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Let $B_t$ be a Brownian motion, let $\sigma > 0$ be fixed and let $X_t$ be a process with fixed beginning value $x_0$ that satisfies $dXt = u_tdt + \sigma X_tdB_t.$

Solve $E\left[\int_0^Tu^2dt+(X_T)^2\right] \rightarrow \min$

Hint: try a value function of the form $\phi(t)x^2$ and note that $1/(a \phi + \phi^2) = (1/a)[1/ \phi - 1/(\phi + a)]$.

I think I need to verify the Hint with a Bellman equation to solve the problem.

I think $u$ is a function of $t$.

janR
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    I do not understand what this question is asking. What is $u$? The way you have written the question makes it appear to be a number. Is $u$ a function? Are you trying to minimize the quantity in that expectation? What does that arrow mean? – Chris Janjigian Nov 21 '13 at 18:28

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