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Mean of a mean-reverting process

HardStochastic ProcessesFinance & Derivatives~10m

A process follows the CIR dynamics dXt=κ(θ−Xt) dt−σXt dWtdX_t = \kappa(\theta - X_t)\,dt - \sigma\sqrt{X_t}\,dW_t with κ=1\kappa = 1, θ=8\theta = 8, and X0=2X_0 = 2 (parameters chosen so Xt>0X_t > 0 almost surely). Find E[XT]\mathbb{E}[X_T] at time T=ln⁡2T = \ln 2.

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  1. Take expectations: the Xt dWt\sqrt{X_t}\,dW_t term vanishes, leaving m′(t)=κ(θ−m(t))m'(t)=\kappa(\theta-m(t)).
  2. So E[Xt]=θ+(X0−θ)e−κt\mathbb{E}[X_t]=\theta+(X_0-\theta)e^{-\kappa t}; with e−ln⁡2=12e^{-\ln 2}=\tfrac12 you get 8−3=58-3=5.

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Asked at: Citadel, Two Sigma

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