# 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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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### 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 ...
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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 ...
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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 $t\geq0$ $$\lim_{n\to\infty}\mathbb{E}(... 2 votes 0 answers 87 views ### Local martingale for a (two-dimensional) diffusion Let X be a two-dimensional diffusion (a solution of dX_t=f(X_t)\,dt+dB_t, with B a standard two-dimensional Brownian motion) living on some open set \Lambda\subset \mathbb{R}^2. Let h:\Lambda ... 2 votes 1 answer 184 views ### Existence of solution for a non-linear SDE Since \exp(\cdot) is locally Lipschitz, the following SDE has a strong solution:$$ \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 votes 0 answers 63 views ### 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 votes 0 answers 146 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 ... 2 votes 0 answers 85 views ### 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 53 views ### 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 86 views ### 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$$s.t. x_0=0 and x_1=0 where dw_t is a Wiener process. I am thinking about ... 6 votes 2 answers 257 views ### Fractional Brownian motion of Riemann-Liouville type is not a semimartingale Given a filtered probability space (\Omega,\mathcal{F},\mathbb{F},\mathbb{P}) satisfying the usual conditions, B a standard one-dimensional Brownian motion and H\in(0,1/2). Consider the process ... 0 votes 0 answers 28 views ### Langevin dynamics or stochastic gradient flow for grand canonical ensemble We know that for a measure exp(-U(X)) (canonical ensemble), we can use the dynamic dX=-DU(X)+ noise to sample the measure as t goes to infinity. Is there any dynamic corresponding to the grand ... 2 votes 1 answer 195 views ### When does a solution to SDE have full support? Suppose an n-dimensional process (X_t)_{0 \leq t \leq 1} satisfies an SDE of the form:$$dX_t = u_t(X_t) \,dt + dB_t, ~~X_0 = 0$$where (B_t)_{t\geq 0} is a Brownian motion with B_1 \sim N(0,K)... 2 votes 0 answers 59 views ### Stochastic differential equations driven by composed Poisson process Consider the stochastic differential equation as follows:$$X_t = x + \int_0^t b(X_s)\,ds + \int_0^t a(X_{s-})\,dL_s,\quad \forall t\ge 0,$$where L=(L_t)_{t\ge 0} denotes some Lévy process. What ... 2 votes 1 answer 276 views ### Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions? \newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\ \mathrm d}Let (\Omega, \mathcal F, \mathbb P) be a probability space. B=(B^1, \ldots, B^N) independent one-dimensional Brownian motions. X=(X_0^... 2 votes 0 answers 159 views ### Ito lemma for SDEs on a Lie group I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group. In doing so i'm trying to define a formula ... 2 votes 1 answer 122 views ### Uniqueness of the solution to stochastic differential equation Let W be a Brownian motion and consider the SDE$$dX_t = b(t,X_t) \, dt + a(t,X_t)\,dW_t,\quad \forall t\ge 0. \tag{$\ast$} $$Assume that x\mapsto b(t,x), a(t,x) are locally Lipschitz in x ... 1 vote 1 answer 115 views ### On a martingale defined via some SDE Let W be a one-dimensional Brownian motion. Consider the stochastic differential equation (SDE)$$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 123 views ### 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 56 views ### 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 views ### 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 36 views ### 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),$$where \boldsymbol{\eta}(t) is white noise and \mathbf{f}(\cdot) is an ... 1 vote 0 answers 49 views ### Continuity in the uniform operator topology of a map I have a question concerning the continuity for t>0 in the uniform operator topology L(X) of the following map:$$t\mapsto A^\alpha R(t) where A is the infinitesimal generator of an analytic ...
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 $t$. ...