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Time Series and Stochastic Differential Equations
Convert Parametric SDE Processes to Equivalent Ito Processes
The canonical form of
ItoProcess
or
StratonovichProcess
in
Mathematica
encodes their defining SDEs
as follows.
WienerProcess
solves the SDE
, where
is the standard Wiener process, also known as the Brownian motion process.
In[1]:=
X
ItoProcess[WienerProcess[\[Mu], \[Sigma]]]
Out[1]=
GeometricBrownianMotionProcess
solves the SDE
.
In[2]:=
X
ItoProcess[ GeometricBrownianMotionProcess[\[Mu], \[Sigma], Subscript[x, 0]]]
Out[2]=
OrnsteinUhlenbeckProcess
solves the SDE
.
In[3]:=
X
ItoProcess[ OrnsteinUhlenbeckProcess[\[Mu], \[Sigma], \[Theta], Subscript[x, 0]]]
Out[3]=
CoxIngersollRossProcess
solves the SDE
.
In[4]:=
X
ItoProcess[ CoxIngersollRossProcess[\[Mu], \[Sigma], \[Theta], Subscript[x, 0]]]
Out[4]=
BrownianBridgeProcess
solves the SDE
.
In[5]:=
X
ItoProcess[ BrownianBridgeProcess[\[Sigma], {Subscript[t, 1], Subscript[a, 1]}, {Subscript[t, 2], Subscript[a, 2]}]]
Out[5]=