# 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.

268
questions

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### 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 ...

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### Is this a “contradiction” on stochastic Burgers' equation? How to understand it?

For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...

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### The probability distribution of “derivative” of a random variable

Disclaimer: Cross-posted in math.SE.
Let me set the stage;
Consider a stochastic PDE, which has to following form
$$\partial_t h(x,t) = H(x,t) + \chi(x,t),$$
where $H$ is a deterministic function, ...

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82 views

### Backward Stochastic Differential Equation

Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and
$$
dX_t=f_tdt+B_tdW_t
$$
where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated ...

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### Extension of probability space problem: Hilbert space valued process V.S. random field

Maybe the question should be "Understanding the measurability: Hilbert space valued process V.S. random field"
Consider the SPDE $${\rm d}u+\cdots{\rm d}t=\sigma(t,u){\rm d}W.$$
Consider the ...

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**1**answer

206 views

### An application of Itô's formula to an SDE on a Lie group

I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...

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73 views

### Sufficient conditions for taking limits in stochastic partial differential problems

Let's say we have a Cauchy problem:
$$
(1) \hspace{0.5cm} u_t (x,t)+A(u) \cdot u_x (x,t)=\nu \cdot u_{xx} (x,t) + \epsilon \cdot f(u) \cdot W, $$
$$(2) \hspace{0.5cm} u(x,0)=u_0(x),
$$
where $x \in ...

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50 views

### Lebesgue Integral in SDE

In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form
$$
\int_0^t f(s,X(s))ds.
$$
If we ...

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61 views

### Defining weak solutions to infinitely many SDEs on the same probability space

Suppose I have an SDE of the form
$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$
which I can solve weakly if I cut off the last integral to range over the set $\{\mid{y}\mid > ...

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### Example of a “very noisy” SDE on a compact manifold with zero maximal Lyapunov exponent

Setting:
Let $M$ be a compact connected $C^\infty$ Riemannian manifold of dimension $D \geq 2$, with $\lambda$ the normalised Riemannian volume measure.
Write $T_{\neq 0}M \subset TM$ for the non-...

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106 views

### How to judge the solution process of an SDE to lie on the sphere?

Consider the following SDE on $\mathbf R^d$:
\begin{equation}\tag{*}
dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d,
\end{equation}
where $W = (W^1,W^2,...

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### Expected Solution of a Stochastic Differential Equation Expressed as Conditional Expectation

On all you geniusses out there: this is a tough one.
Preliminaries and Rigorous Technical Framework
Let $T \in (0, \infty)$ be fixed.
Let $d \in \mathbb{N}_{\geq 1}$ be fixed.
Let $$(\Omega, \...

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**1**answer

133 views

### Simulation of Itô integral processes where integrand depends on terminal (Volterra process)

I need to simulate a process of the form
$$X_t=\int_0^t f(s,t)\mathop{dW_s}$$
where $f$ is deterministic and the integral is an It? integral. I know I can simply take finite It? sums of discrete ...

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60 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 ...

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92 views

### Why should we give special attention to at most polynomially growing solutions of PDEs?

The equation
\begin{gather}
\frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\
u(0,x) = \...