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Maoduan Ran (talk | contribs) Tag: section blanking |
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This family <math>\mathbb{P}^{\xi,\tau}_{a;b}</math> is called the <math>\mathcal{L}_{a;b}</math>-diffusion, where <math>\mathcal{L}_{a;b} = L_{a;b} + \frac{\partial}{\partial t}</math> is the time‐dependent infinitesimal generator.
It is clear that if we have an <math>L_{a;b}</math>-diffusion, i.e. <math>(X_t)_{t \ge 0}</math> on <math>(\Omega, \mathcal{F}, \mathcal{F}_t, \mathbb{P}^{\xi,\tau}_{a;b})</math>, then <math>X_t</math> satisfies the SDE <math>dX_t^i = \sqrt{2v}\,\sum_{k=1}^d \sigma^i_k(X_t)\,dB_t^k + b^i(X_t)\,dt</math>. In contrast, one can construct this diffusion from that SDE if <math>a^{ij}(x,t) = \sum_k \sigma^k_i(x,t)\,\sigma^k_j(x,t)</math> and <math>\sigma^{ij}(x,t)</math>, <math>b^i(x,t)</math> are Lipschitz continuous.
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