Maximize value of exponential

Question

How can i go about maximising the value of the following,

$$ exp\left( -\alpha e^{rT}x -\alpha\sigma\lambda e^{rT}\int_0^T e^{-rs}\pi_sds+\frac{1}{2}(\alpha\sigma)^2 e^{2rT}\int_0^Te^{-2rs}\pi_s^2ds \right) $$

with respect to the process $(\pi_s)_{s>0}$?

Am tempted to differentiate with respect to $\pi_s$, but also not quite sure what's the right way to do that

$\pi_s$ is a process dependent on $s$. I've update the question to reflect this. — Danny, May 08 '15 at 23:26

score 0 · Answer 1 · answered May 10 '15 at 10:57

0

Complete the squares: $$\frac{1}{2}(\alpha\sigma)^2 e^{2rT}\int_0^Te^{-2rs}\pi_s^2ds-\alpha\sigma\lambda e^{rT}\int_0^T e^{-rs}\pi_sds=\frac 12\int_0^T\left(e^{r(t-s)} \pi_s\alpha \sigma +\lambda \right)^2\mathrm ds -\frac{T\lambda^2}2 .$$

answered May 10 '15 at 10:57

Davide Giraudo

172,925

Maximize value of exponential

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