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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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Definite integral

Please I need help with the evaluation of this integral. I've tried with both mathematica and maple, but to no avail. Here is the integral: …
user8280
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Find the expectation of Ito integral and a random process

I need to find the expectation of this stochastic integral. $$E\left[W(t) \int_0^t e^{3W(s)} dW(s)\right]$$ Obviously I cannot put the Expectation inside the integral because it is stochastic. Also I cannot separate $W(t)$ with the integral.…
Shu Tiang wei
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Why is are the definitions of Expectation w.r.t. Probability $P$ and density $\mu$ equivalent?

The expectation of a random variable $X$ is defined by $$\operatorname{E}[X] = \int_\Omega X \, dP = \int_\Omega X(\omega)\,dP(\omega)$$ $X$ defines a probability measure $\mu_X(B) = P(X^{-1}(B))$ and that implies $$\operatorname{E}[X] =…
don-joe
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a question about covariation in stochastic integration

Let H, K be bounded previsibe process. M, N be two local martingales. How can I prove $d_t = H_tK_td_t$ $$ means the quadratic variation of M. Thanks
XXX11235
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Interpretation of the stochastic integral of simple functions

A simple function $h$ (or random staircase function) is defined as $$ h(t) = \sum_0^{n} \xi_i I_{(t_i,t_{i+1}]}(t) $$ where $0
Beginner
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Stochastic integral estimate

I'm trying to derive the estimate $$ E\left[\left|\int_{0}^{t}h_r\,dB_r\right|^4\right] \leq 3C^4t^2,$$ where $h_r$ is continuous, adapted (to the natural Brownian filtration up to time $t$) and bounded (with constant $C$). I almost got this…
KKK
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Profit Maximization

I have listed a homework problem below that I have been working on. How do I get the expected number sold/expected number unsold/expected number lost if I do not have the pdf for the demand? Am I supposed to derive a pdf from the information given…
user161367
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A jump process as an integrand in Itô integral with respect to an Itô process

So, $X_1(s)$ is a jump process, $X_2(s)$ is another jump process, $X_2^c(s)$ is the continuous part of $X_2(s)$. And $\int_0^tX_1(s-)dX_2^c(s) = \int_0^tX_1(s)dX_2^c(s)$, is it because the integration on the jumps of the integrand $X_1$ is just…
Yue Harriet Huang
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stochastic integral and brownian motion

I'm trying to solve a problem similar to Stochastic Integral. I have to evaluate $$ \mathbb{Var}\left(\int_{0}^t ((B_s)^2 + s) \mathrm{d}B_s \right)$$ I have split the problem in two parts: 1) $ \mathbb{Var}\left(\int_{0}^t (B_s)^2 \mathrm{d}B_s…
Brownians
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Questions around the establish of Ito integral

I got a some detailed questions on the Ito integral and hope someone can help. I'm reading Chap 3 of Oksendal's SDE book. There he establishes the Ito integral and the Ito isometry for simple processes which is fine with me. But then he introduced…
user90846
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Calculate the expectation of Stratonovich integral

how to Calculate the expectation of Stratonovich integral as follows? $ \mathbb{E}\left[ \left( \int_0^t{s\mathrm{d}B_{s}^{2}} \right) \right] \text{、}\mathbb{E} \left[ \left( \int_0^t{s\mathrm{d}B_{s}^{2}} \right) ^2 \right]…
Damiao Liu
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Inequality for stochastic integral of elementary process

I have to prove the following inequality: $$ \mathbb{E}\left[ \left( \int_0^t X_s \text{d}B_s \right)^4 \right] \leq 3c^4 t^2 $$ where $(X_t)_{t \geq 0}$ is an elementary process such that $|X_t| \leq c$ for all $t \geq 0$; $(B_t)_{t \geq 0}$ is a…
user515933
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Expectation of the integral of Wiener process?

Background Consider the Wiener process: \begin{equation} W(t)=\int_0^t d W=\int_0^t \xi\left(t^{\prime}\right) d t^{\prime} \end{equation} where $\xi(t)$ is white Gaussian noise: $\langle \xi(t)\rangle = 0$ and $\langle \xi(t)\xi(t')\rangle =…
Matt
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Law of two stochastic integrals

Suppose we are given two filtered probability spaces $(\Omega, \mathcal F, (\mathcal F_t)_t, \mathbb P)$ and $(\Omega', \mathcal F', (\mathcal F'_t)_t, \mathbb P')$. On $\Omega$ (respectively $\Omega'$) we are a given a standard Wiener process $W$…
user590533
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An elementary example of Ito's integral

Let $m(t,\omega)=\sum_{j \ge 0}B_{(j+1)2^{-n}}(\omega)I_{[j.2^{-n},(j+1)2^{-n})}(t)$ where $B(t)$ is the Brownian motion and $I_[.]$ is the standard indicator function, Can some body explain me why $$ E[\int_0^T m(t,\omega)dB_t(\omega)]=\sum_{j \ge…
Ron
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