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$ \newcommand{\RR}{\mathbb{R}} \newcommand{\TT}{\mathbb{T}} \newcommand{\NN}{\mathbb{N}} \newcommand{\PP}{\mathbb{P}} \newcommand{\EE}{\mathbb{E}} \newcommand{\FF}{\mathbb{F}} \newcommand{\PPP}{\...

Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE \begin{equation} dX_t = \mu(X_t)\,dt + \sigma(X_t)\,dW_t\,. \end{equation} Let $f(x,t|x_0,0)$ denote its transition density function. ...

Let $X=(X_t)_{t\in I}$ ($I\subset\mathbb{R}$ a (non-degenerate) interval) be a stochastic process with continuous sample paths and such that $X_t$ admits a continuous Lebesgue density $\chi_t\in C(\...

Let $$\Gamma(f,g):=\frac12f'g'\;\;\;\text{for }f,g\in C^1(\mathbb R),$$ $\mu$ be a probability measure on $(\mathbb R,\mathcal B(\mathbb R))$ with a continuously differentiable and positive density $\...

What are the SDE's satisfied by the following processes? $X_t = B_t^q$ $X_t = (\sin B_t)^q$ $X_t = B_t^q (\sin B_t)^r$ Assume $B_t$ is a standard Brownian motion with $B_0 > 0$ and the equations ...