Some stochastic control problems in finance

Several examples were considered.

Optimal Stopping and free boundary problems. 

Let's consider the following financial market containing a non-risky asset $S^0_t = e^{r t}$, and $d$ risky assets (e.g. stocks) with prices modeled by a $d$-dimensional diffusion process S with dynamics

$$dS_{t}=rS_{t}dt+\mbox{diag}(S_{t})\sigma (t,S_{t})dW_{t\quad },$$

where $W$ is a standard Brownian motion.

An important problem in finance is the pricing of American options. Given a reward function $g$, an American option is a contract that provides to the owner the right to receive (from the seller) the amount $g(S_{t})$ at any time t if exercised before some fixed maturity $T$. This right can be exercised only once during the period $[0,T]$. The price of this option at time t can be expressed as the value function associated to the optimal stopping problem

$$v(t,S_{t})=\sup_{\tau \in {\mathcal T}_{[t,T]}} E[e^{-r(\tau-t)}g(S_{t})\,\vert\,S_{t}]\qquad (1.1.1)$$

where ${\mathcal T}_{[t,T]}$ is the set of all stopping times with values in $[t,T]$.

The associated value function $v$ is the solution of the free boundary problem

$$\begin{eqnarray*} \min\{rv-v_t-\,{\mathcal L}v\;;\;v-g\} &=&0\;\;\;\;\;\;\;\mbox... ...fty)^d \\ v(T,.) &=&g(.)\;\;\;\;\;\;\mbox{ on } \; [0,\infty)^d \end{eqnarray*}$$

where $\cal{L}$ is the generator of the diffusion $S$.

Optimal Investment and Hamilton-Jacobi-Bellman equations. 

An other important issue in finance is that of optimal investment. Denoting by $\nu_t$ the number of stocks held by a given financial agent at time t, the associated wealth-process $X^\nu$ has dynamics given by

$$dX_{t}=\nu_{t}dS_{t}+(X_{t}-\nu_{t}^{*}S_{t})dS_{t}^{0}$$

where $^{*}$ stands for transposition, and $S$ may have a general dynamics of the form

$$dS_{t}=\mu (t,S_{t},\nu_{t})dt+\sigma (t,S_{t},\nu_{t})dW_{t},$$

in order to take into account a possible influence of the agent's financial strategy on the dynamics of the risky assets. Given a concave (utility) function $V$, the agent tries to maximize the expected utility of terminal wealth

$$\max E[V(X^\nu_{T})]\qquad(1.1.2)$$

over a set of admissible financial strategies $(\nu_{t})$ with values in some subset $U$ of $R^{d}$. The associated value function $v$ is the solution of Hamilton-Jacobi-Bellman equation

$$\begin{eqnarray*} -v_{t}-\sup_{\nu \in U} {\mathcal L}^{\nu}v &=&0\;\;\;\;\mbox{... ...x) &=&V(x)\;\;\;\;\mbox{ for} \;(s,x) \in [0,\infty)^{d}\times R \end{eqnarray*}$$

where $L^{\nu}$ is the generator of the diffusion $(S,X^{\nu})$ .

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