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This paper formulates algorithms to upper-bound the maximum Value-at-Risk(VaR) of a state function along trajectories of stochastic processes. The VaRis upper bounded by two methods: minimax tail-bounds(Cantelli/Vysochanskij-Petunin) and Expected Shortfall/ConditionalValue-at-Risk (ES). Tail-bounds lead to a infinite-dimensional Second OrderCone Program (SOCP) in occupation measures, while the ES approach creates aLinear Program (LP) in occupation measures. Under compactness and regularityconditions, there is no relaxation gap between the infinite-dimensional convexprograms and their nonconvex optimal-stopping stochastic problems. Upper-boundson the SOCP and LP are obtained by a sequence of semidefinite programs throughthe moment-Sum-of-Squares hierarchy. The VaR-upper-bounds are demonstrated onexample continuous-time and discrete-time polynomial stochastic processes.

Peak Value-at-Risk Estimation of Stochastic Processes using Occupation Measures