# 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.

124 questions
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### Onsager-Machlup function for special matrix-valued diffusion process

Potentially useful background info For standard vector-valued diffusion processes the following result is well-known: Suppose we have a diffusion $X_{t}$ on $\mathbb{R}^{m}$ given by \begin{align*} ...
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### Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ is standard Wiener. This solution is ...
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### 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 ...
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### 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 ...
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### The existence of stationary measures for certain Markov process

My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...
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### Sobolev Bundle on Wiener Space

Right now I am learning about analysis of stochastic processes and the Malliavin calculus. It seems though, that most of the theory works for Brownian motion in $\mathbb{R}^n$, and it seems non-...
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### 2D Stochastic Navier Stokes equations with Navier boundary condition

For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u - u\cdot \nabla u - \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the ...
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### Gradient bound for the Markov semigroup generated by the solution to an Langevin SDE

Let $h\in C^2(\mathbb R)$ with $$h''\ge\rho\tag1$$ for some $\rho>0$ and $$\int\underbrace{e^{-h}}_{=:\:\varrho}\:{\rm d}\lambda=1$$ $\mu$ be the measure with density $\varrho$ with respect to the ...
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### Singular direction of a particle system

Consider a system of n-sdes in $\mathbb{R}$ ( the formula is not important). The corresponding particle system $X(t)=(X_{1}(t),X_{2}(t),...,X_{n}(t))$ lives in $\mathbb{R}^{n}$ and assume that ...
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### Stationary distribution of gradient dynamics

We consider the gradient dynamics $d X_{t} = d B_{t} - \nabla(U(X_{t}))dt$ in $\mathbb{R}^{d}$. G.Royer in the book "An initiation to logarithmic sobolev inequalities" (p29,30) says that if (1) U ...
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### Why does the correct scaled dimension for SPDEs count time as two dimensions?

In this video, Felix Otto says that the correct way to count dimensions for parabolic equations is $2+\text{number of space dimensions}$. He said nothing about this. In the accompanying notes it is ...
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### Locally Lipschitz sufficiently implies a Gronwall inequality?

In the paper [1], it seems to me the authors implicitly use a local Lipschitz property to deduce a Gronwall's inequality. I am not able to justify/show that this is indeed the case and perhaps someone ...
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### 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, ...
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### Has this type of pathwise (S)DE been studied before?

I thought of a possible type of pathwise-defined nonautonomous/stochastic differential equation, and I was wondering if it has been studied before. Let $(G,\ast)$ be an abelian $C^1$ Lie group....
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### Moment Estimate

Let $\epsilon > 0$ be a small parameter and consider the following lemma. Lemma. Let $B(t)$ be a bounded, continuous, $R^{n \times n}$-valued function defined on a time interval $[0,T]$ such that ...
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### Martingale covariation operator in infinite-dimensions

Let $(\Omega,\mathcal A,(\mathcal F_t)_{t\in[0,\:T]},\operatorname P)$ be a filtered probability space $U,H$ be separable $\mathbb R$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ and $(f_n)_{n\in\mathbb N}$...
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### Associative law of the stochastic integral in Hilbert spaces

Let $(\Omega,\mathcal A,\operatorname P)$ be a complete probability space $T>0$ $I:=(0,T]$ $(\mathcal F_t)_{t\in\overline I}$ be a complete and right-continuous filtration on $(\Omega,\mathcal A)$ ...
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### Domain of a Reflected SDE reference

I am currently investigating the domain of the infinitesimal generator of a reflected stochstic differential equation (for a smooth and bounded domain) with Lipshitz coefficients. Namely SDEs of the ...
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Given a diffusion process $X_t$ on a Riemannian manifold $(M,g)$, with an infinitesimal generator $\mathcal{G}=\Delta_g/2 + b$, the Onsager-Machlup function is well-known to be: $$\mathcal{L}(x,v) =... 0answers 224 views ### Construction of the quadratic variation for Hilbert space valued local martingales Let H be a separable \mathbb R-Hilbert space (e_n)_{n\in\mathbb N} be an orthonormal basis of H (\Omega,\mathcal A,\operatorname P) be a probability space (\mathcal F_t)_{t\ge0} be a ... 0answers 66 views ### Equivalence of Itō and Stratonovich equations and how we ensure that the latter are well-defined Remark: I've asked this question on MSE as well. Let T>0 I:=[0,T] (\Omega,\mathcal A,\operatorname P) be a probability space (\mathcal F_t)_{t\in I} be a complete and right-continuous ... 0answers 74 views ### Ito's formula for jump diffusions Suppose I have dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i where H_i(t) = \mathbb{1}_{\tau_t \leq t} denotes a default indicator process of i. \tau_i is the default time and h_i... 0answers 47 views ### Stochastic Control with Stochastic Cost-functional Is there any literature dealing with a stochastic control problem whose cost-functional J_t is stochastic also? That is, let X_t^u is the solution to a controlled SDE$$ dX_t = \mu(t,u_t,X_t^u)dt ...
I'm trying to transform the first order differential equation $\dot{x} = k x(t)$ into the corresponding set of stochastic differential equations. I have two independent uniformly distributed ...