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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1 vote
0 answers
29 views

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 $(W_t)_t$ ...
3 votes
1 answer
651 views

Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined

Remark: I've asked this question on MSE as well. Let $T>0$ $I:=[0,T]$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\in I}$ be a complete and right-continuous ...
4 votes
1 answer
447 views

A variant to the Fokker–Planck equation

Consider the PDE of $p(t,x)\ge 0$ given as $$\partial_t p = \frac{\partial^2_{xx}p}{(1+m(t))^2} - \partial_x p,\quad \forall t,~x \in (0,\infty)$$ with initial and boundary conditions $p(0,\cdot)=\rho$...
0 votes
1 answer
211 views

Can we define the divergence of a stochastic process?

Suppose I have a stochastic process $(X_t)_{t\in \mathbb{R}^d}$ with infinitesimal generator $\mathcal{A}$, for example $\mathcal{A}f(X) = -\mu f'(X) + \frac{1}{2}\sigma^2f''(X)+\lambda \int (f(X')-f(...
1 vote
0 answers
42 views

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 ...
1 vote
1 answer
280 views

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 $t$. ...
1 vote
0 answers
56 views

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)) :...
1 vote
0 answers
97 views

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 ...
2 votes
0 answers
138 views

Time reversal of infinite-dimensional SDE

Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
6 votes
0 answers
70 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 $n$-...
3 votes
0 answers
69 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-\|...
2 votes
1 answer
190 views

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 ...
3 votes
1 answer
181 views

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 ...
4 votes
1 answer
141 views

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 ...
2 votes
1 answer
349 views

A mean field SDE with hitting time

Let $b\in \mathbb R$ and $\sigma>0$ be given. For a fixed probability distribution $\mu_0$ on $\mathbb R$ s.t. $$\int_{(0,\infty)}\mu_0(dx)=1,$$ consider the mean field SDE : $$dX_t = \mathbf{1}_{\...
3 votes
0 answers
78 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. ...
4 votes
1 answer
554 views

Random time change from Oksendal's SDE textbook

I have two questions related to the random time change introduced in Oksendal's textbook on SDEs (page 155-156). Specifically, for Lemma 8.5.6., I have no clue as to why we should define $t_j$ in ...
1 vote
1 answer
394 views

How to calculate the probability of 2 events happening in time series under only cdf information?

In time domain $0\rightarrow T$, there are two independent events $A$ and $B$. $B$ follows Poisson Process with density $\lambda$. It's easy to get $P_B(t)$ which denotes $P_B(N(\tau+t)-N(\tau)\geq 1)...
4 votes
1 answer
176 views

Invariant measure theory for SPDEs with distributional solutions (Hilbert versus Polish)

SPDEs such as the stochastic heat equation for $d\geq 2$ with space-time white noise and the stochastic quantization equation have distributional solutions and we still try to make sense of their ...
2 votes
1 answer
502 views

Stationary distribution of overdamped Langevin dynamics

Consider the over damped Langevin dynamics: $d X_{t} = d B_{t} - \nabla U(X_{t}) dt $ on $\mathbb{R}^{d}$ where $B_t$ is a standard Brownian motion. On pages 29 and 30 of the following book Royer,...
2 votes
1 answer
209 views

Smoothness of expectation

Suppose that $X_t$ is a strong solution to the SDE, $$dX_t = C_t \,dB_t$$ where $B_t$ is a standard Brownian motion and $C_t \ge 0$ is measurable with respect to the natural filtration generated by ...
2 votes
0 answers
57 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\...
7 votes
1 answer
363 views

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 $\sigma(b)=0$. ...
1 vote
1 answer
123 views

Let $(X, W)$ be a weak solution to a SDE. Is $W$ a Brownian motion w.r.t. $\sigma(X_s : s \le t)$?

