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Finance

  

PoissonProcess

  

create new Poisson process

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

PoissonProcess(lambda)

PoissonProcess(lambda, X)

Parameters

lambda

-

algebraic expression; intensity parameter

X

-

algebraic expression; jump size distribution

Description

• 

A Poisson process with intensity parameter 0<λt, where λt is a deterministic function of time, is a stochastic process N with independent increments such that N0&equals;0 and

PrNt+hNt=1|Nt&equals;lambdath&plus;oh

  

for all 0t. If the intensity parameter λt itself is stochastic, the corresponding process is called a doubly stochastic Poisson process or Cox process.

• 

A compound Poisson process is a stochastic process Jt of the form Jt&equals;i&equals;1NtYi, where Nt is a standard Poisson process and Yi are independent and identically distributed random variables. A compound Cox process is defined in a similar way.

• 

The parameter lambda is the intensity. It can be constant or time-dependent. It can also be a function of other stochastic variables, in which case the so-called doubly stochastic Poisson process (or Cox process) will be created.

• 

The parameter X is the jump size distribution. The value of X can be a distribution, a random variable or any algebraic expression involving random variables.

• 

If called with one parameter, the PoissonProcess command creates a standard Poisson or Cox process with the specified intensity parameter.

Examples

withFinance&colon;

JPoissonProcess1.0&colon;

PathPlotJt&comma;t&equals;0..3&comma;timesteps&equals;50&comma;replications&equals;20&comma;thickness&equals;3&comma;color&equals;red..blue&comma;axes&equals;BOXED&comma;gridlines&equals;true&comma;markers&equals;false

Create a subordinated Wiener process with J as a subordinator.

WWienerProcessJ&colon;

PathPlotWt&comma;t&equals;0..3&comma;timesteps&equals;20&comma;replications&equals;10&comma;markers&equals;false&comma;color&equals;red..blue&comma;thickness&equals;3&comma;gridlines&equals;true&comma;axes&equals;BOXED

Next define a compound Poisson process.

YStatistics&lsqb;RandomVariable&rsqb;Normal0.3&comma;0.5&colon;

&lambda;0.5

&lambda;:=0.5

(1)

XPoissonProcess&lambda;&comma;Y&colon;

PathPlotXt&comma;t&equals;0..3&comma;timesteps&equals;20&comma;replications&equals;10&comma;markers&equals;false&comma;color&equals;red..blue&comma;thickness&equals;3&comma;gridlines&equals;true&comma;axes&equals;BOXED

Compute the expected value of XT for T&equals;3 and verify that this is approximately equal to λT times the expected value of Y.

T3

T:=3

(2)

ExpectedValueXT&comma;replications&equals;104&comma;timesteps&equals;100

value&equals;0.4435049150&comma;standarderror&equals;0.007163535555

(3)

&lambda;TStatistics&lsqb;ExpectedValue&rsqb;Y

0.45

(4)

Here is an example of a doubly stochastic Poisson process for which the intensity parameter evolves as a square-root diffusion.

&kappa;0.354201

&kappa;:=0.354201

(5)

&mu;1.21853

&mu;:=1.21853

(6)

&nu;0.538186

&nu;:=0.538186

(7)

y01.81

y0:=1.81

(8)

ySquareRootDiffusiony0&comma;&kappa;&comma;&mu;&comma;&nu;&colon;

JPoissonProcessyt&colon;

PathPlotyt&comma;t&equals;0..3&comma;timesteps&equals;100&comma;replications&equals;10&comma;thickness&equals;3&comma;color&equals;red..blue&comma;axes&equals;BOXED&comma;gridlines&equals;true

PathPlotJt&comma;t&equals;0..3&comma;timesteps&equals;100&comma;replications&equals;10&comma;thickness&equals;3&comma;color&equals;red..blue&comma;axes&equals;BOXED&comma;gridlines&equals;true

References

  

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

Compatibility

• 

The Finance[PoissonProcess] 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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