Wald's equation holds for discrete stochastic processes. Is there a version of such equation for continuous stochastic processes?
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On Wald's Equations in Continuous Time (1970) summarizes and extends Wald's equation in discrete time to continuous time.
answered Apr 12, 2019 at 18:46
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1$\begingroup$ Two comments: (1) For the Brownian motion, this is a very classical fact, and can be proved by standard approximation of a stopping time by 'discretized' stopping times (taking finitely many values), using the discrete Wald's identity, and passing to the limit. I am sure numerous textbooks on Brownian motion include this result. $\endgroup$ Apr 12, 2019 at 18:53
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1$\begingroup$ (2) For general local martingales $M_t$, $\mathbb{E} M_\tau = 0$ is a consequence of the optional stopping theorem (given appropriate integrability conditions, of course), while $\mathbb{E} M_\tau^2 = \mathbb{E} \mathbb \langle M\rangle_\tau$ is a consequence of $M_t^2 - \langle M\rangle_t$ being a martingale. $\endgroup$ Apr 12, 2019 at 18:53