create new Poisson process
algebraic expression; intensity parameter
algebraic expression; jump size distribution
A Poisson process with intensity parameter 0<λ⁡t, where λ⁡t is a deterministic function of time, is a stochastic process N with independent increments such that N⁡0=0 and
for all 0≤t. If the intensity parameter λ⁡t itself is stochastic, the corresponding process is called a doubly stochastic Poisson process or Cox process.
A compound Poisson process is a stochastic process J⁡t of the form J⁡t=∑i=1N⁡t⁡Yi, where N⁡t 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.
J := PoissonProcess(1.0):
PathPlot(J(t), t = 0..3, timesteps = 50, replications = 20, thickness = 3, color = red..blue, axes = BOXED, gridlines = true, markers = false);
Create a subordinated Wiener process with J as a subordinator.
W := WienerProcess(J):
PathPlot(W(t), t = 0..3, timesteps = 20, replications = 10, markers = false, color = red..blue, thickness = 3, gridlines = true, axes = BOXED);
Next define a compound Poisson process.
Y := Statistics[RandomVariable](Normal(.3, .5)):
lambda := 0.5;
X := PoissonProcess(lambda, Y):
PathPlot(X(t), t = 0..3, timesteps = 20, replications = 10, markers = false, color = red..blue, thickness = 3, gridlines = true, axes = BOXED);
Compute the expected value of X⁡T for T=3 and verify that this is approximately equal to λ⁢T times the expected value of Y.
T := 3;
ExpectedValue(X(T), replications = 10^4, timesteps = 100);
Here is an example of a doubly stochastic Poisson process for which the intensity parameter evolves as a square-root diffusion.
kappa := 0.354201;
mu := 1.21853;
nu := 0.538186;
y0 := 1.81;
y := SquareRootDiffusion(y0, kappa, mu, nu):
J := PoissonProcess(y(t)):
PathPlot(y(t), t = 0..3, timesteps = 100, replications = 10, thickness = 3, color = red..blue, axes = BOXED, gridlines = true);
PathPlot(J(t), t = 0..3, timesteps = 100, replications = 10, thickness = 3, color = red..blue, axes = BOXED, gridlines = true);
Glasserman, P., Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag, 2004.
The Finance[PoissonProcess] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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