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Time Series and Stochastic Differential Equations
Stochastic Differential Equation for Exponential Decay
Define a stochastic process satisfying the Ito stochastic differential equation
. This models exponential decay subject to Wiener noise.
In[1]:=
X
\[ScriptCapitalP] = ItoProcess[\[DifferentialD]x[ t] == x[t] \[DifferentialD]t + \[Sigma] \[DifferentialD]w[t], x[t], {x, x0}, t, w \[Distributed] WienerProcess[]]
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Simulate the process for different values of the variance parameter
.
In[2]:=
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Table[ListLinePlot[ RandomFunction[\[ScriptCapitalP] /. {x0 > 5}, {0, 4, 0.01}, 10], PlotLabel > Row[{"\[Sigma]", "\[Equal]", N@\[Sigma]}]], {\[Sigma], {1/4, 1/2, 1, 2}}]
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The mean function of the process is independent of
.
In[3]:=
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Mean[\[ScriptCapitalP][t]]
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This implies that the mean function coincides with the following deterministic solution.
In[4]:=
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x[t] /. DSolve[{x'[t] == x[t], x[0] == x0}, x[t], t]
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Find the variance function of the process
.
In[5]:=
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Variance[\[ScriptCapitalP][t]]
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Find the probability density function of the value of the process
.
In[6]:=
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PDF[\[ScriptCapitalP][t], x]
Out[6]=