Euler–Maruyama method

In <a href="/facts/It%C3%B4_calculus/2uSR02fY">Itô calculus</a>, the Euler–Maruyama method (also simply called the Euler method) is a method for the approximate <a href="/facts/Numerical_analysis/QT9cYa9z">numerical solution</a> of a <a href="/facts/Stochastic_differential_equation/ol9a9WNq">stochastic differential equation</a> (SDE). It is an extension of the <a href="/facts/Euler_method/7yF5B4z9">Euler method</a> for <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary differential equations</a> to stochastic differential equations named after <a href="/facts/Leonhard_Euler/z2fsbjgj">Leonhard Euler</a> and <a href="/facts/Gisiro_Maruyama/G0FlIv81">Gisiro Maruyama</a>. The same generalization cannot be done for any arbitrary deterministic method.
Consider the stochastic differential equation (see <a href="/facts/It%C3%B4_calculus/2uSR02fY">Itô calculus</a>)

d
        
        
          X
          
            t
          
        
        =
        a
        (
        
          X
          
            t
          
        
        ,
        t
        )
        
        
          d
        
        t
        +
        b
        (
        
          X
          
            t
          
        
        ,
        t
        )
        
        
          d
        
        
          W
          
            t
          
        
        ,
      
    
    {\displaystyle \mathrm {d} X_{t}=a(X_{t},t)\,\mathrm {d} t+b(X_{t},t)\,\mathrm {d} W_{t},}

with <a href="/facts/Initial_condition/sA4ohFP4">initial condition</a> X0 = x0, where Wt denotes the <a href="/facts/Wiener_process/KPh7vYd7">Wiener process</a>, and suppose that we wish to solve this SDE on some interval of time [0, T]. Then the Euler–Maruyama approximation to the true solution X is the <a href="/facts/Markov_chain/5oP5hntk">Markov chain</a> Y defined as follows:

<ul><li>Partition the interval [0, T] into N equal subintervals of width 
 
 
 
 Δ
 t
 >
 0
 
 
 {\displaystyle \Delta t>0}
 
:</li></ul>

0
 =
 
 τ
 
 0
 
 
 <
 
 τ
 
 1
 
 
 <
 ⋯
 <
 
 τ
 
 N
 
 
 =
 T
 
  and 
 
 Δ
 t
 =
 T
 
 /
 
 N
 ;
 
 
 {\displaystyle 0=\tau _{0}<\tau _{1}<\cdots <\tau _{N}=T{\text{ and }}\Delta t=T/N;}

<ul><li>Set Y0 = x0</li>
<li>Recursively define Yn for 0 ≤ n ≤ N-1 by</li></ul>

Y
          
            n
            +
            1
          
        
        =
        
          Y
          
            n
          
        
        +
        a
        (
        
          Y
          
            n
          
        
        ,
        
          τ
          
            n
          
        
        )
        
        Δ
        t
        +
        b
        (
        
          Y
          
            n
          
        
        ,
        
          τ
          
            n
          
        
        )
        
        Δ
        
          W
          
            n
          
        
        ,
      
    
    {\displaystyle \,Y_{n+1}=Y_{n}+a(Y_{n},\tau _{n})\,\Delta t+b(Y_{n},\tau _{n})\,\Delta W_{n},}

where

Δ
        
          W
          
            n
          
        
        =
        
          W
          
            
              τ
              
                n
                +
                1
              
            
          
        
        −
        
          W
          
            
              τ
              
                n
              
            
          
        
        .
      
    
    {\displaystyle \Delta W_{n}=W_{\tau _{n+1}}-W_{\tau _{n}}.}

The <a href="/facts/Random_variable/TwTBXnLT">random variables</a> ΔWn are <a href="/facts/Independent_and_identically_distributed/othIRaWt">independent and identically distributed</a> <a href="/facts/Normal_distribution/UapjjPyQ">normal random variables</a> with <a href="/facts/Expected_value/1XV0JKL8">expected value</a> zero and <a href="/facts/Variance/ULBJKXD1">variance</a> Δt.

Euler–Maruyama method open-in-new

Euler–Maruyama method