Euler–Maruyama method

The Euler- Maruyama method, often called Euler - Maruyama scheme or stochastic Euler scheme is the simplest method for the numerical solution of stochastic differential equations. It was first investigated in the 1950s by the Japanese mathematician Gisiro Maruyama and is based on the originating by Leonhard Euler explicit Euler method for the solution of ordinary ( deterministic ) differential equations.

While the explicit Euler method since its invention constantly improved and developed was ( implicit Euler method, Runge- Kutta methods, multistep methods ) and even has thereby lost its practical importance is the Euler- Maruyama for lack of alternatives still in practice dominant process.

Formulation

Consider a Wiener process as well as to the following stochastic initial value problem (S- AWP ):

To calculate a numerical approximation on the interval with how the ordinary Euler method discrete time points

And with step size selected. In addition, the stochastic differential is determined by the growth

Replaced. From the properties of the Wiener process it follows that the are independent and normally distributed with mean and variance.

The Euler- Maruyama method is used to calculate an approximation of the following:

Then an approximation for.

Convergence of the method

The main theoretical result regarding the Maruyama scheme describes its strong convergence (or convergence in probability ) to the desired solution: A sequence of stochastic processes on a common probability space converges by definition strongly with order against a process when there is a constant such that for all:

In the case of Maruyama scheme can now be shown that the discretization converges strongly with order against the solution of the S- AWP, if for all real numbers and all positive, the following bound:

From weak or convergence in distribution with order one speaks, when applies to a constant:

For all functions that are at least twice continuously differentiable and their derivatives are all bounded by polynomials.

For sufficiently smooth coefficient functions and the Euler - Maruyama method typically has the weak convergence order.

Comments

• There is also a strong solution process of higher order than the Euler - Maruyama method, such as the Milstein method, which usually reaches order 1. However, these methods are numerically complex and not always result in a faster convergence.

• The above condition for the strong convergence of order 0.5 is only slightly more stringent than the condition of a and b, which ensures the existence of solution S. It is therefore almost always fulfilled.

• On strong convergence one is rarely interested in practice, since in most cases not a special solution is sought for a special Wiener process, but rather a sample from the probability distribution of the process as it is needed, for example, Monte - Carlo method.

• An implicit Maruyama scheme as analogous to the implicit Euler method is not possible; this is due to the definition of the ( stochastic ) Ito integrals are defined via the stochastic differential equations and the functions always at the beginning of an interval analyzes ( see below). So Implicit methods converge here to partially completely incorrect results.

• The simulation of a standard Brownian motion, by a Gaussian random walk can be interpreted as an application of the Euler Maruyama scheme is the simple equation.