from the problem…
fahadh4ilyas
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Stochastic Integration of $B^2dB$?

From Oksendal's Stochastic Differential Equations I'm to prove that, for B Brownian motion and using the definition of an Ito integral, $$ \displaystyle{\int_0^t}B_s^2\, dB_s = \frac{1}{3}B_t^3 - \displaystyle{\int_0^t}B_s\,ds $$ However I keep…
GenericUsrnme
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