All Questions

Many books [see below for references] explore the connections between partial differential equations and expectation values. Assume $X$ is a diffusion with generator $A$, then they conclude, that ...

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...

Consider the diffusion process $$ d X = \mu(X, t) dt + \sigma(X, t) dY. $$ When $Y$ is a Brownian motion, we know that the density follows the Fokker-Planck equation. Here I'm considering the general ...

$X_t$ is an Itô diffusion process with continuous version, $\mathbb{L}_X$ is its generator. $D$ is a closed set in $\mathbb{R}$. The stopping time $\tau_D$ is the first entry time of $D$, that is $\...

Consider the scalar diffusion $X=(X_t)_{t\geq 0}$ given by $$\mathrm{d}X_t \ = \ b(X_t)\,\mathrm{d}t \ + \ \sigma(X_t)\,\mathrm{d}B_t, \qquad X_0=\xi,$$ with $b$, $\sigma$ smooth, $\xi$ absolutely ...

Consider the following $n$-dimensional Ito-SDE: \begin{align} dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t \end{align} What are the necessary regularity conditions on $\mu$ and $\sigma$ to ensure that the ...