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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Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$

Let $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ $(\Omega,\mathcal A,\...
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38 views

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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1answer
56 views

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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28 views

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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1answer
2k views

Martingale representation theorem for Levy processes

Is there an equivalent of martingale representation theorem for Levy processes in some form? I believe there is no such theorem in generality, but maybe there are some specific cases?
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68 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
2
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1answer
99 views

Example of progressively measurable process that is not predictable

Is there an example of progressively measurable process that is not predictable? This question is motivated by Revuz-Yor, Continuous Martingales and Brownian Motion http://www.springer.com/gb/book/...
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2answers
629 views

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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1answer
287 views

Generalized Ito's lemma

I have the following quantity: $$ g(t)=(f(t))^{M_{t}}, $$ where $M_{t}$ is a jump process which is neither Markovian nor Levy, and $f(t)$ is a positive, increasing but limited, right-continuous ...
4
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1answer
493 views

weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
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53 views

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 ...
2
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0answers
66 views

Differentiability of a simple value function driven by a diffusion

Consider a diffusion given by, $d X_t = \mu(X_t) dt + \sigma(X_t) dB_t$ $X_0 = x$. Suppose the functions $\mu$ and $\sigma$ are as follows - $f(x) = \mu(x) = \sigma(x) = \begin{cases} 2 & \...
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34 views

Boundary behavior for Ito diffusions

The classification of boundary behavior for a time-homogeneous diffusion satisfying an Ito stochastic differential equation (SDE) is well known. According to the Feller classification, there are four ...
5
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1answer
378 views

'Nonclassical' abstract Wiener space

Is it possible to construct an abstract Wiener space $(W,H,\mu)$ such that $C^{0,\frac{1}{2}}(\Omega)\subset H$ and $W$ is a normed function space such that the convergence in norm implies convergence ...
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0answers
77 views

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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1answer
128 views

A problem about the quotient space of an extended Dirichlet space

Let $(\mathscr{E},\mathscr{F})$ be a recurrent Dirichlet form on $L^2(X;m)$ and $\mathscr{F}_e$ the corresponding extended Dirichlet space, then $1\in\mathscr{F}_e$ and $\mathscr{E}(1,1)=0$. Let ${\...
4
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1answer
165 views

Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$

I am considering the transition semigroup $P_t$ associated with the Ito diffusion process $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ where the coefficients are assumed to be Lipschitz continuous. I hope to ...
3
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1answer
77 views

Decouple system of SDEs / handle scaling problem

Consider $\begin{split} \newcommand{\d}{\mathrm d} \d x &= -yx \d t + x^2 \d B\\ \d y &= -2 y^2 \d t + 2xy \d B. \end{split}$ This is a system of two SDEs driven by the same standard ...
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35 views

Adiabatic elimination of a variable in a system of nonlinear stochastic ODEs?

If this is too basic for MathOverflow... say the word and I shall move it to Math.SE First consider this system of ODEs. Say I have two variables $u$ and $a$, following $$ \dot u = -u + f(a) $$ $$ \...
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1answer
149 views

Can all Local Martingales Be Represented using Only Brownian Motion and Finite Variation Processes?

This is a cross-post of my unanswered (more than a week) question on Math.SE. Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to try to ...
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0answers
102 views

Strong law of large number for semimartingale

I just want to know if for semimartingale $X$ we have $\lim_{t \rightarrow \infty} \frac{X_{t}}{\langle X\rangle_{t}}=0$ or when it is possible. I know it is true for Brownian motion. Thanks
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1answer
95 views

Representation of support of Gaussian measure by kernels of no-variance functionals

Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for $$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$ ...
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92 views

Short time asymptotics for Brownian motion on a compact manifold

Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in ...
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0answers
80 views

Brownian hitting probability of a small body

Consider a Brownian motion $B(t)$ starting from the origin $0$ in $\mathbb{R}^n$. Consider the ball $B(0, r)$ and an open set $V \subset B(0, r)$. If it is known that the probability of the Brownian ...
2
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0answers
67 views

Brownian motion in perturbed (asymptotically flat) metric

Let $g_{\mathbb{R}^n}$ denote the usual Euclidean metric on $\mathbb{R}^n$ and let $B_g(t)$ denote the Brownian motion associated to a complete metric $g$ on $\mathbb{R}^n$. Consider a Brownian motion ...
5
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0answers
82 views

Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term

Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs. I'm reading Stochastic Differential Equations in ...
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0answers
38 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
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2answers
176 views

A Stochastic Taylor Expansion/Asymptotics

Question: Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth ...
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1answer
53 views

Girsanov theorem and the density of a process

I am coming across a paper ( Proposition $1.1$ from http://www.sciencedirect.com/science/article/pii/0304414987901840 ) that claims the following fact which I don't understand why: On a ...
2
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1answer
163 views

Question on Wiener processes not hitting 0

Let $W_t$ be a standard Wiener process, and $0\leq a < b$. Let $\hat{W}_t:=W_{a+t}-W_a$. Then $\hat{W}_t$ is also a standard Wiener process. I think that the following should be true: $$\mathbb P\...
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40 views

Where can I find this article of Doléans-Dade?

I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade. I could not find a pdf version online, and my university library does not have a printed version. Thank ...
2
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0answers
176 views

Expected value and variance of a stochastic process

I would like to ask if there is a way to find the expected value and the variance of the following process $$ dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0 $$ where $a\in (-\infty,+\infty), b&...
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27 views

Smoothness of Value function for SDE with discontinuous coefficients

Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous). I'm interested in the function $v:\...
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0answers
79 views

Construction of a random variable

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the statement:...
3
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1answer
157 views

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 ...
13
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1answer
1k views

surprisingly difficult filtration problem

I am interested in a proof of the following statement which seems intuitive, but is somehow really tricky: Let $X$ be a stochastic process and let $(\mathcal{F}(t) : t \geq 0)$ be the filtration ...
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0answers
58 views

A problem on Markov chains and Dirichlet forms

Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for ...
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0answers
64 views

Basic Monte Carlo Integral Approximation

On the very first page of a well-known book on Monte Carlo techniques, there is the following statement. Let \begin{equation} I = \int_D g(\textbf{x})d\textbf{x}, \end{equation} where $D \subset \...
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0answers
27 views

Fubini's Theorem for Lévy bases

Let $M$ be an infinitely divisible independently scattered and homogeneous random measure on $\mathbb R^d$ (ie a homogeneous Lévy basis). Let $\nu$ be a sigma finite measure on $\mathbb R^k$. Let $f:\...
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1answer
102 views

Limit (Convergence) of stopping times

Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
2
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0answers
82 views

Hypergeometric function

Suppose that $V$ follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$ I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that $F(V)$ must ...
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0answers
40 views

regularity of the conditional expectation: from Markov to Non-Markov

Let $B=(B_t)_{0\le t\le T}$ be a standard Brownian motion and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Let $\xi=\xi(B)$ be a bounded measurable functional. Now let's ...
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37 views

Distribution of stopped Brownian motion in $\mathbb R^2$

Let $B=(B^1_t,B^2_t)_{t\ge 0}$ be a standard Brownian motion in $\mathbb R^2$. Let $U=(U^1,U^2)$ be an independent random variable taking values in a circle $C_1\subset\mathbb R^2$ with uniform ...
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0answers
55 views

limit multiple integral

I want to know if $\lim_{T-> \infty}$ of this integral $$ \frac{\sigma^{4}C_{H,K}^{2}}{4 T^{4HK}e^{2\theta T }}\\ \times \int\limits_{[0,T]^{4}}e^{\theta(t_{1}-s_{1})}e^{\theta(t_{2}-s_{2})}\left\...
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0answers
34 views

Strong Markov vector-valued process from component strong Markov process and independence

I want to prove that if $X$ and $Y$ are (continuous time) independent strong markov $\mathbb{R}$-valued processes w.r.t. their natural filtrations $\mathcal{F}^X_t$ and $\mathcal{F}^Y_t$, that the ...
2
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1answer
195 views

Question about the stochastic integral of martingales

Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $\mathbb{R}$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e....
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36 views

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'...
3
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0answers
191 views

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/...
1
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1answer
126 views

Malliavin derivative under change of measure

Let $\widetilde{B}$ be a Brownian Motion under the measure $\mathbb{P}$. Let $\theta$ be a stochastic process fulfilling the Novikov's condition and $Z_\theta$ the relative Radon–Nikodym derivative ...
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1answer
73 views

Brownian motion increments

We know that if $W_t$ is a Brownian motion, $W_{t+t_0}-W_{t_0}$ is one too. Does the "converse" holds : Let $t_0$ be a positive number. I have a Brownian motion $W_t$ and I seek another Brownian ...