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This thesis is devoted to the study of extreme value statistics in stochasticprocesses and their applications. In the first part, we obtain exact analyticalresults on the extreme value statistics of both discrete-time andcontinuous-time random walks. In particular, we focus on the gap statistics ofrandom walks and exhibit their asymptotic universality with respect to the jumpdistribution in the limit of a large number of steps. In addition, we computethe asymptotic behavior of the expected maximum of random walks in the presenceof a bridge constraint and reveal a rich behavior in their finite-sizecorrection. Moreover, we compute the expected length of the convex hull ofBrownian motion confined in a disk and show that it converges slowly to theperimeter of the disk with a stretched exponential decay. In the second part,we focus on numerically sampling rare trajectories of stochastic processes. Weintroduce an efficient method to sample bridge discrete-time random walks. Weillustrate it and apply it to various examples. We further extend the method toother stochastic processes, both Markovian and non-Markovian. We apply ourmethod to sample surviving particles in the presence of a periodic trappingenvironment. Finally, we discuss several optimization problems in stochasticprocesses involving extreme value statistics. In particular, we introduce a newtechnique to optimally control dynamical systems undergoing a resetting policy.

PhD thesis "Extreme value statistics and optimization problems in stochastic processes"