# 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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questions

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### Regularity of convergent flow of parabolic PDE (Fokker-Planck equation)

Consider the divergence-type 2nd order linear PDE on $\mathbb{R}^d$
$$\partial_t u_t = Lu_t := \nabla\cdot(u_t\,\nabla V)+\Delta u_t,$$
representing the Fokker-Planck evolution equation for the ...

4
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1
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### Reference request: showing that solution of an Ito SDE stays bounded with positive probability

Assume that we have a (well-posed) Ito SDE of the form $$\mathrm{d} X_t = b(X_t)\,\mathrm{d} t + \sigma(X_t)\,\mathrm{d}W_t \label{1}\tag{1},$$ where $b \colon \mathbb{R}^d \to \mathbb{R}^d$, $\sigma \...

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0
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+50

### Freidlin Wentzell for stochastic differential inclusions

Consider the SDI
$$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$
Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?

2
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0
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### Controlling the adjoint variables in a stochastically perturbed control problem

Suppose we have a deterministic control problem
$$dX_t = b(X_t, u_t) \, dt$$
on a finite timeframe with no terminal cost; i.e. the objective functional to be maximised is
$$\mathbb E \left [\int_{0}^T ...

1
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1
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### Stochastic Stokes flow: where to start from?

I would need to get acquainted to the subject of stochastic Stokes flows, so studying Stokes equations under some noise of some kind, let's say an additive white noise to begin with.
The problem is ...

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0
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### Laplace transform of a stochastic process

Let $R := (R_1, R_2)$ be a two-dimensional diffusion process defined by the following SDE:
$$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t} \, \mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \, \mathrm{d}W_t$...

2
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1
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### Stability results for general linear stochastic ODE

I am interested in the following time-invariant multivariate SDE:
\begin{equation}
dx_i = \sum_{j} a_{ij} x_j\,dt + \sum_{j,k} b_{ijk} x_k \, dW_j
\end{equation}
Despite its simplicity the general ...

3
votes

1
answer

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### Each diffusion SDE is associated to a *unique* family of transition kernels

I consider an SDE of the form $dX_t=b(X_t) \, dt + \sigma(X_t) \, dW_t$, with $b$ and $\sigma$ globally Lipschitz on $\mathbb{R}^n$.
How can I prove that there exists a unique family of transition ...

2
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0
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### Equivalence of score function expressions in SDE-based generative modeling

I am studying the paper "Score-Based Generative Modeling through Stochastic Differential Equations" (arXiv:2011.13456) by Yang et al. The authors use the following loss function (Equation 7 ...

2
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0
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### What is the state of the art for rough path regularity on coefficients?

Consider the rough differential equation
$$dY_t=b(Y_t,t) \, dt+\sigma(Y_t,t) \, d\mathbf X_t,$$
where $\mathbf X$ is a $p$-rough path with $1\leq p<3$. If $b$ and $\sigma$ are $C^3_b$ then we have ...

1
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0
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### Uniqueness of the weak solution to stochastic differential equation

Consider the stochastic differential equation
$$dX_t = {\bf 1}_{\{0<X_t<1\}} a(t,X_t)dW_t, \quad \forall t\in [0,T],$$
where $a$ is continuous on $[0,T)\times [0,1]$ and is Holder continuous ...

2
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0
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### How much is known about the action functional for small noise diffusions with general volatility coefficients?

Let $W$ be a d-dimensional Brownian motion, and for every $\varepsilon > 0$, let $X^\varepsilon$ be the solution to the SDE
$$dX^\varepsilon_t = b(X^\varepsilon_t) \, dt + \varepsilon \sigma (X^\...

1
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1
answer

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### Linear response for SDE

Consider a family of stochastic processes $dX^h_t=(g(X^h_t)+h(s))\,dt+dW_t$ and a functional $I_f:h(s) \rightarrow E[f(X_t^h)] $. I would like to compute the kernel of the derivative of this ...

2
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0
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### Ergodicity of the solution to some SDE

Consider the SDE (stochastic differential equation) as follows:
$$dX_t=X_t\big(b(X_t)dt+a(X_t)dW_t\big)$$
where $b,a:\mathbb R\to\mathbb R$ are Lipschitz and bounded and $W$ is a real-valued Brownian ...

2
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0
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### Bounding from below the distance between SDE started from different initial conditions

Let $W$ be a standard one dimensional Brownian motion, and let $X$ be the solution to the SDE
$$dX_t = \mu(X_t) \, dt + \sigma(X_t) \, dW_t$$
with $\mu, \sigma: \mathbb R \to \mathbb R$ Lipschitz ...

3
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0
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### 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
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0
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### Lower bound of $\mathbb P[\sup_{t-\theta\le s\le t}|X_s-x|\le \varepsilon \mid X_t=x]$ (without observing history)

Let $X$ be the solution to some stochastic differential equation
$$dX_t =b(X_t) \, dt+a(X_t) \, dW_t,\quad \forall t>0.$$
Here $b,a: \mathbb R^d \to\mathbb R^d$ are bounded and Lipschitz and $W$ ...

