Financial markets with discontinuities

University Of California-Santa Barbara, Santa Barbara CA

Investigators

Abstract

Ichiba 1313373 The main objective of the project is to develop new stochastic models for important financial systems such as equity, foreign exchange, bond, and credit markets through stochastic differential equations (SDEs) with discontinuous coefficients and with degeneracies. Discontinuous coefficients introduce a rich family of probabilistic models that capture abrupt transitions from one phase to another phase of random phenomena. The degeneracy of the diffusion coefficients brings to the system some interesting mixtures of stochastic dynamics and deterministic dynamics in the neighborhood around the domain of degeneracy. A prototypical tractable example is a system of SDEs with piece-wise constant drift/diffusion coefficients, which can be applied to financial mathematics in the context of stochastic portfolio theory. The investigator tackles new problems in SDEs with piece-wise constant coefficients, works towards more general models with degeneracy and discontinuous coefficients, and applies these mathematical results to improve understanding of the complicated global financial system. The problems include boundary behaviors, ergodicity, large deviations, weak/strong solvability of forward and backward SDEs with discontinuous coefficients and boundary conditions, large-scale optimal portfolio choice, and long-term stochastic optimizations. The investigator studies financial systems by developing and analyzing new models of stochastic dynamics that may capture elastic changes and aberrant risky phenomena in the system. Those anomalies are caused by uncertainty inherent in the financial system. It is important to understand the properties of the models and to propose good safeguards, in order to increase efficiency of the financial system and to prevent its failure. Another feature of the proposed work is to capture, describe, and predict the evolution of the financial system. The greater understandings of the financial market mechanism that result from these studies can potentially increase efficiency of long-term financial investments. In addition, mathematical methods developed in the project may contribute to several areas of mathematics -- partial differential equations, combinatorics, geometry, and general dynamical systems.