## Introduction

In this post I am going to explain the Euler-Maruyama method for approximating stochastical differential equations.

## Euler method

First, the Euler method. Suppose you have a (deterministic) ordinary differential equation \begin{align*} X'(t)=f(X(t),t) \end{align*} where $X'$ is the derivative of $X$, $f$ is a computable function and the domains of definition are so that everything “works fine”. Suppose we want to approximate the solution on the closed interval $[a,b]$. We also need to have a boundary condition, e.g., that $X(a)=X_0$ to ensure uniqueness of the solution.

Let us now approximate the result. First choose a partitioning of the interval $[a,b]$. We take contant stepsizes, but you can also use any other partitioning. So choose a $n>0$ “big” and then define the stepsize as $\Delta t:=\frac{b-a}{n}$ and the different steps as $t_i:=a+i\cdot \Delta t$ for $0\leq i\leq n$.

We know from our boundary condition that $X(t_0)=X(a)=X_0$.
What is now a meaningful value for $X(t_1)=X(t+\Delta t)$?
We know that $X'(t_0)=f(X(t_0),t_0)=f(X_0,t_0)$ which we can compute. Thus we can approximate the “real” solution $X(t)$ in a neighborhood of $t_0$ by a line. This works in a practical sense if $n$ was chosen “big enough”, since then $\Delta t$ is small.

So a good guess for $X(t_1)$ would be $X(t_1)=X(t_0)+f(X_0,t_0)\cdot\Delta t$. We take the correct value and gradient at $t_0$ and walk $\Delta t$. More generally we can define an approximation $Y(t)$ of $X(t)$ via

If this is not clear to you I suggest you stop here and draw a picture.

## Euler-Maruyama method

The Euler-Maruyama method is for SDEs the same as the Euler method is for ODEs. Suppose we have the SDE

where $B(t)$ is brownian motion.

We want to do the same, namely approximate $X(t)$ on the closed interval $[a,b]$. Again we need a boundary condition like $X(a)=X_0$. And we also take a steplength $\Delta t:=\frac{b-a}{n}$ and define the different points as $t_i:=a+i\cdot \Delta t$ for $0\leq i\leq n$. Basically we do everything exactly as before. We can also again approximate $X(t_i)$ in a small neighborhood by a line but this time the gradient of the line is stochastical in nature. So we get as approximation the markov chain

where $\Delta B_i=B(t_i)-B(t_i-1)$ which are i.i.d. random variables which follow a $\mathcal{N}(0,\Delta t)$ distribution.

What does Markovian mean in this context? It means that for the conditional expected values $E(Y(t_i)|Y(t_l),l\leq i)=E(Y(t_i)|Y(t_{i-1}))$ so the next expected value depends only on the previous one.

14 July 2015

Lesson