Questions tagged [stochastic-calculus]
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.
933
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Existence of a solution to an infinite dimensional Stratonovich SDE
Let
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
$U_0:=Q^{1/2}U$
$(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...
1
vote
1
answer
151
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Hilbert-Space Values SDE in terms of Basis
Suppose:
$$
dX_t = a(t,X_t)dt + b(t,X_t)dW^H_t
$$
is an SDE with values in a separable Hilbert Space $H$, and $W^H_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $...
1
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1
answer
636
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Properties of the trace term in the Itō formula
Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where
$U,H$ are separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ is nonnegative ...
2
votes
1
answer
645
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Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces
I'm reading the paper On the relation between the Itō and Stratonovich integrals in Hilbert spaces and there is something I don't understand.
In the notation of the paper, let
$H,H_1$ be separable $\...
1
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0
answers
132
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infinite dimensional funtional ito calculus
I've been reading into functional Ito calculus and everything I've come across deals with processes generated by finite dimensional semimartingales. In Dupire's 2009 landmark paper he speaks about ...
1
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1
answer
124
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Reference for convergence of Hilbert-space valued SDEs
I'm fairly familiar with the literature dealing with convergence of SDEs in $\mathbb{R}^d$ but recently I've needed to use extended results dealing with convergence of SDEs in Hilbert Spaces. However ...
5
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0
answers
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Second order calculus and rough paths
In Emery's book "Stochastic calculus in manifolds", he shows how to make sense of integrals of the form
$$ \int \langle\Theta_t, \mathbf{d} X_t\rangle,$$
where $X$ is a semimartingale on a manifold $M$...
3
votes
1
answer
611
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Asymptotic behavior of an integral of OU process
Let $X=(X_t)_{t\ge 0}$ be a stochastic process (Ornstein-Uhlenbeck process) determined by
$$dX_t=-aX_tdt+\sigma dW_t,$$
where $X_0=0$, $a>0$ and $\sigma>0$ are constants, and $W=(W_t)_{t\ge 0}$...
1
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0
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123
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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,\...
1
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0
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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 ...
2
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2
answers
825
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!
3
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1
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Calculate Moments of SDE
I have posted a similar question on math.stackexchange (https://math.stackexchange.com/questions/1848492/calculate-mean-of-sde), but didn't find anyone who could help.
I'm interested in the one-...
1
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1
answer
864
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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 \in ...
2
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0
answers
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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 ...
3
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1
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159
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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 & \...
4
votes
1
answer
481
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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 ...
3
votes
0
answers
183
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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 $ ...
2
votes
0
answers
256
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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
$$
\dot u = -u + f(a)
$$
$$
\...
6
votes
1
answer
2k
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Can all local martingales be represented using only Brownian motion and finite variation processes?
This is a cross-post of my unanswered (more than a week) question on Math.SE. Since it covers topics from my graduate-level course on stochastic processes, I thought it might be appropriate to try to ...
3
votes
1
answer
156
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Representation of support of Gaussian measure by kernels of no-variance functionals
Let $\mu$ be a Gaussian measure on a separable Banach space $X$ and $q$ is the covariance operator of $\mu$. I am reading a proof for
$$\operatorname {supp} \mu = \bigcap_{q(f, f) = 0} \ker f =: E$$
...
4
votes
0
answers
120
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Short time asymptotics for Brownian motion on a compact manifold
Consider a compact Riemannian manifold $(M, g)$. Choose a ball $B(p, r)$ inside $M$, and a quasi-isometric ball $B(q, s)$ in $\mathbb{R}^n$, in the image of a coordinate chart containing $B(p, r)$ (in ...
1
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0
answers
85
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Brownian motion in perturbed (asymptotically flat) metric
Let $g_{\mathbb{R}^n}$ denote the usual Euclidean metric on $\mathbb{R}^n$ and let $B_g(t)$ denote the Brownian motion associated to a complete metric $g$ on $\mathbb{R}^n$. Consider a Brownian motion ...
