Stability results for martingale representations: The general case

Abstract

In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the martingales and the associated filtrations converge in the extended sense, and that the limiting martingale is quasi-left-continuous and admits the predictable representation property. Then, we prove that each component in the martingale representation of the sequence converges to the corresponding component of the martingale representation of the limiting random variable relative to the limiting filtration, under the Skorokhod topology. This extends in several directions earlier contributions in the literature, and has applications to stability results for backward SDEs with jumps and to discretisation schemes for stochastic systems. 1 2 A. PAPAPANTOLEON, D. POSSAMAÏ, AND A. SAPLAOURAS A.3. Young functions 47 A.4. Proof of Corollary 3.4 51 REFERENCES 53 Once the semimartingales considered have more structural properties, other interesting results can be obtained. Barrieu, Cazanave, and El Karoui [6], for instance, and later Barrieu and El Karoui [5] were interested in what they coined "continuous quadratic semimartingales" (see also related articles by Mocha and Westray [54], still in the continuous case, and recent extensions to jump processes by Ngoupeyou [58] and El Karoui, Matoussi, and Ngoupeyou [25]), for which, roughly speaking, the bounded variation process part in the semimartingale X is absolutely continuous with respect to the quadratic variation of the martingale part of X.[5, 6] obtained associated stability results for these processes.An important common feature of the articles mentioned so far, is that they actually only consider the strong framework we described at the beginning of this introduction, in the sense that there is a always a fixed probability space and all processes (meaning here mainly X n and X ∞ ) are adapted to the same fixed filtration G. An important exception is Słomiński [67], where the probability space is fixed, but not the filtration. However, for practical purposes, and especially for the analysis of numerical schemes, it is well-known that the weak framework is also of paramount importance, as illustrated for instance by the famous Donsker theorem. There has thus been a certain number of studies of stability properties for semimartingale or martingale decompositions when the underlying filtration itself is also allowed to change. In that direction, Antonelli and Kohatsu-Higa [3], followed by Coquet, Mackevičius, and Mémin [14, 15], Ma, Protter, San Martín, and Torres [47], Briand, Delyon, and Mémin [9], Briand, Delyon, and Mémin [10], and then Cheridito and Stadje [11], studied such stability properties for continuous backward stochastic differential equations (BSDEs for short), a type of non-linear martingale representation. Mémin [52] looked into the stabi...

All adapted topologies are equal

“…The related question of stability of stochastic differential equations (as well as their backwards version) with respect to the driving noise has particularly seen a burst of activity in the last two decades. For brevity's sake we only refer to the recent article by Papapantoleon, Posamaï, and Saplaouras [50] for an overview of the many available works in this direction.…”

Estimating processes in adapted Wasserstein distance

“…Applications to stability of SDEs/BSDEs have particularly seen a burst of activity in the last two decades. We refer to the recent article [28] for an overview of the many available works in this direction.…”

Stability of backward stochastic differential equations: the general case

“…The former was introduced and successfully used by Briand, Delyon, and Mémin [18,19] in order to efficiently control the doubly indexed sequence obtained by the sequence of Picard schemes; we will follow the same approach. The latter is presented in Papapantoleon, Possamaï, and Saplaouras [81]. It comes as no surprise that the stability of martingale representations comes into play, since the existence and uniqueness of the solution of the BSDE (1.1) with stochastically Lipschitz generator is obtained by means of a martingale representation argument.…”

“…Theorem A.7 provides a characterisation of weak convergence of measures to an atomless measure on the real (half-)line and its corollaries are going to guarantee the convergence of the Lebesgue-Stieltjes integrals when the integrators are regular enough. In view of the above brief comments, it is natural to combine the frameworks of [80,81] 3 and to properly enrich them in order to obtain the framework suitable for our purpose. We postpone the detailed description and the required technicalities until Subsection 3.1.…”