# 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.

364 questions
Filter by
Sorted by
Tagged with
0answers
20 views

0answers
32 views

### Quadratic variation of generalized stochastic integrals

My question is based on this paper: https://pdfs.semanticscholar.org/0b5a/e41096a3b16d0756a1d36da55143d861ed7c.pdf. In summary, this talks about the generalization of stochastic integrals to a two ...
1answer
170 views

### Path integral presentation of solutions of Dirac equation

It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold). Is there a way to present solutions of the Dirac equation using path ...
1answer
82 views

### Tightness on a set $A$ implies tightness on a set $B$ where $A\subset B$?

From the book Billingsley - Convergence of probability measures, 1999, we have the following definitions of tightness and relative compactness and the Prohorov's theorem: Tightness: Let $\Pi$ be a ...
0answers
58 views

1answer
271 views

0answers
27 views

### Generator of a Hilbert space valued Wiener process from the solution of a martingale problem

Let $H$ be a separable $\mathbb R$-Hilbert space, $Q\in\mathfrak L(U)$ be nonnegative and self-adjoint with $\operatorname{tr}Q<\infty$ and $(W_t)_{t\ge0}$ be a $H$-valued Wiener process on a ...
0answers
79 views

### Estimating the hitting time for a SDE solution

Consider a the following OU process in one dimension, $$dX = -\theta(X -x_0)dt + \sqrt{s}dW$$ Now one can define the time $t_x$ as the time it takes for the solution to reach the point $x$. Then ...
0answers
33 views

### Convergence of the probability that hitting times being infinity

Let $X^n=(X^n_t)_{t\ge 0}$ and $X=(X_t)_{t\ge 0}$ be RCLL (right-continuous with left limits) processes such that $$\lim_{n\to\infty}X^n=X,\quad \quad \mbox{almost surely},$$ where this convergence ...
0answers
88 views

### Path dependent Markov property

Let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded \begin{align*} \Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty) \end{align*} Then my question is:...
1answer
68 views

### Show an SDE's solution has positive probability to visit every set in the state space

Let $(\Omega, \mathcal{F},\mathbb{P})$ be a filtered probability space, let $b:[0,T]\times \mathbb{R}^n\to \mathbb{R}^n$ be a continuous function and Lipschitz continuous in the space variable. For ...
0answers
211 views

### Probability of a particle surviving forever

Consider a particle whose position is driven by the following equation: $$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$ where $y>0$, $0<C<1$...
0answers
52 views