Modern Stochastics: Optional Processes and their Applications

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The cornerstone of modern stochastic analysis is a probability space equipped with filtration as a non-decreasing family of sigma-algebras. The well-developed theory of stochastic processes assumes "usual conditions", when filtration is complete and right-continuous. This theory generated many important results in probability theory, statistics, mathematical finance etc. In 1975 famous experts in stochastic processes Doob and Dellacherie initiated studies of stochastic processes without this technical assumption. Further developments were done by Lepingle, Horowitz, Lenglart, and mostly by Galtchouk. Their parallel version of stochastic analysis deals with optional processes admitting regular trajectories. The existence of such initial theory calls for a new initiative for its further developments and applications today with its new challenges. The goal of the proposal is to take a new look at optional processes bringing new methods, techniques and results to mathematical finance and related areas. In the proposal, we are going to investigate stochastic differential equations with respect to optional semimartingales in the sense of existence of strong solutions and their path-wise comparison properties. The results will be applied to approximate option price bounds and other financial quantities in the markets driven by optional processes. The problem of approximate pricing will be also investigated with the help of extensions of probability distributions of stock returns using orthogonal polynomials and the Pade rational approximations. We are going to use the technique of optional processes in option pricing problem in the area of mergers and acquisitions, where jump processes promise to create an adequate pricing model. We investigate a possibility to obtain a version of the uniform Doob-Meyer decomposition of optional supermartingales. Its fundamental role in mathematical finance is well-established due to its application to superhedging problem in incomplete markets, markets with transaction costs and other market restrictions. We also want to show how this decomposition can be exploited to construct an optimal filter in the filtering problem for optional semimartingales which covers many well-known models. Besides optional decomposition, we will derive a version of the Galtchouk-Kunita-Watanabe representation for optional martingales with further applications to mean-variance hedging problem. Another fundamental problem known in mathematical finance as insider trading will be treated based on the calculus of optional processes. The parameter estimation problem for optional semimartingales will be investigated. These results are reasonable to provide an adequate calibration in the markets driven by optional processes and to create a general framework for many regression models exploited in mathematical finance and statistics. The Proposal is wide enough to accommodate a number of student research projects of master's and PhD levels.