Solving stochastic differential equations through genetic programming and automatic differentiation

This paper investigates the potential of evolutionary algorithms, developed using a combination of genetic programming and automatic differentiation, to obtain symbolic solutions to stochastic differential equations. Using the MATLAB programming environment and based on the theory of stochastic calculus, we develop algorithms and conceive a new methodology of resolution. Relative to other methods, this method has the advantages of producing solutions in symbolic form and in continuous time and, in the case in which an equation of interest is completely unknown, of offering the option of algorithms that perform the specification and estimation of the solution to the equation via a real database. The last advantage is important because it determines an appropriate solution to the problem and simultaneously eliminates the difficult task of arbitrarily defining the functional form of the stochastic differential equation that represents the dynamics of the phenomenon under analysis. The equation for geometric Brownian motion, which is usually applied to model prices and returns from financial assets, was employed to illustrate and test the quality of the algorithms that were developed. The results are promising and indicate that the proposed methodology can be a very effective alternative for resolving stochastic differential equations.