# SuanShu, a Java numerical and statistical library

com.numericalmethod.suanshu.stats.stochasticprocess.univariate.sde.discrete

## Class MilsteinSDE

• java.lang.Object
• com.numericalmethod.suanshu.stats.stochasticprocess.univariate.sde.discrete.MilsteinSDE
• All Implemented Interfaces:
DiscreteSDE

public class MilsteinSDE
extends Object
implements DiscreteSDE
Milstein scheme is a first-order approximation to a continuous-time SDE. It adds a term to the Euler scheme by expanding both the drift and diffusion terms to O(dt). $dX_t = \mu * dt + \sigma * \sqrt{dt} * Z_t + \frac{1}{2} \frac{d\sigma}{dt} * \sigma * dt * (Z_t^2 - 1)$
Wikipedia: Milstein method
• ### Constructor Summary

Constructors
Constructor and Description
MilsteinSDE(SDE sde)
Discretize a continuous-time SDE using the Milstein scheme.
• ### Method Summary

All Methods
Modifier and Type Method and Description
double db(Ft ft)
$\frac{d\sigma}{dt}$
double dXt(Ft ft)
This is the SDE specification of a stochastic process.
Ft getNewFt()
Get an empty filtration of the process.
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### MilsteinSDE

public MilsteinSDE(SDE sde)
Discretize a continuous-time SDE using the Milstein scheme.
Parameters:
sde - a continuous-time SDE
• ### Method Detail

• #### dXt

public double dXt(Ft ft)
This is the SDE specification of a stochastic process.

This is an implementation of the Milstein scheme. $dX_t = \mu * dt + \sigma * \sqrt{dt} * Z_t + \frac{1}{2} \frac{d\sigma}{dt} * \sigma * dt * (Z_t^2 - 1)$

Specified by:
dXt in interface DiscreteSDE
Parameters:
ft - a filtration
Returns:
the increment of the process in dt
• #### getNewFt

public Ft getNewFt()
Description copied from interface: DiscreteSDE
Get an empty filtration of the process.
Specified by:
getNewFt in interface DiscreteSDE
Returns:
an empty filtration
• #### db

public double db(Ft ft)
$\frac{d\sigma}{dt}$
Parameters:
ft - a filtration
Returns:
$$\frac{d\sigma}{dt}$$