All Questions
178 questions with no upvoted or accepted answers
7
votes
0
answers
304
views
Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories
Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...
- 167
6
votes
0
answers
88
views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
- 5,405
6
votes
0
answers
245
views
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 $M$...
- 9,853
6
votes
0
answers
774
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 ...
- 167
5
votes
0
answers
400
views
Uniform bound for the occupation time of a diffusion
Note: We denote by $\mathcal L(U)$ the Lebesgue measure of a set $U$.
Let $\mu: \mathbb R^d \to \mathbb R^d$ and $\sigma: \mathbb R^{d} \to \mathbb R^{d \times d}$ be Borel functions.
Suppose the ...
- 6,215
4
votes
0
answers
122
views
Finiteness of the moments of the Malliavin derivative of the stochastic heat equation
I am studying section 2.4.2 from Nualart's book "The Malliavin calculus and related topics" on the stochastic heat equation. I have some questions on the validity of some estimates for the ...
- 61
4
votes
0
answers
328
views
Convergence to unique stationary distribution for SDEs and Markov processes
I am interested in understanding the behavior of solutions to stochastic differential equations (SDEs) and continuous-time Markov processes with constant coefficients. Specifically, I would like to ...
- 807
4
votes
0
answers
306
views
A notion of SDE via the martingale representation theorem
$\newcommand{\d}{\mathrm{d}}$It is well-known that differentiating stochastic processes with respect to time is usually impossible in the usual sense. For instance, a Brownian motion $W$ on a ...
- 11.8k
4
votes
0
answers
167
views
Occupation time of SDE
Let $b:\mathbb{R}^d\to\mathbb{R}^d$ be locally Lipschitz and assume that, for any $x\in\mathbb{R}^d$ and any $f\in C^{\infty}([0,1],\mathbb{R}^d)$, the equation
$$
X_t^{x,f}=x+\int_0^t b(X_s^{x,f})\,...
- 93
4
votes
0
answers
95
views
Reference request on theory about Stochastic Riemann problem
I am trying to find references in the literature that deal with the Stochastic Riemann problem. Let me explain it a bit.
On one hand, in the literature it is not hard to find books or papers that deal ...
- 657
4
votes
0
answers
145
views
Regularity of martingales with respect to spatial parameters
In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...
- 167
4
votes
0
answers
276
views
Exit time of a stochastic process defined by a SDE
Let $\mathcal{P}$ be a "small particle" trapped in a $n$-dimensional potential. We will assume the dynamics of $\mathcal{P}$ are well described by the stochastic differential equation
\begin{align*}
\...
- 205
4
votes
0
answers
223
views
Optimal control of SDEs
I've set up a system of stochastic differential equations that I'd like to control. I'm new to optimal control theory and SDEs (and, admittedly, weak on PDEs), so I'm not certain if I've set this up ...
- 41
4
votes
0
answers
414
views
Definition of the Stratonovich integral in Hilbert spaces
Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...
- 167
4
votes
0
answers
112
views
Feynman-Kac formula and time-ordering for vector bundles
Let $M$ be a compact Riemannian manifold and let $\mathrm{d}\mathbb{W}^{yx;T}(\gamma)$ denote the Brownian Bridge measure, i.e. the Wiener measure on the paths that travel from $x$ to $y$ in time $T$ (...
- 9,853
3
votes
0
answers
74
views
Reference for PDEs from system of SDEs
I'm working with a system of SDEs
\begin{align*}
dX_t &= b(X_t, t) + \sigma dB_t\\
dY_t &= c(X_t, Y_t, t) + \sigma dB_t.
\end{align*}
Here, the Brownian motion is the same.
I know that ...
3
votes
0
answers
54
views
Unique weak solution of an SDE for a general initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\eps}{\varepsilon}
\newcommand{\diff}{\...
- 835
3
votes
0
answers
80
views
Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...
- 167
3
votes
0
answers
86
views
Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. ...
3
votes
0
answers
122
views
Slow points of diffusion processes
Let $W$ be a standard $d$-dimensional Brownian motion, and $X$ the solution to the SDE
$$dX_t = \mu(X_t) dt + \sigma(X_t) \, dW_t,$$
with $\mu$ and $\sigma$ Lipschitz continuous.
