Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### The maximality principle revisited : on certain optimal stopping problems

Auteur(s):

Code(s) de Classification MSC:

• 60G40 Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Résumé: We investigate in detail works of Peskir [13] and Meilijson [9] and develop a link between them. We consider the following optimal stopping problem: maximize $V_\tau=\e\Big[\phi(S_\tau)-\int_0^\tau c(B_s)ds\Big]$ over all stopping times with $\e\int_0^\tau c(B_s)ds<\infty$, where $S=(S_t)_{t\geq 0}$ is the supremum process associated with real valued Brownian motion $B$, $\phi\in C^1$ is non-decreasing and $c$ is continuous. From work of Peskir [13] we deduce that this problem has a unique solution if and only if the differential equation $$g'(s)=\frac{\phi'(s)}{2c(g(s))(s-g(s))}$$ admits a maximal solution $g_*(s)$ such that $g_*(s)\leq s$ for all $s\geq 0$. The stopping time which yields the highest payoff can be written as $\tau_*=\inf\{t\geq 0: B_t\leq g_*(S_t)\}$. The problem is actually solved in a general case of a real-valued, time homogeneous diffusion $X=(X_t:t\geq 0)$ instead of $B$. We then proceed to solve the problem for more general functions $\phi$ and $c$. Explicit formulae for payoff are given. We apply the results to solve the so-called optimal Skorokhod embedding problem. We give also a sample of applications to various inequalities dealing with terminal value and supremum of a process.