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Suppose $b : \mathbb{R} \to \mathbb{R}$ and $\sigma: \mathbb{R}\to \mathbb{R}$ are Lipschitz and that $(X_t)_{t\ge0}$ is a diffusion with $X_0 = x_0$ and $dX_t = b(X_t)dt + \sigma(X_t)dW_t$ .

Consider the PDE $$b(x)v'(x) + \frac{\sigma(x)^2}{2}v''(x) = 0.$$

If $v$ is a classical supersolution to the problem, i.e. $v$ is twice differentiable and $$b(x)v'(x) + \frac{\sigma(x)^2}{2}v''(x) \le 0,$$ then $(v(X_t))_{t\ge 0}$ is a local supermartingale, by Ito's lemma.

In the case that $v$ is a lower semicontinuous viscosity supersolution, does this still hold?

Ben
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  • @ ben derrett : I think I remember that the two notions coincide under mild condition (implying the result you want) but I couldn't find exact reference, maybe you should review literature about Stochastic Optimal Control. Best regards – The Bridge Nov 22 '13 at 11:46
  • @TheBridge Thanks. Yes, they coincide if $v$ is twice differentiable. I haven't spotted this in my (limited) overview of the SOC literature. – Ben Nov 22 '13 at 13:08
  • migrated here by OP request. – Willie Wong Nov 26 '13 at 13:38
  • Just thinking out loud: Let $\phi$ be a test function such that $v-\phi$ has a local minimum at $(v-\phi)(x)=0$. Then, $$v(x)=\phi(x)=\mathbb{E}\left[\int_t^\tau (-\mathcal{L}\phi)(X_s) ds + \phi(X_\tau)\right] \geq \mathbb{E}[ \phi(X_\tau) ].$$ Because $v$ is LSC (and hence first Baire), we can approximate it by continuous functions from below. Can we pick this sequence of smooth functions $(\phi_m)_m$ so that we can plug them into this inequality and take limits? I would be interested in finding out if anybody has an idea to fill in the blanks. – parsiad Oct 05 '16 at 18:49

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