Given a (...
- 6,215
3
votes
0
answers
77
views
Inverse comparison principle for stochastic differential equations
Consider two SDEs (stochastic differential equations) as follows:
$$dX_t=b^-(t,X_t) \, dt+a(t,X_t) \, dW_t;\quad dY_t = b^+(t,Y_t)\,dt+a(t,Y_t)\,dW_t,$$
where $b^-,b^+,a$ are Lipschitz such that $b^-&...
- 1,399
3
votes
0
answers
122
views
Dealing with noise that is white in time, colored in space numerically
I am broadly working on a dynamic process where we want to see how a field $\rho(r)$ changes in space in time with thermal noise. The system is biased around a thermodynamic saddle point dictated by $...
- 31
3
votes
0
answers
201
views
Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?
In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
- 177
3
votes
0
answers
235
views
Probability of a particle surviving forever
Consider a particle whose position is driven by the following equation:
$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$
where $y>0$, $0<C<1$...
user128095
3
votes
0
answers
569
views
Domain of the Generator of a Bessel process
Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$
\begin{align}
\rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t}
\end{align}
where $(W_{t})_{t\geq ...
3
votes
0
answers
90
views
Mutual dependencies of BSDE solutions with markovian drivers with different starting points
Let $(\Omega,\mathcal F, P)$ be a complete probability space with a Brownian motion $(W_t)_{0\le t\le T}$ and the Brownian standard filtration $(\mathcal F_t)_t$ with $\mathcal F_T = \mathcal F$.
Let ...
- 335
3
votes
0
answers
89
views
Generators and Covariance Operators of Diffusions
For a constant coefficient Ornstein-Uhlenbeck process, how should I think about the relationship between the infinitesimal generator of the process and the covariance operator of the process (or, ...
- 379
3
votes
0
answers
89
views
Why is the Jain Monrad condition the right condition on general Gaussian processes?
Consider a covariance function $\sigma^2(s,t)=E((X_t-X_s)^2)$, where $X\colon I\to \Bbb R^d$ is a Gaussian process.
Given a $\rho\ge 1$ and a superadditive function $\omega(s,t)$ we say that Jain ...
user69208
3
votes
0
answers
201
views
Reference request on connection between PDE problems
I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study deterministic and stochastic conservation laws. Problems that I am interested ...
- 657
3
votes
0
answers
73
views
Uniqueness of a stationary measure for a simple ODE
How to prove uniqueness of the stationary measure using coupling approach to the following process: $u_{n+1}=S(1)u_n+\eta_{n+1}(\omega)$, where
a) $S(1)y_0:=y(1)$, $y(t)$ solves the following Cauchy ...
- 383
3
votes
0
answers
122
views
Monte-carlo estimation on the drift of SDE
On any probability space $(\Omega, \mathcal{F} , \mathbb{P})$ with a Brownian motion $W$, we consider the following one-dimensional SDE:
$$dX_t = F(t, X_t) \, dt + dW_t,$$
where $F(t,x) = \mathbb{...
- 357
3
votes
0
answers
231
views
I've found a representation of the Itō-Stratonovich correction term and don't understand the used notion of a "trace"
Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a ...
- 167
3
votes
0
answers
78
views
Perscribed/Inverting Conditional Expectation
I'm having difficulty finding papers which deal with the following inversion problem.
Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...
- 5,405
3
votes
0
answers
186
views
When we integrate with respect to a $Q$-Wiener process on $U$, why do we restrict integrands to be operators on $Q^{1/2}U$ (instead of $U$)?
When we integrate with respect to a $Q$-Wiener process $(W_t)_{t\ge 0}$ ($Q$ being a bounded, linear, nonnegative and self-adjoint operator on a separable $\mathbb R$-Hilbert space $U$ with finite ...
- 167
3
votes
0
answers
276
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 ...
- 131
3
votes
0
answers
240
views
Using compactness method to prove the existence of a pathwise solution to an SPDE
For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...
3
votes
0
answers
235
views
Strong solution to an SDE with a discontinuous diffusion term
I am having an SDE for which I would be in trouble if there were no strong solution.
