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Time Series and Stochastic Differential Equations
Ito and Stratonovich Solutions of the Linear Growth Model
Define
ItoProcess
and
StatonovichProcess
for the SDE
.
In[1]:=
X
ipr = ItoProcess[\[DifferentialD]x[t] == x[t] (r \[DifferentialD]t + \[Sigma] \[DifferentialD]w[t]), x[t], {x, x0}, t, w \[Distributed] WienerProcess[]]
Out[1]=
In[2]:=
X
spr = StratonovichProcess[\[DifferentialD]x[t] == x[t] (r \[DifferentialD]t + \[Sigma] \[DifferentialD]w[t]), x[t], {x, x0}, t, w \[Distributed] WienerProcess[]]
Out[2]=
Find the mean and variance functions for the Ito process.
In[3]:=
X
{Mean[ipr[t]], Variance[ipr[t]]} // Simplify
Out[3]=
The mean and variance functions for the Stratonovich process are different.
In[4]:=
X
{Mean[spr[t]], Variance[spr[t]]} // Simplify
Out[4]=
When
, the Ito solution converges to zero almost surely.
In[5]:=
X
Assuming[0 < r < \[Sigma]^2/2 && \[Sigma] > 0 && x0 > 0, Limit[Probability[ 0 < \[FormalX][t] <= 1/t, \[FormalX] \[Distributed] ipr], t > \[Infinity]]]
Out[5]=
Confirm the convergence to zero using simulations.
In[6]:=
X
ListLogPlot[ RandomFunction[ ipr /. {r > 1, \[Sigma] > 2, x0 > 1}, {0, 20., 0.002}, 6, Method > "KPS"], PlotRange > All, Joined > True, ImageSize > 300]
Out[6]=
When
, the Stratonovich solution, however, diverges almost surely.
In[7]:=
X
Assuming[0 < r < \[Sigma]^2/2 && \[Sigma] > 0 && x0 > 0, Limit[Probability[\[FormalX][t] > t, \[FormalX] \[Distributed] spr], t > \[Infinity]]]
Out[7]=
Confirm the divergence using simulations.
In[8]:=
X
ListLogPlot[ RandomFunction[ spr /. {r > 1, \[Sigma] > 2, x0 > 1}, {0, 20., 0.002}, 6, Method > "KPS"], PlotRange > All, Joined > True, ImageSize > 300]
Out[8]=