Vector Ito process

Questions
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1

Hello,

I would like to simulate an Ito process on a vector, where the drift and the volatility differ, but the results of the simulation are both based on the same realization of a (0,1) wiener process. I tried this one:

proc = ItoProcess[{{10 - x[t], 0}, {Sqrt[1 + x[t]^2], 1}, {x[t], 
    b[t]}}, {{x, b}, {1, 0}}, {t, 0}]

The aim for this try was to check if I can recover the wiener proces to check see (manually, for the first value of the simulation) wether the realization of the wiener process fits the simulation of the other process. After varous tries I think this is not the case.

Does someone have any suggestions?

2

Hi Rodrigo,

in the minimum you seem to have to declare equations, not only right hand sites, the basic example from the manual (ref/ItoProcess) is

proc = ItoProcess[\[DifferentialD]x[t] == -x[t] \[DifferentialD]t + 
    Sqrt[1 + x[t]^2] \[DifferentialD]w[t], x[t], {x, 1}, t, 
  w \[Distributed] WienerProcess[]]

if you do so, the flavour

ItoProcess[sdeqns, expr, x, t, w \[Distributed] dproc]
represents an Ito process specified by a stochastic differential equations sdeqns, output expression eypr, with state x and time t, driven by w following the process dproc

should work it out. See also the details Common specifications for coefficients a and b include … I would try to deliver 2 equations to ItoProcess[], at least to be sure.
If you give 0 in the first vector and 1 in the second one this would not really erect a second stochastic differential equation, but rather some trivial expression … ?