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Zbl 1078.60058
El Karoui, Nicole; Föllmer, Hans
A non-linear Riesz respresentation in probabilistic potential theory.
(English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 41, No. 3, 269-283 (2005). ISSN 0246-0203

Let $(X_t)_{t\geq 0}$ be a Hunt process on a locally compact metric space $S$. Define the nonlinear potential operator $\overline{G}$ by $\overline{G}f(x)=E_x\left(\int_0^\zeta \sup_{0\leq s\leq t} f(X_s) dt\right)$. The main purpose of this paper is to show the Riesz decomposition theorem relative to this potential, that is, for a given regular function $u$, to decompose $u$ as $u=\overline{G}f +h$ with a harmonic function $h$. For the purpose, as in [{\it P.~Bank} and {\it N.~El Karoui}, Ann.~Probab. 32, No. 1B, 1030--1067 (2004; Zbl 1058.60022)], assume that any excessive function is lower-semicontinuous and $g(x)=E_x(\zeta)$ is bounded and continuous. Furthermore, assume that $u$ is a continuous function of class (D) and that $h(x)=E_x\left(\lim_{t\uparrow \zeta} u(X_t)\right)$ is continuous. Under these hypotheses, the Riesz decomposition theorem is shown with $$f=\underline{D}u(x)=\inf_{T\in {\cal T}(x)} (u(x)-P_Tu(x))/E_x(T),$$ where ${\cal T}(x)$ is the family of relatively compact open neighbourhoods of $x$ and $p_Tu(x)=E_x\left(u(X_T),T<\zeta\right)$. For the proof, a relation between the réduite and the value function of the optimal stopping problem is used. As a corollary, it is shown that $\sup_{T} E_x\left(u(X_T)+cT\right)$ is attained by the first entrance time into $A_c=\{R u_c=u_c\}$ and the value is equal to $\overline{G}(\underline{D}u \vee c)$.
[Yoichi Oshima (Kumamoto)]
MSC 2000:
*60J45 Probabilistic potential theory
60G40 Optimal stopping
31C05 Generalizations of harmonic (etc.) functions

Keywords: Potential operator; Harmonic function; Optimal stopping

Citations: Zbl 1058.60022

Cited in: Zbl 1184.60030

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