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Expected level of a mean-reverting square-root process

Question

A process YY starts at Y0=x>0Y_0=x>0 and satisfies

dYt=κ(θ−Yt) dt−σYt dWtdY_t=\kappa(\theta-Y_t)\,dt-\sigma\sqrt{Y_t}\,dW_t

for a standard Brownian motion WW. Assume the parameters keep YY strictly positive, so the square root is always defined (no proof needed). Find E[YT]\mathbb{E}[Y_T] in terms of x,κ,θ,σ,Tx,\kappa,\theta,\sigma,T and evaluate at κ=0.2\kappa=0.2, θ=2\theta=2, σ=0.1\sigma=0.1, x=5x=5, T=10T=10. Writing the value as A+BeCA+Be^{C} with integers A,B,CA,B,C, give ABCABC.

Short answer

−12-12. Applying the product rule to eκtYte^{\kappa t}Y_t cancels YY in the drift and leaves a mean-zero stochastic integral, so E[YT]=θ+(x−θ)e−κT=2+3e−2\mathbb{E}[Y_T]=\theta+(x-\theta)e^{-\kappa T}=2+3e^{-2}, and ABC=2⋅3⋅(−2)ABC=2\cdot3\cdot(-2). The volatility plays no role in the mean.

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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