# Tagged Questions

Stochastic calculus provides a consistent theory of integration for stochastic processes and is used to model random systems. Its applications range from statistical physics to quantitative finance.

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### Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories. More concretely, I want to obtain a SDE of type as ...
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### About Ito integral of power of brownian motion

Using Ito's lemma, one can get the following expression for Ito integral of monomials: $\int_0^TW(t)^ndW(t) = \frac{1}{n+1}W(t)^{n+1} - \frac{n}{2}\int_0^TW(t)^{n-1}dt.$ What can we say about the ...
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### A difficult integral [closed]

Is there any analytical result on the following integral? $$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{1+e^{-(x-\mu)}}dx$$ Thanks a lot!
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### Calculate Moments of SDE

I have posted a similar question on math.stackexchange (http://math.stackexchange.com/questions/1848492/calculate-mean-of-sde), but didn't find anyone who could help. I'm interested in the one-...
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### Stochastic process with discontinuous drift

While studying a portfolio optimization problem, I came across the process $$dX(t) = X(t)\,\Big(\,\big(\mu - \alpha\,1_{\{X(t)\,\geq\,C\}}\big)\,dt + \sigma\,dW(t) \Big)$$ which has a discountinuous ...
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### Feynman-Kac for heat equation on a compact manifold with boundary

It is known that for any open $\Omega \subset \mathbb{R}^n$, given $f \in L^2(\Omega)$, $x \in \Omega$, one has $$e^{t\Delta}f(x) = \mathbb{E}_x(f(\omega(t))\psi_\Omega(\omega, t)),$$ where $\Delta$ ...
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### How to calculate the PSD of a stochastic process

This question was asked on math.stackexchange about 2 months ago, but it hasn't been very successful in attracting answers yet, so I'm posting it here. Say we have a stochastic process described by a ...
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### Are the elementary predictable processes dense in $L^2([M])$ for $M$ a local martingale?

The question is the one from the title. I know this is true when $M$ is an $L^2$ bounded martingale (which is often used in the classical approach to the construction of the stochastic integral) but I'...
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### Proof of Feynman Kac formula

I am trying to write a complete proof of the Feynman Kac formula in the multi-dimensional case. My starting point was the proof of the univariate form on wikipedia, at https://en.wikipedia.org/wiki/...
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### Stochastic calculus in $L^1$

Does there exist a more general (than Malliavin or Itô) "Stochastic calculus" defined on $L^1$ space, or some Orlicz space between $L^2$ and $L^1$? For examples: are there: Ito Isometry(-...
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### Brownian motion - probability of striking a sphere in $\mathbb{R}^n$ (a clarification)

This is primarily in reference to this question on MO. Serguei Popov's answer gives an explicit formula for the probability of a Brownian particle starting at the origin in $\mathbb{R}^n$ hitting the ...
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### Special random variables and monotone class theorem

I am currently reading a proof where the $\pi-\lambda$ Lemma and the monotone class theorem are applied to show a certain property for bounded random variables. The author of the book always shows the ...
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### Processes with the same finite dimensional distributions as the solutions to SDEs

Consider a sequence of stochastic processes $\{\tilde{x}^n\}$, $\tilde{x}^n = \tilde{x}^n_t(\omega)$, and Brownian motions $\{\tilde{w}^n\}$. Suppose that for each $\tilde{x}^n$ solves the stochastic ...
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### Weak convergence of process

Background: I am trying to compute the weak limit of the following model from mathematical biology that is supposed to exist: Let L(f)(\eta)= \sum_{x \in \mathbb{Z}}\frac{1}{2}\left(1_{\eta(x+1) \...