Hypercomplexification of SDEs Exact integration of some systems of stochastic differential equations through hypercomplexification

Abstract

We leverage commutative hypercomplex analysis to find closed-form solutions of some systems of stochastic differential equations. Specifically, we obtain necessary and sufficient conditions under which a system of stochastic differential equations can be transformed into a scalar one involving processes valued in a commutative hypercomplex. In the event the targeted scalar stochastic differential equation is solved by quadratures, we recover the solution of the original system by projecting the solution of the scalar stochastic differential equation along the units of the underlying commutative hypercomplex. The conversion of a system of stochastic differential equations involving real-valued processes into a scalar one written in terms of hypercomplex-valued processes is termed hypercomplexification. Both hypercomplexification and its reverse are mediated by the analyticity of stochastic differential equations data. They may be iterated in order to generate higher-dimensional integrable systems of stochastic differential equations and solve them. We showcase the utility of hypercomplexification by treating several examples including linear, and linearizable systems of stochastic differential equations and stochastic Lotka-Volterra systems. Although we consider only random systems driven by white noises, hypercomplexification is fundamentally algebraic, and it readily extends to stochastic systems involving other types of disturbances.