All Questions

Let $V: \mathbb R^n \to \mathbb R^n$ be a Lipschitz vector field. Consider a one dimensional Brownian motion $W$ and the SDE $$dX_t = V(X_t) \, dW_t,$$ where we identify $V(X_t) \in \mathbb R^n$ with ...

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ is standard Wiener. This solution is ...

Consider $\partial_{t}u=\partial_{xx}u$ with Neumann boundary condition $u_{x}(0,t)=u_{x}(1,t)=0$ and initial condition $u(x,0)=f(x)\geqslant0$. Then up to time $T$, the maximal value of $u$ should be ...

Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...