# Questions tagged [stochastic-differential-equations]

Stochastic ordinary and partial differential equations generalize the concepts of ordinary and partial differential equations to the setting where the unknown is a stochastic process.

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### Uniqueness of the solution to switching coefficient SDEs

Consider the following SDE driven by real-valued Brownian motion $W=(W_t)_{t\ge 0}$:
$$dX_t = \left(\sigma {\bf 1}_{\{X_t>1\}} + \sigma' {\bf 1}_{\{0<X_t\le 1\}}\right)dW_t,\quad \forall t>0,$...

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### Can a regime switching, but otherwise autonomous diffusion converge to a point almost surely?

Let $W$ be a standard $d$-dimensional Brownian motion on a probability space $\Omega$, and $X$ the solution to the SDE
$$dX_t = \mu_{\mathbb i(t)} (X_t) \, dt + \, \sigma_{\mathbb i(t)} (X_t) \, dW_t,$...

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### 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 ...

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### Are smooth dynamical systems stabilised by "sufficient noisiness"?

Preliminaries.
(See [1] for further details.)
Let $M$ be a compact connected $C^\infty$ Riemannian manifold.
We say that a list $\sigma_1,\ldots,\sigma_n$ ($n \in \mathbb{N}$) of $C^\infty$ vector ...

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### 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}{\...

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### Equivalence of Wind Forces: Intensity vs. Duration [closed]

The strongest tornado in the world happened recently in Greenfield Iowa with winds over 318 mph: https://www.facebook.com/watch/?v=2176728102678237&vanity=reedtimmer2.0
I am curious, are less ...

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### Pathwise Hölder continuity of Ito diffusions - is this result written anywhere?

Let $X$ be the solution to the multidimensional SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t,$$
with $W$ a Brownian motion, and $\mu, \sigma$ Lipschitz continuous with $\sigma$ nowhere zero. I'm ...

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1
answer

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### Combination of the Dirichlet and Cauchy problems, find the PDE by which $\mathbb{E}_x M(X_{\tau_D \wedge t})$ is met

$X_t$ is an Itô diffusion process with continuous version, $\mathbb{L}_X$ is its generator. $D$ is a closed set in $\mathbb{R}$. The stopping time $\tau_D$ is the first entry time of $D$, that is $\...

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### 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}{\...

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### Markov process on a torus with prescribed invariant distribution

In Euclidean space, $\mathbb R^d$, the Langevin diffusion $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1,$$ where $\sigma:\mathbb R^d\to\mathbb R^{d\times k}$, $$b:=\frac{\Sigma+U}2\nabla\ln p+...

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### Comparison theorem for SDEs driven by a continuous martingale

Consider the well-known comparison theorem for SDEs, versions of which appear in several textbooks, e.g., Karatzas and Shreve, Proposition 5.2.18, or Revuz and Yor, Theorem IX.3.7.
The result states ...

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### Smoothness of resolvent of the infinitesimal generator of an Ito diffusion acting on bounded continuous function

Let $dX_t=\sigma(X_t)\,dW_t+\mu(X_t)\,dt$ be an Ito diffusion with Lipschitz coefficients and $\sigma(x)>0$. Let $f(x)$ be a continuous and bounded and non decreasing function. Can we prove that ...

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### 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 ...

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### Best textbooks/resources for "advanced" probability theory?

When I say "Advanced Probability", I mean for a person acquainted with the measure-theoretic foundations of probability theory, that wants to learn about Stochastic Processes from there, in ...

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### Ornstein Uhlenbeck process with discontinuous drift

This question is a modified version of this unanswered question asked on MSE, which mainly concerns an Ornstein-Uhlenbeck process with discontinuous drift on $\mathbb R^n$(for simplicity let $n=2$ for ...

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### 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}{\...

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### Conditional expectation w.r.t filtration of Brownian motion as a continuous map of its paths

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space on which we define Brownian motion $B$ and let us denote by $\mathcal{F}_t$ its natural filtration. Assume we have Ito process $dX_t = \...

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### Rate of convergence of mollified distributions in Besov spaces with negative regularity

Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...

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### 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+...

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### A stochastic optimal control problem with filtering-like dynamics

I want to extend the following stochastic optimal control problem with randomized feedback control to the continuous time case
\begin{align}
\text{minimize}\quad \mathbb{E}_{\mathbb{H}}&\bigg[\...

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### Regularity structures vs Renormalization

What are the substantial differences in the theory of "Regularity Structures" versus perturbative renormalization from Quantum Field Theory?
The idea that to treat divergences inherent to ...

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### 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 \...

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1
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154
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### Perturbation methods for stochastic/partial differential equations

I'm asking for a good reference on perturbation methods for stochastic and/or partial differential equations.
Something like this: Perturbation of a stochastic differential equation
I'm familiar with ...

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149
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### Estimating $p$th moment bound of error between small noise SDE and ODE

For a $d$-dimensional standard Brownian motion $W$, and a locally Lipschitz function $b: \mathbb{R}^d \rightarrow \mathbb{R}^d$, consider an SDE:
$$dX_t^\varepsilon = b(X_t) dt + \varepsilon^t dW_t,\...

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### The limit ratio of two Markov Chain Probability

Suppose there are two given SDE in $\mathbb{R}^d$:
$$
\begin{align}
\left\{
\begin{aligned}
dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...

