On explicit solutions to Ito diffusions

TL;DR

This paper characterizes when stochastic differential equations have explicit solutions, linking their structure to differential equations and commutation relations, and provides construction methods with applications in filtering and finance.

Contribution

It establishes a three-way equivalence for explicit solutions of Ito diffusions and introduces construction theorems using diffeomorphisms for simpler solution forms.

Findings

Explicit solutions exist iff the coefficients satisfy commutation relations.

Construction theorems enable explicit solution derivation via diffeomorphisms.

Applications demonstrated in filtering and option pricing.

Abstract

Strong solutions of p-dimensional stochastic differential equations that can be represented locally in explicit simulation form are considered. The following three-way equivalence is established: 1) There exists such a representation from all starting points, 2) the representation pair satisfies a set differential equations, and 3) the stochastic differential equation coefficients satisfy commutation relations. Next, construction theorems, based on a diffeomorphism between the original equation solutions and the strong solutions to a simpler Ito integral equation, with a possible deterministic component, are given. Finally, motivating examples are provided and reference to its importance in filtering and option pricing is given.