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RLC Circuit Driven by Periodic Signal and White Noise

Consider an electrical circuit consisting of a resistor, inductor, and capacitor connected in series. The circuit is under external voltage, which is a superposition of a periodic signal and a white noise.

    
Wolfram Language code: CapacitorChargeModel = ItoProcess[{ ⅆ𝒬[t] == ℐ[t]ⅆt, ℒ ⅆℐ[t] + ℛ ℐ[t]ⅆt + (1/𝒞)𝒬[t]ⅆt == Sin[ω t]ⅆt + σ ⅆw[t]}, 𝒬[t], {{ℐ, 𝒬}, {0, 0}}, t, wWienerProcess[]]
Define the parameters of the model:
Wolfram Language code: params = {ℒ -> 2, 𝒞 -> 2, ℛ -> 1 / 2, ω -> 3 / 2, σ -> 1 / 4};
Compute the mean function, i.e. the mean value of the charge of the capacitor as a function of time:
Wolfram Language code: mqf = Simplify[Mean[CapacitorChargeModel[t] /. params]]
Solve the deterministic problem in the absence of noise and compare with the mean function:
Wolfram Language code: detq = (Q[t] /. First[DSolve[ℒ Q''[t] + ℛ Q'[t] + Q[t] / 𝒞 == Sin[ω t]∧Q[0] == 0 && Q'[0] == 0 /. params, Q[t], t]]);
Wolfram Language code: Simplify[detq == mqf]
Compute the variance function:
Wolfram Language code: vf = Simplify[Variance[CapacitorChargeModel[t] /. params]]
Simulate the model using a strong order 1.5 Kloeden–Platen–Schurz scheme:
Wolfram Language code: path = RandomFunction[CapacitorChargeModel /. params, {0., 22., 0.002}, 12, Method -> "KloedenPlatenSchurz"]
Visualize simulated paths:
Wolfram Language code: pls = ListLinePlot[path, PlotRange -> All, ImageSize -> 400]
Visualize a 95% confidence band about the mean function:
Wolfram Language code: epl = With[{q = Quantile[NormalDistribution[], ((1 + .95)/2)]}, Plot[{mqf - qSqrt[vf], mqf, mqf + qSqrt[vf]} /. params//Evaluate, {t, 0, 22}, PlotRange -> All, PlotStyle -> Directive[Thick, Cyan], FillingStyle -> Directive[Lighter[Blue, .6], Opacity[.15]], Filling -> {1 -> {3}}, ImageSize -> 400]]
Combine the plots together.
Wolfram Language code: Show[{epl, pls}, PlotRange -> All]