The SDE is -
$ dX = \mu(x) dt + \sigma_1 (x) db_{1t} + \sigma_2(x) db_{2t}$
where $b_1$ and $b_2$ are two ...
- 553
2
votes
0
answers
67
views
The unique weak solution to some SDE yields the unique strong solution?
For some filtered probability space $\big(\Omega,\mathcal F, (\mathcal F_t),\mathbb P\big)$, consider a stochastic differential equation (driven by a real-valued Brownian motion $W$) for $X=(X_t)$, ...
- 1,399
2
votes
0
answers
41
views
Approximate the adjoint generator of the discretization of an SDE
Let
$d\in\mathbb N$;
$\sigma\in\mathbb R^{d\times d}$;
$p\in C^1(\mathbb R^d)$ be positive with $$c:=\int p(x)\;{\rm d}x<\infty\tag1$$ and $$b:=\frac12\Sigma\nabla\ln p;$$
$(X_t)_{t\ge0}$ denote ...
- 167
2
votes
0
answers
82
views
Existence of SDE solution under integrability of Lipschitz coefficients
I am reading the paper Lan and Wu, Stoch. Process. Appl., 2014, on sufficient conditions weaker than Lipschitzianity for the existence of strong solutions of time-inhomoegneous $d$-dimensional SDEs. ...
- 135
2
votes
0
answers
42
views
Diffusions vs elliptic operators with dkp coefficients
I am wondering if there is any literature on the relationship between diffusions and elliptic equations. In particular I am interested in literature concerning operators with Dahlberg–Kenig–Pipher ...
- 121
2
votes
0
answers
89
views
Malliavin calculus for the regularity of the density of the supremum of a process
I am reading Chapter 2 from Nualart's book 'The Malliavin calculus and related topics'.
Proposition 2.1.10 gives the conditions for the law of the supremum of a process to have a density. Condition (...
- 61
2
votes
0
answers
80
views
Stability of Hölder constants of frozen Itô stochastic integrals
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 835
2
votes
0
answers
89
views
Are speed, scale function and killing measures of Itô diffusion absolutely continuous respect to Lebesgue measure and do have smooth derivative?
In Borodin and Salminen's Handbook of Brownian motion (MR1912205, Zbl 1012.60003), pages 16–17, they mention the fact that if the three basic characteristics (speed measure, scale function and killing ...
- 41
2
votes
0
answers
66
views
Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?
$
\newcommand{\RR}{\mathbb{R}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\PPP}{\...
- 835
2
votes
0
answers
119
views
How does the first hitting time depend on the drift of drifted Brownian motion?
Let $W$ be a standard Brownian motion, and $a,b:\mathbb R_+\times \mathbb R\to\mathbb R$ be Lipschitz. Consider the stochastic differential equations:
$$X_t=1+\int_0^ta(s,X_s)ds + W_t,\quad\quad Y_t=1+...
- 1,334
2
votes
0
answers
95
views
Brownian bridge as a limit of SDEs
Let $B$ be a Brownian motion and with respect to some probability measure $\mathbf{P}$ and filtration $(\mathcal{F})_{t \geq 0}$ and let $S_\epsilon = \{B_1 \in (-\epsilon,\epsilon)\}$.
For every $t \...
- 141
2
votes
0
answers
78
views
SDE driven by Lévy processes
Consider a stochastic differential equation (SDE) on some filtered probability space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$ : for all $t>0$
$$dX_t = u_tf(X_{t-})dt+ u_t g(X_{t-})dW_t + u_t\...
- 1,399
2
votes
0
answers
81
views
Assumptions for uniform measure of SDE on manifolds
Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
2
votes
0
answers
90
views
How to estimate the difference between two Ito diffusions?
Suppose $𝑏:\mathbb R^d \to \mathbb R^d, \sigma:\mathbb R^d \to \mathbb R^{d\times d}$ are measurable functions and satisfy
\begin{equation*} 2\langle 𝑥−𝑦,𝑏(𝑥)−𝑏(𝑦)\rangle +\|\sigma(𝑥)−\sigma(�...
- 622