Solving Stochastic Ordinary Differential Equations by Monte Carlo and Polynomial Chaos

In this project, we study and compare two methods to solve stochastic ordinary differential equations. The first is the Monte Carlo method and the second uses Polynomial Chaos. In the first part, we will solve a stochastic ordinary differential equation by both a crude Monte Carlo method and a Quasi-Monte Carlo method. Convergence analysis of the two different methods is performed. Generation of samples according to different probability distributions is studied in detail. In the second part, we will approximate functions by orthogonal polynomial. Several classical orthogonal polynomials are introduced and the property of orthogonality is checked for the first few polynomials. Approximation for different functions leading to different convergence results is carried out. In particular, the Gibbs phenomenon is analyzed. This will be useful for the polynomial chaos expansion which approximate the solution of a stochastic ordinary differential equation by orthogonal polynomials and calculate its expectation by quadrature formula. We will give examples of several type of polynomial chaos and applies them to solve stochastic ordinary differential equations. These two methods being different, we are interested in study their rate of convergence. In fact, we will see that the Monte Carlo method has a polynomial convergence rate and the polynomial chaos achieves an exponential convergence rate for our test example.