imported>Nestor at 00:48, 20 November 2012

2012-11-20T00:48:59Z

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 {{stub}}
-(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
+(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*}
@@ Line 7: / Line 7: @@
 \end{equation*}
-and whenever $f \in C^2(\mathbb{R}^2)$ we have that
+and whenever $f \in C^2(\mathbb{R}^d)$ we have that
 \begin{equation*}
@@ Line 13: / Line 13: @@
 \end{equation*}
-is a [[Martingale| Local Martingale]] under $\mathbb{P}^{x_0}$
+is a [[Martingale| Local Martingale]] under $\mathbb{P}^{x_0}$. If for any $x \in \mathbb{R}^d$ there exists a unique $\mathbb{P}^x$ satisfying the above conditions we say that the Martingale Problem for $\mathcal{L}$ is well posed.

imported>Nestor: Created page with "{{stub}} (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 fin..."

2012-11-20T00:35:47Z

Created page with "{{stub}} (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 fin..."

New page

{{stub}}

(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 [[Martingale| Local Martingale]] under $\mathbb{P}^{x_0}$

Martingale Problem - Revision history

imported>Nestor at 00:48, 20 November 2012

imported>Nestor: Created page with "{{stub}} (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 fin..."