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 $h_{2} .[30 \%]$
第一问按照定义白给
第二问用Fokker–Planck equation+Stochastic control问题的刻画得到h1，h2，h3表达式
第三问用Ito formula算一下
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.
Stochastic Analysis Suggested reading list
Suggested reading/resource (link to My Reading Lists)
