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There are very simple counterexamples to your claim. e.g. if $g\equiv-1$, $R_t=-t$, then the optimizer is $\tau=\infty$. Even if you assume $g, R \geq 0$ it does not hold, for instance taking $R_t=1$ for $t=0,1$ and $R_t=0$ for $t \geq 2$. Then the optimizer $\tau$ will be $0$ or $1$ but it will not be in general the hitting time of a singleton.
If the supremum on the lhs is attained at $t^*$, then the inequality is strict unless almost surely $G(t^*,X) = \sup_t G(t,X)$. So the obvious condition that the maximum is always attained at the same $t$ is also almost necessary.
@Bridge : what I am looking for is $W$ and $N$ being BM and Poisson wrt to the same filtration. So the fact that $W$ is BM when forgetting about $N$ is not enough. Do you think that this goal is achievable with a copula construction like you described ?
Thanks for the suggestion and the reference. I am not familiar with copulas so it is not very clear to me how your construction works. In the reference, there is no construction of a bivariate stochastic process from two independent processe and a fixed copula, but only construction of one process from marginals and copulas for $t_1,\ldots,t_k$. In particular it is not obvious to me why the "new" $N$ and $W$ remain Poisson and Brownian.
