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Maoduan Ran (talk | contribs) |
Maoduan Ran (talk | contribs) |
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Let <math>E = \mathbb{R}^d</math> be the state space with Borel <math>\sigma</math>-algebra, and let <math>\Omega = C([0,\infty), \mathbb{R}^d)</math> denote the canonical space of continuous paths. A family of probability measures <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> (for <math>\tau \geq 0</math>, <math>\xi \in \mathbb{R}^d</math>) solves the diffusion problem for coefficients <math>a^{ij}(x,t)</math> (uniformly continuous) and <math>b^i(x,t)</math> (bounded, Borel measurable) if:
<math>\mathbb{P}^{\xi,\tau}_{a;b}\big( \psi \in \Omega : \psi(t) = \xi \text{ for } 0 \leq t \leq \tau \big) = 1.</math>
<math>M_t^{[f]} = f(\psi(t), t) - f(\psi(\tau), \tau) - \int_{\tau}^t \left( L_{a;b} + \frac{\partial}{\partial s} \right) f(\psi(s), s) \, ds</math>
is a local martingale under <math>\mathbb{P}^{\xi,\tau}_{a;b}</math>, where
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