# 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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:...
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### 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 ...
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 $y>0$, $0<C<1$...