# Maximise the integral w.r.t. probability measure.

Let $(Z_t)_{0\leq t\leq T}$ be a stochastic process. Then $Z_T$ is a r.v. and $F_{Z_T}$ a corresponding cdf.

Suppose $\mathbb{E}[|e^{Z_t}|]<\infty$ for all $t\geq0$. Also \begin{equation} \mathbb{E}\big[e^{Z_t} | \mathcal{F}_s\big]=1=e^{Z_0} \quad\forall 0\leq s\leq t \end{equation}

(i.e. $(e^{Z_t})_{t\geq0}$ is a martingale) and thus \begin{equation} \int_{\mathbb{R}}^{}e^zdF_{Z_T}(z)=1 \end{equation}

Consider the problem:

\begin{equation} \max_{F_{Z_T}}\int_{lnK}^{\infty}(e^z-K)dF_{Z_T}(z) \end{equation}

subject to \begin{equation} \int_{\mathbb{R}}^{}z^2dF_{Z_T}(z)\leq Q \end{equation}

where $Q\in\mathbb{R}^+$ is some fixed constant.

I thought of Euler - Lagrange approach from Calculus of Variations but I am not sure if that will work since there is no information whether $F$ is differentiable or even continuous ($Z_T$ may be discrete r.v.)

I would be grateful for a hint on how these kind of problems should be approached.

• what about integration by parts ? what about (low-pass) filtering $F$ ? – reuns Jan 2 '16 at 22:05