When we hear symmetries we think of geometry. A symmetry of a shape is a manipulation or transformation of the shape the preserves said shape. However, for an equation a symmetry is transformation that transforms one solution into another solution. By determining a symmetry of an equation we are able to obtain unknown solutions, from known ones, that might otherwise be difficult to obtain.
In the late 19th century Sophus Lie developed the theory of symmetries for a particular type of equation called partial differential equations. Partial differential equations are used to model various physical phenomena and understanding their symmetries leads to a powerful tool for their study.
While the symmetries of Partial Differential Equations have been well studied since Sophus Lie’s time, there is another type of equation whose symmetries are known to a lesser degree. These equations are known as stochastic differential equations and can be considered partial differential equations with random component. This means that the solution to a stochastic differential equation is a random process that returns random values every time it is given an input.
Immediately one can see that the problem of determining symmetries to stochastic differential equations is harder than that of partial differential equations. However, there is a system (more than one that must be satisfied at the same time) of partial differential equations that is associated with any given stochastic differential equation. This system of partial differential equations is called the Itô system and it provides a specific form to the solution of it’s corresponding stochastic differential equation.
It is via the Itô system of a stochastic differential equation that we may examine the symmetries of said stochastic differential equation. Indeed, we may ask what happens to the solution of a stochastic differential equation when one applies a symmetry to the corresponding Itô system?
University of Technology Sydney