📂Stochastic Differential Equations

Euler-Maruyama Method Derivation

Methods ¹

$$d X(t) = f \left( X_{t} \right) dt + g \left( X_{t} \right) d W_{t} \qquad , t \in [t_{0}, T]$$ Let’s say that the Itô process is given as a solution to the autonomous stochastic differential equation as shown above. For the equidistant points $\left\{ t_{i} \le T : t_{i+1} = t_{i} + h \right\}_{i=0}^{N}$ with a constant interval of $h$, $Y_{i} := Y \left( t_{i} \right)$ calculated as follows is the numerical solution to the given differential equation. $$Y_{i+1} = Y_{i} + f \left( Y_{i} \right) h + g \left( Y_{i} \right) \sqrt{h} Z$$ Here, $Z$ is a random variable that follows the standard normal distribution.

Convergence

This solution converges strongly to the $\gamma = 1/2$th order and weakly to the $\beta = 1$th order.

Description

The Euler-Maruyama Approximation Scheme is the simplest method for numerically solving stochastic differential equations, conceptually no different from the Euler method for solving ordinary differential equations, and essentially just involves looking at the equation and writing down the simulation as it is.

Derivation

$$X_{t+h} = X_{t} + \int_{t}^{t+h} f(s) ds + \int_{t}^{t+h} g(s) d W_{s}$$ Let’s consider the integral form of the Itô process. The drift term is $$\int_{t}^{t+h} f(s) ds \approx f \left( X_{t} \right) \int_{t}^{t+h} ds = f \left( X_{t} \right) h$$ and the diffusion term is $$\begin{align*} \int_{t}^{t+h} g(s) dW_{s} \approx& g \left( X_{t} \right) \int_{t}^{t+h} dW_{s} \\ =& g \left( X_{t} \right) \left( W_{t+h} - W_{t} \right) \\ =& g \left( X_{t} \right) \Delta W_{t} \end{align*}$$ Since the increment of the Wiener process $W_{t+h} - W_{t}$ follows the normal distribution $N \left( 0, h \right)$, it is $\Delta W_{t+h} \sim \sqrt{h} N \left( 0 , 1 \right)$. Therefore, for the random variable $Z$ that follows the standard normal distribution, it can be represented as follows. $$X_{i+h} = X_{i} + f \left( X_{i} \right) h + g \left( X_{i} \right) \sqrt{h} Z$$

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