t | \mathcal{F}_s] \le X_s\), representing "unfavorable" or decreasing processes. Optional Stopping and Stopping Times A stopping time \(\tau\) is a random time at which a decision is made based on current info
bability measures and integrals, setting the stage for advanced stochastic analysis. Martingales, introduced in the 20th century by Jean Ville and later formalized by Paul Lévy, emerged as a powerful class of stochastic processes embodyi
d Sequential Analysis In sequential hypothesis testing, martingale techniques assist in controlling error probabilities and designing efficient procedures. Williams's contributions improve the robustness of these methods. Stochastic Control an
ifically, for a Brownian filtration, every martingale \( M_t \) admits a representation: \[ M_t = M_0 + \int_0^t \phi_s \, dB_s \] where \( \phi_s \) is an adapted process satisfying integrability conditions.
