Problem 1.

Let $\left{X_{t}^{\pi}\right}_{t \in[0, T]}$ be a controlled process satisfying
$$d X_{t}^{\pi}=\left(\mu+\pi_{t}\right) d t+\sigma d B_{t}$$
and let $H:[0, T] \times \mathbb{R}$ be the dynamic value function associated with the maximization of
$$\mathbb{E}\left[\left(X_{T}^{\pi}\right)^{2}-c \int_{0}^{T} \pi_{t}^{2} d t\right]$$
where $c>0$.
(a) Write the Hamilton-Jacobi-Bellman equation with terminal conditions that is satisfied by the function $H$. $[30 \%]$
(b) Assume that $H$ is of the form
$$H(t, x)=h_{0}(t)+h_{1}(t) x+h_{2}(t) x^{2}$$
Substitute this expression into the PDE from part a) and write the system of ODEs and terminal conditions satisfied by $h_{0}, h_{1}$, and $h_{2}$. Do not solve these equations. $[40 \%]$
(c) Write the optimal control $\pi^{*}$ in terms of the functions $h_{0}, h_{1}$, and $h_{2} .[30 \%]$

Proof .

## Stochastic Analysis Module description

Syllabus

Normal random variables and Gaussian processes; martingales; Brownian motion, stochastic integral, rules for stochastic calculus (Ito, martingale representation, Levy characterisation);applications: stochastic differential equations; martingale representation and Girsanov’s change of measure.

Prerequisites

7CCMFM01, real analysis, basic probability theory.

## Stochastic Analysis Assessment details

Assessment

2 hr written examination, class test, or alternative assessment

## Stochastic Analysis Educational aims & objectives

Aims

You will acquire a sophisticated understanding of several modern concepts and results in the theory of stochastic processes, including stochastic calculus and the theory of Brownian motion.

## Stochastic Analysis Teaching pattern

Four hours of lectures, 1 tutorial and 2 walk-in tutorials per week for the second half of Semester.