5
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1
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### How can we prove that a stochastic process converges to a deterministic value?

As an illustrative example, consider a modified O-U process $dX_t = -X_tdt + \exp(-t)dW_t$. It is not too hard to understand that after a while the behaviour is dominated by the deterministic ...

2
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0
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### Asymptotic behaviour of the solution to some delayed stochastic differential equation

Consider the delayed stochastic differential equation as below:
$$dX_t^\theta=X_{(t-\theta)^+}^\theta(1-X_{(t-\theta)^+}^\theta)(dt+dW_t),\quad \forall t>0$$
$$dY_t^\theta=Y_{(t-\theta)^+}^\theta(1-...

1
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0
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### Characteristic function of stochastic integral of a pure jump Lévy process with respect to another pure jump Lévy process

(I am cross-posting this question here from MSE: https://math.stackexchange.com/questions/4725734/characteristic-function-of-stochastic-integral-of-a-pure-jump-l%c3%a9vy-process-with. I apologize if ...

2
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0
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### Can a diffusion process admit an invariant measure with a non-differentiable density?

The precise domain of the generator $A$ of an Itō diffusion on a Hilbert space $H$ (assume $H=\mathbb R^d$, if that's easier for you to work with) can usually not be determined explicitly$^1$. Usually,...

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1
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### joint density of two relevant random variables

It seems that for most of the examples to derive the joint density of two or more random variables, the random variables themselves need to be independent. Is it possible to get the joint density of ...

1
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1
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### Integral of $M^\text{*} - M$ with respect to $M^\text{*}$ is zero for $M^\text{*}$ the running maximum of $M$ a continuous local martingale

Given $M$ a continuous local martingale, and $M^\text{*} = \sup_{0 \leq s \leq t} M_s$ its running maximum, we consider the finite variation integral
$$
I_T:= \int_0^T (M^\text{*}_s - M_s) \, \text{d}...

3
votes

0
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### 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 $...

6
votes

0
answers

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### Fokker-Planck equation for SDEs on manifold

Let $M_d$ be the set of $d\times d$ complex matrices and $S_d\subset M_d$ be its subset of density matrices, i.e. $A\in S_d$ iff $A\ge 0$, $A^*=A$ and $tr(A)=1$, where $A^*$ denotes the conjugate ...

3
votes

1
answer

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### Strong blow up limits for SDE

Note: This is a strengthening of the following result, motivated by the need for strong convergence in applications.
Let $W$ be a one dimensional standard Brownian motion, and let $X$ be the solution ...

1
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0
answers

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### Eigenvalues/eigenfunctions of a diffusion generator

Consider the following symmetric second order diffusion operator, defined, for $\phi \in \mathcal{C}^{2,1}_c\left(\mathbb{R}\times \mathbb{R}_+\right)$, by:
$$L\phi := \lambda_1 \partial_{R_1}(R_1 \...

5
votes

1
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### Does the entropy of a SDE with nondegenerate noise always increase?

Let $W$ be a standard Brownian motion, and let $X$ be the solution to the one dimensional SDE
$$dX_t = \sigma(t, X_t) \, dW_t$$
with initial condition $X_0 = x_0$ a.s. for some $x_0 \in \mathbb R$. We ...

1
vote

1
answer

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### Convergence of the quadratic variation process

Suppose we are given a sequence of stochastic processes $X^n, n\in\mathbb{N},$ with finite quadratic variations and a stochastic process $X$ such that for every $t\geq0$
$$
\lim_{n\to\infty}\mathbb{E}(...

2
votes

0
answers

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### Local martingale for a (two-dimensional) diffusion

Let $X$ be a two-dimensional diffusion (a solution of $dX_t=f(X_t)\,dt+dB_t$, with $B$ a standard two-dimensional Brownian motion) living on some open set $\Lambda\subset \mathbb{R}^2$. Let $h:\Lambda ...

2
votes

1
answer

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### Existence of solution for a non-linear SDE

Since $\exp(\cdot)$ is locally Lipschitz, the following SDE has a strong solution:
$$
\mathrm{d}X_s=\exp(X_s) \, \mathrm{d}B_s,\quad X_0=1,
$$
where $B$ is a standard Brownian motion. I wonder if the ...

0
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0
answers

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### Reference request: Gaussian estimates for SDE with discontinuous diffusion coefficient

Let $b:\mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ and $\sigma:\mathbb R_+ \times \mathbb R^d \to \mathcal M_{d \times d}^{\text{sym}} (\mathbb R)$ be bounded measurable where $\sigma$ is ...

4
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0
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### 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 ...

2
votes

0
answers

85
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### Feynman-Kac for PIDEs: to jump or not to jump?

