Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

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

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article Article Journal Journal Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]