Let $(X, W)$, $(\Omega, \mathcal{F}, \mathbb{P})$, $\{\mathcal{F}_t\}$ be a weak solution to an SDE. Per definition $W$ is an $\mathcal{F}_t$-Brownian motion and both $X$, $W$ are adapted to $\mathcal{...
0 votes
1 answer
192 views

Solving SDE with sign function in drift term?

Consider the following SDE with $X_0 = 1$, $$ dX_t = X_t\operatorname{sign}(X_t) \, dt + X_t \, dW_t, $$ where $\operatorname{sign}(x) = \mathbb{1}\{x \ge 0\}$. How am I supposed to solve this SDE?
2 votes
0 answers
69 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 ...
3 votes
1 answer
282 views

Core of the generator of squared bessel process in $L^2(\mathbb{R}_+)$

Consider the squared bessel process with generator $$Gf(x)=xf''(x)+f'(x), \ \ x\in\mathbb{R}_+.$$ It is known that the Lebesgue measure is an invariant measure for this process and thus, can be ...
3 votes
2 answers
215 views

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 ...
5 votes
2 answers
1k views

Textbooks or lecture notes about mean field games

I am looking for a good introductory level textbook (or lecture notes) on mean field games that would be suitable for a graduate course. Ideally, it would include some brief words about optimal ...
5 votes
1 answer
311 views

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 ...
1 vote
0 answers
62 views

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. ...
2 votes
0 answers
83 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(�...
3 votes
0 answers
173 views

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 ...
2 votes
0 answers
55 views

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 ...
2 votes
0 answers
193 views

Algebraic normalisation of regularity structures: can there be a explicit expression of g?

This is related to Bruned, Hairer, and Zambotti - Algebraic renormalisation of regularity structures. In the method of re-normalization the functional $g$ shown in page 6 plays a major role. However, ...
2 votes
1 answer
481 views

Time interval of existence of an SDE solution with locally Lipschitz drift

Consider the stochastic ODE $$ dX = F(X) \, dt + dB $$ where $B$ is Brownian motion. If the drift $F$ is locally Lipschitz, then the solution exists and is unique over $[0,T]$ where $T$ is an "...
5 votes
1 answer
551 views

Differentiable dependence on the initial condition of the solution of a SDE

Let $b,\sigma:\mathbb R\to\mathbb R$ be differentiable and Lipschitz continuous $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a complete and right-...
4 votes
1 answer
185 views

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 $d$-...
3 votes
0 answers
72 views

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 ...
7 votes
2 answers
452 views

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}\...
23 votes
5 answers
3k views

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
198 views

Weighted Lebesgue space with exponential weights: smoothing effect and properties

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

SDE driven by fractional Brownian motion

Let $B^H$ be a fraction Brownian motion of Hurst parameter $H$. Consider the SDE driven by $B^H$ as below: $$dX_t = b(t,X_t)dt + a(t,X_t)dB^H_t,\quad \forall t\ge 0.$$ I am looking for references that ...
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 ...
3 votes
1 answer
2k views

Expected value of a stochastic integral expression

I am wondering if the following expression can be processed a bit analytically, $$ E \left[ e^{aX} \int_0^X e^{bu}dW(u)\right], $$ where $W_u$ is the normal Brownian motion (1D Wiener process), and $...
3 votes
1 answer
203 views

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
146 views

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 ...
1 vote
0 answers
139 views

Marcus-SDE to Itô-SDE

In the field of stochastic calculus, everyone knows the Itô and Stratonovich integrals, as well as the conversion from Stratonovich to Itô SDEs. The Stratonovich integration has the particularity of ...
1 vote
0 answers
65 views

Estimating the entropy of the solution to an SDE

Forgive me for the poorly researched question. I'm currently working on a computer science project involving training a neural stochastic differential equation, and I've run into a problem while ...
0 votes
1 answer
368 views

Can derivatives of 2 stochastic processes be multiplied?

We understand SDEs like "$dX_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$" for Brownian process $B$ to be formally the same as "$\frac{dX_t}{dt} = b(t,X_t) + \sigma(t,X_t)W_t$" where $W$ is ...

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