ItoProcess - Maple Help
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Finance

  

ItoProcess

  

create new Ito process

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

ItoProcess(, mu, sigma)

ItoProcess(, mu, sigma, x, t)

ItoProcess(X, Sigma)

Parameters

-

the initial value

mu

-

the drift parameter

sigma

-

volatility parameter

X

-

Vector of one-dimensional Ito processes

Sigma

-

matrix

Description

• 

The ItoProcess command creates a new one- or multi-dimensional Ito process, which is a stochastic process  governed by the stochastic differential equation (SDE)

where

– 

 is the drift parameter

– 

 is the diffusion parameter

and

– 

 is the standard Wiener process.

• 

The parameter  defines the initial value of the underlying stochastic process. It must be a real constant.

• 

The parameter mu is the drift. In the simplest case of a constant drift mu is real number (that is, any expression of type realcons). Time-dependent drift can be given either as an algebraic expression or as a Maple procedure. If mu is given as an algebraic expression, then the parameter t must be passed to specify which variable in mu should be used as a time variable. A Maple procedure defining a time-dependent drift must accept one parameter (the time) and return the corresponding value for the drift.

• 

The parameter sigma is the diffusion. Similar to the drift parameter, the volatility can be constant or time-dependent.

• 

One can use the ItoProcess command to construct a multi-dimensional Ito process with the given correlation structure. To be more precise, assume that  is an -dimensional vector whose components , ...,  are one-dimensional Ito processes. Let ,...,, and ,..., be the corresponding drift and diffusion terms. The ItoProcess(X, Sigma) command will create an -dimensional Ito process  such that

where  is an -dimensional Wiener process whose covariance matrix is Sigma. Note that the matrix Sigma must have numeric coefficients.

Examples

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You can generate sample paths for this stochastic process (in order to do this, we must assign numeric values to mu and sigma).

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Here is an example of a multi-dimensional Ito process.

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In this example, construct a two-dimensional Ito process using two one-dimensional projections and a given covariance matrix.

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References

  

Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.

  

Hull, J., Options, Futures, and Other Derivatives, 5th. edition. Upper Saddle River, New Jersey: Prentice Hall, 2003.

Compatibility

• 

The Finance[ItoProcess] command was introduced in Maple 15.

• 

For more information on Maple 15 changes, see Updates in Maple 15.

See Also

Finance[BlackScholesProcess]

Finance[CEVProcess]

Finance[Diffusion]

Finance[Drift]

Finance[ExpectedValue]

Finance[GeometricBrownianMotion]

Finance[ItoProcess]

Finance[PathPlot]

Finance[SamplePath]

Finance[SampleValues]

Finance[StochasticProcesses]

Finance[WienerProcess]

 


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