1
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0
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65
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System of stochastic equations
I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$
$$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
6
votes
0
answers
726
views
Relationship between the Itō formula for a Q-Wiener process and the Itō formula for a cylindrical Wiener process. A question on the trace term
Remark: Even when this question is about stochastic PDEs, it can be answered by someone who has no knowledge about probability theory or PDEs.
I'm reading Stochastic Differential Equations in ...
1
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1
answer
1k
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Girsanov theorem and the density of a process
I am coming across a paper ( Proposition $1.1$ from http://www.sciencedirect.com/science/article/pii/0304414987901840 ) that claims the following fact which I don't understand why:
On a probability ...
5
votes
2
answers
414
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A Stochastic Taylor Expansion/Asymptotics
Question:
Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and
$$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$
$(\mu,\sigma)$ obeys the linear growth ...
4
votes
1
answer
739
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Transition semigroup of Ito diffusion on $L^2(\mathbb{R})$
I am considering the transition semigroup $P_t$ associated with the Ito diffusion process
$$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$
where the coefficients are assumed to be Lipschitz continuous.
I hope to ...
0
votes
0
answers
56
views
Where can I find this article of Doléans-Dade?
I need to find the article "Intégrales stochastiques dépendant d’un paramètre" by Doléans-Dade.
I could not find a pdf version online, and my university library does not have a printed version.
Thank ...
2
votes
0
answers
95
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Smoothness of Value function for SDE with discontinuous coefficients
Let $\mu: \mathbb{R}\to \mathbb{R}$, $f: \mathbb{R}\to \mathbb{R}$, and $r: \mathbb{R}\to [1, \infty)$ be bounded measurable functions (which may be discontinuous).
I'm interested in the function $v:\...
2
votes
1
answer
2k
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Expected value and variance of a stochastic process
I would like to ask if there is a way to find the expected value and the variance of the following process
$$
dv_t=(a-be^{\alpha v_t})dt+\sigma dW_t, \quad v_t=v_0
$$
where $a\in (-\infty,+\infty), b&...
2
votes
0
answers
126
views
Construction of a random variable
I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the statement:...
1
vote
0
answers
302
views
A problem on Markov chains and Dirichlet forms
Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy
$$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 ...
6
votes
1
answer
2k
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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 ...
2
votes
1
answer
246
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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:
$$\mathbb P\...
1
vote
1
answer
1k
views
Limit (convergence) of stopping times
Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of ...
3
votes
0
answers
153
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Hypergeometric function
Suppose that $V$ follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$
I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that $F(V)$ must ...
2
votes
0
answers
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limit multiple integral
I want to know if $\lim_{T-> \infty}$ of this integral
$$ \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\...
1
vote
0
answers
72
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
votes
0
answers
3k
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
vote
1
answer
291
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 ${\...
3
votes
2
answers
334
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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 ...
0
votes
1
answer
147
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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 ...
1
vote
0
answers
301
views
Does the martingale property holds after changing filtration?
Let $\Omega$ be the space of continuous real-valued functions $\omega=(\omega_t)_{t\ge 0}$ starting at zero, i.e. $\omega_0=0$. Let $\Lambda=\Omega\times \mathbb R_+$ and denote by $\lambda=(\omega,\...
3
votes
1
answer
176
views
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(-types)...
3
votes
1
answer
750
views
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 ...
3
votes
0
answers
237
views
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 ...
0
votes
1
answer
231
views
Transition probabilities for the symmetric random walk on the integers
I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random ...
23
votes
1
answer
994
views
Does a theory of stochastic differential algebras exist?
My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...
0
votes
1
answer
146
views
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) \...
3
votes
1
answer
476
views
Malliavin differentiability of solutions to SDEs
In Bass's book on Diffusions and Elliptic Operators, the author gives a brief introduction into Malliavin Calculus. He calls a functional $F:C([0,1],\mathbb{R})\rightarrow \mathbb{R}$ $L^p-$smooth if ...