Using a new inequality on semimartingale local time, we give a new light and slight generalizatio... more Using a new inequality on semimartingale local time, we give a new light and slight generalizations to some results by Emery and Yor (Lecture Notes in Mathematics 1681, Springer, Berlin, pp. 306-312) in terms of local times of functionals of semimartingales.
ABSTRACT Etant donnée une suite (M n ) de martingales de carré intégrable convergeant vers un mou... more ABSTRACT Etant donnée une suite (M n ) de martingales de carré intégrable convergeant vers un mouvement brownien, on étudie, en fonction de la vitesse de convergence des variations quadratiques de cette suite, la vitesse de convergence de solutions d'équations différentielles conduites par les M n .
A class of stochastic processes, called "weak Dirichlet processes", is introduced and its propert... more A class of stochastic processes, called "weak Dirichlet processes", is introduced and its properties are investigated in detail. This class is much larger than the class of Dirichlet processes. It is closed under C 1 -transformations and under absolutely continuous changes of measure. If a weak Dirichlet process has finite energy, as defined by Graversen and Rao, its Doob-Meyer type decomposition is unique. The methods developed here have been applied to study generalized martingale convolutions.
From a general definition of nonlinear expectations, viewed as operators preserving monotonicity ... more From a general definition of nonlinear expectations, viewed as operators preserving monotonicity and constants, we derive, under rather general assumptions, the notions of conditional nonlinear expectation and nonlinear martingale. We prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation, and in particular admits continuous paths. In other words, it is a g-martingale.
We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends... more We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C 1 transformation.
We study some properties of the weak convergence of filtrations, in particular, its behavior unde... more We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of cr-algebras to filtrations generated by different discretizations of a single process.
In , Z. Chen proved that, if for each terminal condition ξ, the solution of the BSDE associated t... more In , Z. Chen proved that, if for each terminal condition ξ, the solution of the BSDE associated to the standard parameter (ξ, g 1 ) is equal at time t = 0 to the solution of the BSDE associated to (ξ, g 2 ) then we must have g 1 ≡ g 2 . This result yields a natural question: what happens in the case of an inequality in place of an equality? In this paper, we try to investigate this question and we prove some properties of "g-expectation", notion introduced by S. Peng in [8].
For a given weakly convergent sequence fX n g of Dirichlet processes we show weak convergence of ... more For a given weakly convergent sequence fX n g of Dirichlet processes we show weak convergence of the sequence of the corresponding quadratic variation processes as well as stochastic integrals driven by the X n values provided that the condition UTD (a counterpart to the condition UT for Dirichlet processes) holds true. Moreover, we show that under UTD the limit process of fX n g is a Dirichlet process, too.
Using a new inequality on semimartingale local time, we give a new light and slight generalizatio... more Using a new inequality on semimartingale local time, we give a new light and slight generalizations to some results by Emery and Yor (Lecture Notes in Mathematics 1681, Springer, Berlin, pp. 306 -312) in terms of local times of functionals of semimartingales.
Using a new inequality on semimartingale local time, we give a new light and slight generalizatio... more Using a new inequality on semimartingale local time, we give a new light and slight generalizations to some results by Emery and Yor (Lecture Notes in Mathematics 1681, Springer, Berlin, pp. 306-312) in terms of local times of functionals of semimartingales.
ABSTRACT Etant donnée une suite (M n ) de martingales de carré intégrable convergeant vers un mou... more ABSTRACT Etant donnée une suite (M n ) de martingales de carré intégrable convergeant vers un mouvement brownien, on étudie, en fonction de la vitesse de convergence des variations quadratiques de cette suite, la vitesse de convergence de solutions d'équations différentielles conduites par les M n .
A class of stochastic processes, called "weak Dirichlet processes", is introduced and its propert... more A class of stochastic processes, called "weak Dirichlet processes", is introduced and its properties are investigated in detail. This class is much larger than the class of Dirichlet processes. It is closed under C 1 -transformations and under absolutely continuous changes of measure. If a weak Dirichlet process has finite energy, as defined by Graversen and Rao, its Doob-Meyer type decomposition is unique. The methods developed here have been applied to study generalized martingale convolutions.
From a general definition of nonlinear expectations, viewed as operators preserving monotonicity ... more From a general definition of nonlinear expectations, viewed as operators preserving monotonicity and constants, we derive, under rather general assumptions, the notions of conditional nonlinear expectation and nonlinear martingale. We prove that any such nonlinear martingale can be represented as the solution of a backward stochastic equation, and in particular admits continuous paths. In other words, it is a g-martingale.
We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends... more We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C 1 transformation.
We study some properties of the weak convergence of filtrations, in particular, its behavior unde... more We study some properties of the weak convergence of filtrations, in particular, its behavior under elementary set operations. We also derive relations between the convergence of filtrations generated by point processes with a single jump and the convergence of their compensators or distributions of their jump moments. Finally, we apply a lemma on the intersection of cr-algebras to filtrations generated by different discretizations of a single process.
In , Z. Chen proved that, if for each terminal condition ξ, the solution of the BSDE associated t... more In , Z. Chen proved that, if for each terminal condition ξ, the solution of the BSDE associated to the standard parameter (ξ, g 1 ) is equal at time t = 0 to the solution of the BSDE associated to (ξ, g 2 ) then we must have g 1 ≡ g 2 . This result yields a natural question: what happens in the case of an inequality in place of an equality? In this paper, we try to investigate this question and we prove some properties of "g-expectation", notion introduced by S. Peng in [8].
For a given weakly convergent sequence fX n g of Dirichlet processes we show weak convergence of ... more For a given weakly convergent sequence fX n g of Dirichlet processes we show weak convergence of the sequence of the corresponding quadratic variation processes as well as stochastic integrals driven by the X n values provided that the condition UTD (a counterpart to the condition UT for Dirichlet processes) holds true. Moreover, we show that under UTD the limit process of fX n g is a Dirichlet process, too.
Using a new inequality on semimartingale local time, we give a new light and slight generalizatio... more Using a new inequality on semimartingale local time, we give a new light and slight generalizations to some results by Emery and Yor (Lecture Notes in Mathematics 1681, Springer, Berlin, pp. 306 -312) in terms of local times of functionals of semimartingales.
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