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Ito and Stratonovich Solutions of the Linear Growth Model

    
Define Ito and Stratonovich processes with the same SDE :
Wolfram Language code: ipr = ItoProcess[ⅆx[t] == x[t](r ⅆt + σ ⅆw[t]), x[t], {x, x0}, t, wWienerProcess[]]
Wolfram Language code: spr = StratonovichProcess[ⅆx[t] == x[t](r ⅆt + σ ⅆw[t]), x[t], {x, x0}, t, wWienerProcess[]]
Find mean and variance functions for the Ito process:
Wolfram Language code: {Mean[ipr[t]], Variance[ipr[t]]}//Simplify
The mean and variance functions for the Stratonovich process are different:
Wolfram Language code: {Mean[spr[t]], Variance[spr[t]]}//Simplify
When , the Ito solution almost surely converges to zero, i.e. the large limit of probability that process value is does not exceed equals :
Wolfram Language code: Assuming[0 < r < σ^2 / 2 && σ > 0 && x0 > 0, Limit[Probability[[t] ≤ 1 / t, ipr], t -> ∞]]
Confirm it using simulations:
Wolfram Language code: ListLogPlot[RandomFunction[ipr /. {r -> 1, σ -> 2, x0 -> 1}, {0, 20., 0.002}, 6, Method -> "KPS"], PlotRange -> All, Joined -> True]
When , the Stratonovich solution, however, almost surely diverges, i.e. the large limit of the probability that process value exceeds equals :
Wolfram Language code: Assuming[0 < r < σ^2 / 2 && σ > 0 && x0 > 0, Limit[Probability[[t] > t, spr], t -> ∞]]
Confirm it using simulations:
Wolfram Language code: ListLogPlot[RandomFunction[spr /. {r -> 1, σ -> 2, x0 -> 1}, {0, 20., 0.002}, 6, Method -> "KPS"], PlotRange -> All, Joined -> True]