Use TransformedProcess to Create a Custom Process 

Define compensated Poisson process.

In[1]:=

X poisson\[ScriptCapitalP] = PoissonProcess[\[Lambda]]; \[ScriptCapitalP]1 = TransformedProcess[p[t] - Mean[poisson\[ScriptCapitalP][t]], p \[Distributed] poisson\[ScriptCapitalP], t];

Simulate the process with unit rate for 30 seconds.

In[2]:=

X data1 = RandomFunction[\[ScriptCapitalP]1 /. \[Lambda] -> 1, {0, 30, 0.1}, 3];

In[3]:=

X ListLinePlot[data1]

Out[3]=

Compute the covariance function for the process.

In[4]:=

X CovarianceFunction[\[ScriptCapitalP]1, s, t]

Out[4]=

Define a Bessel process in two dimensions.

In[5]:=

X \[ScriptCapitalP]2 = TransformedProcess[Sqrt[ x[t]^2 + y[t]^2], {x \[Distributed] WienerProcess[], y \[Distributed] WienerProcess[]}, t];

Simulate the process for 10 seconds.

In[6]:=

X data2 = RandomFunction[\[ScriptCapitalP]2, {0, 10, 0.01}, 3];

In[7]:=

X ListLinePlot[data2, PlotRange -> All]

Out[7]=

Compute the mean and variance functions for the process.

In[8]:=

X {Mean[\[ScriptCapitalP]2[t]], Variance[\[ScriptCapitalP]2[t]]}

Out[8]=