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# 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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### 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 ...
0answers
61 views

### 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, ...
1answer
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### 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 ...
0answers
58 views

### 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 ...
1answer
199 views

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 $$dg(t)... 0answers 35 views ### Sufficient conditions for taking limits in stochastic partial differential problems Let's say we have a (parabolic) Cauchy problem:$$ (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), $$... 0answers 49 views ### Lebesgue Integral in SDE In the context of proving existence of solutions of S(P)DEs, I've found that few (if any) texts offer significant mention to the deterministic drift term of the form$$ \int_0^t f(s,X(s))ds. $$If we ... 0answers 59 views ### Defining weak solutions to infinitely many SDEs on the same probability space Suppose I have an SDE of the form$$dX_t=b(X_t)dt+\sigma (X_t)dB_t+\int_{\mathbb{R}}G_{t-}(y)N(dtdy)$$which I can solve weakly if I cut off the last integral to range over the set \{\mid{y}\mid > ... 0answers 24 views ### Example of a “very noisy” SDE on a compact manifold with zero maximal Lyapunov exponent Setting: Let M be a compact connected C^\infty Riemannian manifold of dimension D \geq 2, with \lambda the normalised Riemannian volume measure. Write T_{\neq 0}M \subset TM for the non-... 0answers 103 views ### How to judge the solution process of an SDE to lie on the sphere? Consider the following SDE on \mathbf R^d: \begin{equation}\tag{*} dX_t^i = -\frac{d-1}{2}X_t^i dt + \sum_{j=1}^d(\delta^{ij}-X_t^iX_t^j)dW_t^j, \quad i=1,2,...,d, \end{equation} where W = (W^1,W^2,... 0answers 39 views ### Expected Solution of a Stochastic Differential Equation Expressed as Conditional Expectation On all you geniusses out there: this is a tough one. Preliminaries and Rigorous Technical Framework Let T \in (0, \infty) be fixed. Let d \in \mathbb{N}_{\geq 1} be fixed. Let$$(\Omega, \...
1answer
131 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 ...
0answers
57 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 ...
0answers
87 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) = \...
1answer
68 views

### Conditioning on an irrelevant variable in a martingale control problem

Suppose I have two independent Brownian motions $B^1_t, B^2_t$ and $\mathbb F_t$ be the natural filtration generated by them. Let $T > 0$ be a fixed finite number. Let $q_t$ be a $[-1,1]$ valued \$\...

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