zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer’s type. (English) Zbl 0953.60059

This paper studies the backward stochastic differential equation (BSDE) of the form

y t =y T + t T g(s,y s ,z s )ds+(A T -A t )- t T z s dW s ,t[0,T],

where W is a Brownian motion and g is a non-anticipative Lipschitz-continuous function. As usual, a process y is called a supersolution of the BSDE if it is of the above form for some adapted right-continuous, increasing process A and some predictable, square-integrable process z. The main result of the paper is a theorem which asserts that, under suitable integrability conditions, the pointwise monotone limit of a sequence of supersolutions is again a supersolution of the BSDE. Moreover, it is shown that the corresponding integrands z converge weakly in L 2 and strongly in each L p with p<2; the processes A converge weakly in L 2 . As an application of this result, the author proves a generalization of the classical Doob-Meyer decomposition to so-called g-supermartingales. These processes refer to a suitably defined nonlinear expectation operator in essentially the same way as usual supermartingales to usual expectations. As a second application, the author shows that there exists a minimal supersolution of the BSDE which is subject to fairly general state- and time-dependent constraints.

60H99Stochastic analysis
60H30Applications of stochastic analysis
60G48Generalizations of martingales