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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?

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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
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migrated from math.stackexchange.com Nov 26 '13 at 13:38

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