Consider the following Cauchy problem for a $\mathscr{C}^2$ function $F$ characterized by a PIDE:
\begin{align}
\begin{cases}
& F_t(t,x)+\alpha(t,x)F_x(t,x)+\frac{1}{2}\beta^2(t,x)F_{xx}(t,x)
\\
&...

5
votes

0
answers

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### Feynman-Kac statement with no boundedness condition

Theorem 5.3 of Friedman (1975, Volume I) and its version in Theorem 7.6 of Karatzas & Shreve (1991) both establish conditions under which the Feynman-Kac formula holds, namely there is a ...

1
vote

1
answer

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### Phase space Brownian bridge

I understand the concept of the 1 dimensional Brownian bridge with the form of:
$$dx_t=\frac{-1}{1-t}x_t \, dt + dw_t$$
s.t. $x_0=0$ and $x_1=0$
where $dw_t$ is a Wiener process.
I am thinking about ...

6
votes

2
answers

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### Fractional Brownian motion of Riemann-Liouville type is not a semimartingale

Given a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ satisfying the usual conditions, $B$ a standard one-dimensional Brownian motion and $H\in(0,1/2)$. Consider the process $...

0
votes

0
answers

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### Langevin dynamics or stochastic gradient flow for grand canonical ensemble

We know that for a measure exp(-U(X)) (canonical ensemble), we can use the dynamic dX=-DU(X)+ noise to sample the measure as t goes to infinity.
Is there any dynamic corresponding to the grand ...

2
votes

1
answer

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### When does a solution to SDE have full support?

Suppose an $n$-dimensional process $(X_t)_{0 \leq t \leq 1}$ satisfies an SDE of the form:
$$dX_t = u_t(X_t) \,dt + dB_t, ~~X_0 = 0$$
where $(B_t)_{t\geq 0}$ is a Brownian motion with $B_1 \sim N(0,K)$...

2
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0
answers

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### Stochastic differential equations driven by composed Poisson process

Consider the stochastic differential equation as follows:
$$X_t = x + \int_0^t b(X_s)\,ds + \int_0^t a(X_{s-})\,dL_s,\quad \forall t\ge 0,$$
where $L=(L_t)_{t\ge 0}$ denotes some Lévy process. What ...

2
votes

1
answer

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### Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions?

$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\ \mathrm d}$Let
$(\Omega, \mathcal F, \mathbb P)$ be a probability space.
$B=(B^1, \ldots, B^N)$ independent one-dimensional Brownian motions.
$X=(X_0^...

2
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0
answers

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### Ito lemma for SDEs on a Lie group

I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group.
In doing so i'm trying to define a formula ...

2
votes

1
answer

122
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### Uniqueness of the solution to stochastic differential equation

Let $W$ be a Brownian motion and consider the SDE
$$dX_t = b(t,X_t) \, dt + a(t,X_t)\,dW_t,\quad \forall t\ge 0. \tag{$\ast$} $$
Assume that $x\mapsto b(t,x), a(t,x)$ are locally Lipschitz in $x$ ...

1
vote

1
answer

115
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### On a martingale defined via some SDE

Let $W$ be a one-dimensional Brownian motion. Consider the stochastic differential equation (SDE)
$$dX_t = C(t)(1-X_t)dW_t,\quad \forall t\ge 0,$$
where $C$ is a continuous and bounded function. Under ...

4
votes

1
answer

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### Finite number of ergodic random Dirac measures

Let $\Omega$ be a Polish locally compact space and $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Consider a measurable map
\begin{align*}
\theta\colon T\times \Omega &\to \Omega\\
(t,...

1
vote

1
answer

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### How to obtain this differential relation about moments of a stochastic process?

$\newcommand{\Ex}{\mathbb E}$ I'm reading an argument in the proof of Proposition 3.8. in the paper Nonlinear self-stabilizing processes - I Existence, invariant probability, propagation of chaos.
...

0
votes

0
answers

28
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### How can I obtain a SDE with an advection function that contains the difference in covariates?

Suppose that $\mathbf{s}(t)\in S$ denotes the spatial location of a process at time $t$. Further, let $\mathbf{x}(\mathbf{s}(t))$ denote covariates at the location $\mathbf{s}(t)$. My goal is to write ...

0
votes

0
answers

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### Can I use a derivative in my SDE's advection function?

Suppose that I have the following SDE:
$$\frac{d\mathbf{x}(t)}{dt}=\mathbf{f}(\mathbf{x}(t)) + \boldsymbol{\eta}(t),$$ where $\boldsymbol{\eta}(t)$ is white noise and $\mathbf{f}(\cdot)$ is an ...

1
vote

0
answers

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### Continuity in the uniform operator topology of a map

I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...

0
votes

1
answer

131
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### Existence of linear stochastic differential equation given solution

Normally if you have a linear SDE given such as
$dx_t = (A(t)x_t + a(t))dt + \sigma(t) dW_t$, we want to find $x_t$, more precisely we want to find the mean and variance of $x_t$ at each timestep $t$. ...