# 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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### Stochastic stability of “open” continuous-time stochastic systems: reference request

I'm looking for results on the stability of stochastic systems, e.g. SDEs, whose coefficients depend on a different process that is not necessarily stable. I'm calling those systems "open" here, but ...

**2**

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**1**answer

455 views

### Existence of a solution to an infinite dimensional Stratonovich SDE

Let
$U,H$ be separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with finite trace
$U_0:=Q^{1/2}U$
$(\Omega,\mathcal A,(\mathcal F_t)_{t\ge 0},\operatorname P)$ ...

**1**

vote

**1**answer

528 views

### Properties of the trace term in the Itō formula

Let's consider the SDE $${\rm d}X_t=u_t(X_t){\rm d}t+\xi_t(X_t){\rm d}W_t\;\;\;\text{for all }t\ge 0\tag 1$$ where
$U,H$ are separable $\mathbb R$-Hilbert spaces
$Q\in\mathfrak L(U)$ is nonnegative ...

**20**

votes

**1**answer

452 views

### Does a theory of stochastic differential algebras exist?

My question is motivated primarily by finance, where a non-technical student will learn how to approach SDEs using the symbolic manipulation of Itô calculus and the few basic rules of Brownian motion, ...

**5**

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**3**answers

306 views

### Perturbation of a stochastic differential equation

Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively
\begin{align}
dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\
dx &= -(k_0(t)+\epsilon ...

**2**

votes

**0**answers

100 views

### Gaussian free field limiting distribution of additive Stochastic heat eqn bounded domain

Hairer in his spdes notes on pg.6, says that GFF is the stationary solution of $u_{t}(z)=\Delta u(z)+\xi(z,t)$, where
$\xi$ is the space-time white noise
$$\xi(x,t)=\sum \sqrt{\lambda_{k}} B_{k}(t)e_{...

**3**

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**1**answer

433 views

### Strong solution for geometric brownian motion with varying drift and volatility

I have an equation of the form:
$$dX_{t}=\mu(X_{t})X_{t}dt+\sigma(X_{t})X_tdZ_{t}$$
I know that if I wrote it as $dX_{t}=\mu(X_{t})dt+\sigma(X_{t})dZ_{t}$, I would need strong assumptions on the ...

**5**

votes

**2**answers

313 views

### Reference for Feynman-Kac

I would like to have a reference with more in deep explanation of Feynman-Kac than in Evan's An Introduction to Stochastic Differential Equations and, if possible, example of solution for equations ...

**3**

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**1**answer

165 views

### Conditional stochastic integration

Let's say we have two functions $h(s)$ and $g(s)$. We can easily simulate a stochastic integral, e.g.
$$t \mapsto \int_0^t h(s) dB(s) \sim \mathcal{N}\bigg(0, \int_0^t h(s)^2 ds \bigg). $$
What is the ...

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99 views

### Singularity of the solution of a PDE whose coefficients have zeros

The following PDE arises in a problem of finding the stationary measure of a 2d system of stochastic differential equations (see this math.stackexchange post):
$$\mathcal{A}p=0, \quad p\in C^2(\...

**1**

vote

**1**answer

214 views

### Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces

I'm reading the paper On the relation between the Itō and Stratonovich integrals in Hilbert spaces and there is something I don't understand.
In the notation of the paper, let
$H,H_1$ be separable $\...

**1**

vote

**1**answer

164 views

### Reflected SDE with non-Lipschitz coefficients

I have an equation of the form:
$$dX_t=\mu(X_t)dt+\sigma(X_t)dZ_t+dL_t, \quad X_0=x_0\in (-\infty,a]$$
where, $L_t$ is the reflection function (as in Skorokhod, 1961). This reflection does not allow ...

**5**

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**0**answers

243 views

### Derivation of a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories

Let
$d\in\left\{2,3\right\}$
$\mathcal V_t\subseteq\mathbb R^d$ be the bounded domain occupied by an incompressible Newtonian fluid at time $t\ge 0$
$\Phi_t:\mathcal V_0\to\mathcal V_t$ such that $\...

**5**

votes

**1**answer

237 views

### Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?

Many books [see below for references] explore the connections between partial differential equations and expectation values.
Assume $X$ is a diffusion with generator $A$, then they conclude, that ...

**3**

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**0**answers

91 views

### Regularity of martingales with respect to spatial parameters

In Stochastic Flows and Stochastic Differential Equations, Kunita is proving in Theorem 3.1.2 that a family $M(t,x)$ of continous local martingales depending on a spatial parameter $x$ takes values in ...

**3**

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**1**answer

152 views

### Walk with randomised boosts

The classical random walk can be described as the evolution of the position $X_t$ of a walker for integers $t \geqslant 0$, where $X_0 = 0$ and $X_t = X_{t-1} + V_t$ for $t \geqslant 1$, where the "...

**2**

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**0**answers

182 views

### Definition of the Stratonovich integral in Hilbert spaces

Let
$T>0$
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$\mathcal F=(\mathcal F_t)_{t\in[0,\:T]}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$
$B$ be a (standard, real-...