InItô calculus, theEuler–Maruyama method (also simply called theEuler method) is a method for the approximatenumerical solution of astochastic differential equation (SDE). It is an extension of theEuler method forordinary differential equations to stochastic differential equations named afterLeonhard Euler andGisiro Maruyama. The same generalization cannot be done for any arbitrary deterministic method.^[1]

Consider the stochastic differential equation (seeItô calculus)

${\displaystyle \mathrm{d}{X}_t=a(X_t,t)\mathrm{d}t+b(X_t,t)\mathrm{d}{W}_t,}$

withinitial conditionX₀ = x₀, whereW_t denotes theWiener process, and suppose that we wish to solve this SDE on some interval of time [0, T]. Then theEuler–Maruyama approximation to the true solutionX is theMarkov chainY defined as follows:

• Partition the interval [0, T] intoN equal subintervals of width${\displaystyle \mathrm{\Delta }t>0}$:

• SetY₀ = x₀
• Recursively defineY_n for 0 ≤ n ≤ N-1 by

${\displaystyle Y_{n+1}=Y_n+a(Y_n,{\tau }_n)\mathrm{\Delta }t+b(Y_n,{\tau }_n)\mathrm{\Delta }{W}_n,}$

where

${\displaystyle \mathrm{\Delta }{W}_n=W_{{\tau }_{n+1}}-W_{{\tau }_n}.}$

Therandom variables ΔW_n areindependent and identically distributednormal random variables withexpected value zero andvariance Δt.

The Euler-Maruyama formula can be derived by considering the integral form of the Itô SDE

${\displaystyle X_{{\tau }_{n+1}}=X_{{\tau }_n}+{\int }_{{\tau }_n}^{{\tau }_{n+1}}a(X_s,s)ds+{\int }_{{\tau }_n}^{{\tau }_{n+1}}b(X_s,s)d{W}_s}$

and approximating${\displaystyle a(X_s,s)\approx a(X_n,{\tau }_n)}$ and${\displaystyle b(X_s,s)\approx b(X_n,{\tau }_n)}$ on the small time interval${\displaystyle [{\tau }_n,{\tau }_{n+1}]}$.

Like other approximation methods, the accuracy of the Euler–Maruyama scheme is analyzed through comparison to an underlying continuous solution.Let${\displaystyle X}$ denote an Itô process over${\displaystyle [0,T]}$, equal to

${\displaystyle X_t=X_0+{\int }_0^t\mu (s,X_s)ds+{\int }_0^t\sigma (s,X_s)d{W}_s}$

at time${\displaystyle t\in [0,T]}$, where${\displaystyle \mu }$ and${\displaystyle \sigma }$ denote deterministic "drift" and "diffusion" functions, respectively, and${\displaystyle W_t}$ is theWiener process.As discrete approximations of continuous processes are typically assessed through comparison between their respective final states at${\displaystyle T>0}$, a natural convergence criterion for such discrete processes is

${\displaystyle \underset{N\to \mathrm{\infty }}{lim}\mathbb{E}\left[\left|{\hat{X}}_N-X_T\right|\right]=0.}$

Here,${\displaystyle {\hat{X}}_N}$ corresponds to the final state of the discrete process${\displaystyle \hat{X}}$, which approximates${\displaystyle X_T}$ by taking${\displaystyle N}$ steps of length${\displaystyle \mathrm{\Delta }t=T/N}$.^[2]Iterative schemes satisfying the above condition are said to strongly converge to the continuous process${\displaystyle X}$, which automatically implies their satisfaction of the weak convergence criterion,

${\displaystyle \underset{N\to \mathrm{\infty }}{lim}\mathbb{E}\left[\left|g({\hat{X}}_N)-g(X_T)\right|\right]=0,}$

for any smooth function${\displaystyle g}$.^[3]More specifically, if there exists a constant${\displaystyle K}$ and${\displaystyle {\gamma }_s,{\delta }_0>0}$ such that

${\displaystyle \mathbb{E}\left[\left|{\hat{X}}_N-X_T\right|\right]\le K{\delta }_0^{{\gamma }_s}}$

for any${\displaystyle \delta \in (0,{\delta }_0)}$, the approximation converges strongly with order${\displaystyle {\gamma }_s}$ to the continuous process${\displaystyle X}$; likewise,${\displaystyle \hat{X}}$ converges weakly to${\displaystyle X}$ with order${\displaystyle {\gamma }_w}$ if the same inequality holds with${\displaystyle g({\hat{X}}_N)-g(X_T)}$ in place of${\displaystyle {\hat{X}}_N-X_T}$.Strong order${\displaystyle {\gamma }_s}$ convergence implies weak order${\displaystyle {\gamma }_w\ge {\gamma }_s}$ convergence:exemplifying this, it was shown in 1972^[4]that the Euler–Maruyama method strongly converges with order${\displaystyle {\gamma }_s=1/2}$ to any Itô process, provided${\displaystyle \mu ,\sigma }$ satisfy Lipschitz continuity and linear growth conditions with respect to${\displaystyle x}$, and in 1974, the Euler–Maruyama scheme was proven to converge weakly with order${\displaystyle {\gamma }_w=1}$ to Itô processes governed by the same such${\displaystyle \mu ,\sigma }$,^[5]provided that their derivatives also satisfy similar conditions.^[6]

A simple case to analyze isgeometric Brownian motion, which satisfies the SDE

${\displaystyle d{X}_t=\lambda X_{t}dt+\sigma X_{t}d{W}_t}$

for fixed${\displaystyle \lambda }$ and${\displaystyle \sigma }$. ApplyingItô’s lemma to${\displaystyle \ln X_t}$ yields the closed-form solution

${\displaystyle X_t=X_0\exp \left(\left(\lambda -\frac{1}{2}{\sigma }^2\right)t+\sigma W_t\right)}$

Discretising with Euler–Maruyama gives the time-step updates

${\displaystyle Y_{n+1}=\left(1+\lambda \mathrm{\Delta }t+\sigma \mathrm{\Delta }{W}_n\right)Y_n=Y_0\prod _{k=0}^n\left(1+\lambda \mathrm{\Delta }t+\sigma \mathrm{\Delta }{W}_k\right)}$

By using aTaylor series expansion of the exponential function in the analytic solution, we can get a formula for the exact update in a time-step.

Summing the local errors between the analytic and Euler-Maruyama solutions over each of the${\displaystyle N=T/\mathrm{\Delta }t}$ steps gives the strong error estimate

${\displaystyle \mathbb{E}\left[|X_T-Y_N|\right]=O\left(\sqrt{\mathrm{\Delta }t}\right)}$

confirming strong order${\displaystyle 1/2}$ convergence.

Another numerical aspect to consider is stability. The path's second moment is${\displaystyle \mathbb{E}|X_t{|}^2\propto \exp \left((2\lambda +{\sigma }^2)t\right)}$, so long-time decay of the solution occurs only when. The Euler–Maruyama scheme preserves variance decay in this case provided that${\displaystyle \mathrm{\Delta }t\le \frac{-1}{{\lambda }^2}\left(2\lambda +{\sigma }^2\right)}$.

An area that has benefited significantly from SDEs ismathematical biology. As many biological processes are both stochastic and continuous in nature, numerical methods of solving SDEs are highly valuable in the field.

• Numerical differential equations
• Stochastic differential equations
• Leonhard Euler

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