[Math] Expectation and Variance of Stochastic Differential Equations

expectationstochastic-calculusstochastic-differential-equations

  1. Consider the SDE
    $$dr_t=\kappa(\theta-r_t)\,dt+\sigma dW_t,\ r_0=x,$$
    where $\kappa$, $\theta$ and $\sigma$ are constants. You are given that the solution is
    $$r_t=\theta+(x-\theta)e^{-\kappa t}+\sigma\int_0^te^{-\kappa(t-s)}\,dW_s.$$
    Calculate the mean and variance of $r_t$. You may use the result
    $$\mathbb E\left[\left(\int_0^tY_s\,dW_s\right)^2\right]=\mathbb E\left[\int_0^tY_s^2\,ds\right],$$
    in the calculation of the variance.

Hi, I was wondering if somebody could tell me how to calculate the expectation of an SDE? I believe the expectation of a constant is just equal to the constant.

Best Answer

To compute the expectation and the variance, in addition to the given hint (Ito Isometry), you need to know that if the integrator $W_t$ is an arbitrary martingale, and the integrand $f$ is bounded, then the integral is a martingale, and the expectation of the integral is again zero (proof). Then we can proceed.

For: $$r_t=\theta+(x-\theta)e^{-\kappa t}+\sigma\int_0^te^{-\kappa(t-s)}\,dW_s$$

The Expectaction of $r_t$ $$\begin{align} \mathbb{E}[r_t]&=\mathbb{E}\left[\theta+(x-\theta)e^{-\kappa t}+\sigma\int_0^te^{-\kappa(t-s)}\,dW_s\right]\\ &=\theta+(x-\theta)e^{-\kappa t}+\sigma\mathbb{E}\left[\int_0^te^{-\kappa(t-s)}\,dW_s\right]\\ &=\theta+(x-\theta)e^{-\kappa t} \end{align}$$

The variance of $r_t$

$$\begin{align} Var[r_t]&=\mathbb{E}[r_t^2]-\left(\mathbb{E}[r_t]\right)^2\\ &=\mathbb{E}\left[\left(\theta+(x-\theta)e^{-\kappa t}+\sigma\int_0^te^{-\kappa(t-s)}\,dW_s\right)^2\right]-\left(\theta+(x-\theta)e^{-\kappa t}\right)^2\\ &=(\theta+(x-\theta)e^{-\kappa t})^2+2\sigma(\theta+(x-\theta)e^{-\kappa t})^2\mathbb{E}\left[\int_0^te^{-\kappa(t-s)}\,dW_s\right]+\sigma^2\mathbb{E}\left[\left(\int_0^te^{-\kappa(t-s)}\,dW_s\right)^2\right]-(\theta+(x-\theta)e^{-\kappa t})^2\\ &=\sigma^2\mathbb{E}\left[\int_0^te^{-2\kappa(t-s)}\,ds\right]\\ &=\sigma^2\int_0^te^{-2\kappa(t-s)}\,ds\\ &=\dfrac{\sigma^2}{2\kappa}(1-e^{-2\kappa t}) \end{align}$$

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