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

\[\begin{align*} Y(t_0)=X_0 \text{ and } Y(t_i)=Y(t_{i-1})+f(Y(t_{i-1}),t_{i-1})\cdot\Delta t \end{align*}\]

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

\[\begin{align*} d X(t)= f(X(t)) dt + g(X(t)) dB(t) \end{align*}\]

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

\[\begin{align*} Y(t_0)=X_0 \text{ and } Y(t_i)=Y(t_{i-1})+f(X(t_i))\cdot\Delta t + g(X(t_i))\cdot\Delta B_i \end{align*}\]

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