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Fix $\rho>0, C_2>C_1>0$ real numbers.

Assume $dX_t=b(X_t,t)dt+\sigma(X_t,t)dW_t$ where $W_t$ is the standard Brownian motion, $X_0=x$, and $b_1\leq s_1\leq b_2\leq s_2\ldots\to\infty$ be a sequence of stopping times.

And let $\alpha=(b_1,s_1,b_2,s_2\ldots)$, now define $$ R(x,\alpha)=\mathbb{E}\Big[\sum_{n=1}^{\infty} e^{-\rho s_n}e^{X_{s_n}}C_1-e^{-\rho b_n}e^{X_{b_n}}C_2 \Big] $$

and let $v(x)=\sup_{\alpha}R(x,\alpha)$

My question is : what is the dynamical programming principle associated to $v(x)$ in this set up ?

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