Stochastic processes form a key methodology for analyzing random interactions in today's financial and information systems. This project incorporates the fields of stochastic analysis, stochastic comparison, and convex analysis in order to develop new mathematical methods for understanding, computing, and optimizing stochastic processes related to discontinuous and nonlinear models in finance and engineering. Expected theoretical contributions of the project include approximation formulas for complex functionals of Lévy processes, new comparison techniques for stochastic processes using binary relations, algorithms for finding sharp reversible bounds for Markov processes, and new existence and uniqueness criteria for general stochastic optimization problems. Expected contributions to applied sciences include quantitative methods for pricing of financial instruments in illiquid markets, criteria for the existence of arbitrage in financial markets with small transaction costs, and nearly optimal control of data networks.
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