Mean times variance of an arithmetic diffusion

Question

An asset value evolves as $dX_t=5\,dt+2\,dW_t$ with $X_0=2$ , where $W$ is standard Brownian motion. At horizon $T=0.2$ , let $\mu_T$ and $\sigma_T^2$ be the mean and variance of $X_T$ . Report the product $\mu_T\,\sigma_T^2$ .

Short answer

$2.4$ . Integrating gives $X_T=2+5T+2W_T$ , which is normal with mean $2+5(0.2)=3$ and variance $4(0.2)=0.8$ . The product is $3\times0.8=2.4$ .

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Lessons behind this question

The Four Faces of Brownian Motion
Covariance and Correlation
Itô's Formula

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