Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 60G40 Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
- 60J60 Diffusion processes [See also 58J65]

**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.

**Mots Clés:** *Optimal stopping ; maximality principle ; optimal Skorokhod embedding ; maximum process*

**Date:** 2004-05-13

**Prépublication numéro:** *PMA-912*