Milstein method

The Milstein method of stochastic analysis describes a method for the numerical solution of stochastic differential equations (SDGL), named after the Russian mathematician Grigori Noichowitsch Milstein ( Gorky State University of the Urals ).

algorithm

Look at the Itō SDGL

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

with initial condition , denoting the Wiener process . If a solution is to be found on the interval , the Milstein method gives an approximation for the true solution on an equidistant grid: ${\displaystyle X_{0}=x_{0}}$ ${\displaystyle W_{t}}$ ${\displaystyle [0,T]}$ ${\displaystyle Y}$ ${\displaystyle X}$

• Break down the interval into equally long sub-intervals of length : ${\displaystyle [0,T]}$ ${\displaystyle N}$ ${\displaystyle \delta >0}$

${\displaystyle 0=\tau _{0}<\tau _{1}<\dots <\tau _{N}=T}$and . ${\displaystyle \delta ={\tfrac {T}{N}}}$

• Set . ${\displaystyle Y_{0}:=x_{0}}$

• Define for through ${\displaystyle Y_{n+1}}$ ${\displaystyle 0\leq n<N}$

${\displaystyle Y_{n+1}:=Y_{n}+a(Y_{n})\delta +b(Y_{n})\Delta W_{n}+{\frac {1}{2}}b(Y_{n})b'(Y_{n})\left((\Delta W_{n})^{2}-\delta \right),}$

in which

${\displaystyle \Delta W_{n}=W_{\tau _{n+1}}-W_{\tau _{n}}}$

and the derivation of relative is. Note that the random variables are independently normally distributed with expectation value 0 and variance . ${\displaystyle b'}$ ${\displaystyle b(x)}$ ${\displaystyle x}$ ${\displaystyle \Delta W_{n}}$ ${\displaystyle \delta }$

convergence

With the above notation applies to and all , which is why one speaks of first- order convergence . is a Landau symbol . ${\displaystyle E[|Y_{n}-X(\tau _{n})|]={\hbox{o}}(\delta )\;}$ ${\displaystyle \delta \to 0}$ ${\displaystyle n=0,...,N}$ ${\displaystyle {\hbox{o}}}$

See also

• Euler-Maruyama method

credentials

• Peter E. Kloeden, Eckhard Platen : Numerical Solution of Stochastic Differential Equations . Springer, Berlin, 1999, ISBN 3-540-54062-8 .