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Stochastic Differential Equation for Exponential Decay

    
Define a stochastic process satisfying Ito stochastic differential equation , describing exponential decay subject to Wiener noise:
Wolfram Language code: 𝒫 = ItoProcess[ⅆx[t] == -x[t]ⅆt + σ ⅆw[t], x[t], {x, x0}, t, wWienerProcess[]]
Simulate the process for different values of variance parameter :
Wolfram Language code: Table[ListLinePlot[RandomFunction[𝒫 /. {x0 -> 5}, {0, 4, 0.01}, 10], PlotLabel -> Row[{"σ", "==", N@σ}]], {σ, {1 / 4, 1 / 2, 1, 2}}]
Find the mean function of the process:
Wolfram Language code: Mean[𝒫[t]]
Its independence of the noise variance suggests that it coincides with the deterministic solution:
Wolfram Language code: x[t] /. DSolve[{x'[t] == -x[t], x[0] == x0}, x[t], t]
Find the variance function of the process :
Wolfram Language code: Variance[𝒫[t]]
Value of the process at time is a random variable. Find the probability density function of the value of the process :
Wolfram Language code: PDF[𝒫[t], x]