# Tagged Questions

This tag is used for questions about stochastic integrals - especially for calculations . For questions related to more theoretic aspects of stochastic integrals such as its construction. Stochastic-analysis may be a more appropriate tag.

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### Applying Ito formula to Ito process

I would like to simplify the expression $\left(\phi(s_{1})\cdot(X_{s_{1}}-X_{s_{2}})+\phi(s_{2})\cdot(X_{s_{2}}-X_{s_{3}})+\ldots+\phi(s_{n-1})\cdot(X_{s_{n-1}}-X_{s_{n}})\right)^{2}$ where $X_{t}$ ...
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### Simulating a Stochastic Integral of OU process

The stochastic integral I want to simulate is $$\int_{0}^{1}J_c(s)dJ_c(s)$$ where $J_c(s) = \int_{0}^{s}e^{-c(s-r)}dB(r)$, is an OU process. I simulate the data using Matlab and the sample codes are ...
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### Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
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### How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
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Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that $$\limsup_{\... 2answers 82 views ### Integral of Wiener Squared process I don't have a background of stochastic calculus. It is known fact that definite integral of standard Wiener process from 0 to t results in another Gaussian process with slice distribution that ... 0answers 31 views ### Lebesgue-Stieltjes integral and related topics The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ... 0answers 26 views ### Integrating over random boundary What are some correct stochastic integral notions or theories which make formal sense of the problem of "integrating a function over the boundary of random domain"? 0answers 13 views ### Solving the following SDE with a constant Given is the stochastic differential equation: \frac{dX(t)}{X(t)}=\mu+\sigma \theta dt+ \sigma dW(t), where W(t) is the standard Wiener process and X(0)=x_0 I try to solve this by the Itos ... 0answers 43 views ### Some Kind of Generalized Brownian Bridge Let \displaystyle X(t) = \int_0^t f(s)dB(s) where B(t) is a Brownian motion and f(t)\in L^2[0,1]. What is a simple representation for Y(t):=(X(t)|X(1)) in terms of B(t)? Note, I am not ... 0answers 47 views ### Can Stochastic Integration be Further Generalized? Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ... 0answers 480 views ### Expectation of Exponential of Stochastic Integral Let z be the standard Brownian motion, \omega an element of the sample space. Is it true that$$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf E\bigg[\exp\Big(\...
I need to compute some Integrals for my stochastic course. And i have the following problem: $$\frac{\lambda^n}{\Gamma(n)} \int_0^{\infty} \exp(-\frac{\lambda}{y}) \frac{1}{y^n} dy = \star$$ so i ...
I have a big problem with such a task: Calculate $\text{Cov} \, (X_t,X_r)$ where $X_t=\int_0^ts^3W_s \, dW_s$, $t \ge 0$. I've tried to do this in this way: setting up $t \le r$ \text{Cov} \, (...