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### Wellposedness of SDE with switching diffusion

Let $b:\mathbb R\to [-1,1]$ and $a_1, a_2:\mathbb R\to [1,2]$ be Lipschitz functions. Consider the stochastic differential equation (SDE) as follows :
$$dX_t = b(X_t)dt + a(X_t)dW_t,$$
where $(W_t)_t$ ...

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0
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### Derivative with respect to initial condition for the solution of an SDE

Suppose we have an SDE (assuming the Lipschitz continuous conditions required for the existence of the solution):
\begin{align}
dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t
\end{align}
and define its ...

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0
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### Solutions to ODE/SDE with singular coefficients $dX_t = -X_t/t \, dt + g\,dW_t$

I encountered a question regarding the solutions to SDEs with singular drifts. I searched the literature but had a hard time figuring out the intuition behind these analytic results assuming different ...

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1
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### Carnot–Carathéodory norm and the inner product norm

It is well-known that given the extended tensor algebra $T((\mathbb{R}^d))$ one may extract a separable Hilbert space by considering the subset
$$T^1((\mathbb{R}^d)) := \left\{h \in T((\mathbb{R}^d)) :...

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### Decay estimate of moment of an SDE

We consider an SDE
$$
d X_t = b(t, X_t) \, dt + \sigma(t, X_t) \, d B_t,
$$
where $(B_t)$ is a $d$-dimensional Brownian motion on $\mathbb R^d$. We fix $p \in [1, \infty)$. Here $b, \sigma$ are ...

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### 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 $n$-...

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### 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-\|...

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### Solution of SDE at finite time, continuity of pdf

I'm looking at the Langevin dynamics described by the following SDE
$$d X_t = - \nabla U(X_t) \, d t + \sqrt {2 \Sigma} \, d B_t,$$
where $X_t \in \mathbb R^d$, $\nabla U(\cdot)$ has some regularity ...

3
votes

0
answers

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### 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. ...

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### 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\...

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### What happens when the diffusion term in an SDE becomes zero?

Consider this time-homogeneous SDE, in the Ito sense:
$$dX_t= -(X_t-a)\,dt+\sigma(X_t)\,dW_t,$$
where $W_t$ is standard Brownian motion, $a<b\in\mathbb{R}$, $X_0\leq b$ a.s., and $\sigma(b)=0$. ...

2
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### 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 ...

3
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### Statistically stationary properties of expectations conditioned on the value of an Ornstein–Uhlenbeck process

Consider the modified Ornstein–Uhlenbeck process
$$\mathop{dx_t}=\theta(y_t-x_t)\, dt+{}\sigma\,dW_t$$
for a standard Brownian motion $W_t$ and $\theta,\sigma\in\mathbb{R}_{>0}$. Let's define the ...

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### Elliptic PDEs in Finance

In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...

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### Estimation of past knowing present

Let $X$ be the solution to some stochastic differential equation (unidimensional or multidimensional) :
$$dX_t = b(t,X_t)\,dt + a(t,X_t)\,dW_t\quad \forall t\ge 0,$$
where $b, a$ are both Lipschitz. ...

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### 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(�...

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### Autocovariance of harmonic oscillator in fluid (Langevin Equation)

I am looking to work out an analytical solution (if it is known) for the autocovariance $Cov[X_s,X_t]$ of a particle which behaves according to the Langevin equation for a Harmonic Oscillator in a ...

4
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201
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### Flow property for semimartingale driven SDE at a stopping time

Let $S$ be an $n$-dimensional semimartingale such that the SDE
$$dX_t = \sigma(X_t, t) \, dS_t$$
with $\sigma$ Lipschitz continuous admits a globally defined unique strong solution on $[0, T]$.
For $t ...

3
votes

0
answers

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### Cylindrical Wiener processes or SPDE that can make use of Banach valued rough paths?

Rough paths theory has an often advertised perk that it mostly works for general Banach spaces. I am trying to think of some nice examples that actually use this feature, and am coming up stuck.
The ...

4
votes

1
answer

219
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### Weak uniqueness of an SDE with locally Lipschitz drift and additive noise?

Consider the $d$-dimensional SDE, $d > 1$,
$$dX_t = b(X_t) \, dt + \sqrt 2 \, dW_t$$
where
$b$ is locally Lipschitz such that $|b(x)| \le C |x|^2$ for $x \in \mathbb R^d$.
$W$ is a standard $d$-...

7
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2
answers

597
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### Interpretation of second order term in Fokker-Planck equation

Let $G:\mathbb{R}^d\to\mathbb{R}^{d\times d}$ be a matrix-valued smooth function. Let us define a quantity by
$$
\begin{align*}
\nabla^2\cdot G(x)
&=\sum\limits_{i=1}^{d}\sum\limits_{j=1}^{d}\...

3
votes

1
answer

252
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### Weighted Lebesgue space with exponential weights: smoothing effect and properties

I am researching whether there are weighted Lebesgue spaces of the type
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...

23
votes

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### What phenomena are better modelled by SDE instead of ODE?

Both stochastic differential equations (SDE) and ordinary differential equations (ODE) can be used to model a variety of different phenomena, whether physical or otherwise. Most deterministic ODE ...

3
votes

1
answer

228
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### Do regularity structures involve infinite "Taylor" series?

I have been learning about the theory of regularity structures, for which the common motivation is Taylor series. However, I keep seeing direct sums in the definition of a regularity structure, which ...

3
votes

1
answer

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### Stochastic representation of Laplace equation with Neumann boundary condition

Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$.
What if ...