Martingale Problem

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(Classical Martingale Problem) Given a (local) linear operator $\mathcal{L}: C^2(\mathbb{R}^n) \to C(\mathbb{R}^n)$ Martingale Problem for $\mathcal{L}$ consists in finding for each $x_0 \in \mathbb{R}^d$ a probability measure $\mathbb{P}^{x_0}$ over the space of all continuous functions $X: [0,+\infty) \to \mathbb{R}^d$ such that

\begin{equation*} \mathbb{P}^{x_0}\left ( X(0)=x_0 \right ) = 1 \end{equation*}

and whenever $f \in C^2(\mathbb{R}^2)$ we have that

\begin{equation*} f(X(t))-f(X(0))-\int_0^t \mathcal{L} f(X(s)) \;ds \end{equation*}

is a Local Martingale under $\mathbb{P}^{x_0}$