Stability results for martingale representations: The general case

Papapantoleon, Antonis; Possamaï, Dylan; Saplaouras, Alexandros

doi:10.1090/tran/7880

Stability results for martingale representations: The general case
HTML articles powered by AMS MathViewer

by Antonis Papapantoleon, Dylan Possamaï and Alexandros Saplaouras PDF

Trans. Amer. Math. Soc. 372 (2019), 5891-5946 Request permission

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 stochastic differential equations with jumps, and to discretization schemes for stochastic systems.

References

D. Aldous, Weak convergence and the general theory of processes, Department of Statistics, University of California, 1981. Incomplete draft of a monograph.
Charalambos D. Aliprantis and Kim C. Border, Infinite dimensional analysis, 3rd ed., Springer, Berlin, 2006. A hitchhiker’s guide. MR 2378491
Fabio Antonelli and Arturo Kohatsu-Higa, Filtration stability of backward SDE’s, Stochastic Anal. Appl. 18 (2000), no. 1, 11–37. MR 1739298, DOI 10.1080/07362990008809652
Martin T. Barlow and Philip Protter, On convergence of semimartingales, Séminaire de Probabilités, XXIV, 1988/89, Lecture Notes in Math., vol. 1426, Springer, Berlin, 1990, pp. 188–193. MR 1071540, DOI 10.1007/BFb0083765
Pauline Barrieu and Nicole El Karoui, Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs, Ann. Probab. 41 (2013), no. 3B, 1831–1863. MR 3098060, DOI 10.1214/12-AOP743
Pauline Barrieu, Nicolas Cazanave, and Nicole El Karoui, Closedness results for BMO semi-martingales and application to quadratic BSDEs, C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 881–886 (English, with English and French summaries). MR 2441926, DOI 10.1016/j.crma.2008.06.010
V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
Philippe Briand, Bernard Delyon, and Jean Mémin, Donsker-type theorem for BSDEs, Electron. Comm. Probab. 6 (2001), 1–14. MR 1817885, DOI 10.1214/ECP.v6-1030
Philippe Briand, Bernard Delyon, and Jean Mémin, On the robustness of backward stochastic differential equations, Stochastic Process. Appl. 97 (2002), no. 2, 229–253. MR 1875334, DOI 10.1016/S0304-4149(01)00131-4
Patrick Cheridito and Mitja Stadje, BS$\Delta$Es and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness, Bernoulli 19 (2013), no. 3, 1047–1085. MR 3079306, DOI 10.3150/12-BEJ445
Samuel N. Cohen and Robert J. Elliott, Stochastic calculus and applications, 2nd ed., Probability and its Applications, Springer, Cham, 2015. MR 3443368, DOI 10.1007/978-1-4939-2867-5
François Coquet and Leszek Słomiński, On the convergence of Dirichlet processes, Bernoulli 5 (1999), no. 4, 615–639. MR 1704558, DOI 10.2307/3318693
François Coquet, Vigirdas Mackevičius, and Jean Mémin, Stability in $\textbf {D}$ of martingales and backward equations under discretization of filtration, Stochastic Process. Appl. 75 (1998), no. 2, 235–248. MR 1632205, DOI 10.1016/S0304-4149(98)00013-1
François Coquet, Vigirdas Mackevičius, and Jean Mémin, Corrigendum to: “Stability in $\bf D$ of martingales and backward equations under discretization of filtration” [Stochastic Processes Appl. 75 (1998), no. 2, 235–248; MR1632205 (99f:60109)], Stochastic Process. Appl. 82 (1999), no. 2, 335–338. MR 1700013, DOI 10.1016/S0304-4149(99)00020-4
F. Coquet, J. Mémin, and V. Mackevičius, Some examples and counterexamples of convergence of $\sigma$-algebras and filtrations, Lith. Math. J. 40 (2000), no. 3, 228–235.
François Coquet, Jean Mémin, and Leszek Słominski, On weak convergence of filtrations, Séminaire de Probabilités, XXXV, Lecture Notes in Math., vol. 1755, Springer, Berlin, 2001, pp. 306–328. MR 1837294, DOI 10.1007/978-3-540-44671-2_{2}1
F. Delbaen and W. Schachermayer, A compactness principle for bounded sequences of martingales with applications, Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996) Progr. Probab., vol. 45, Birkhäuser, Basel, 1999, pp. 137–173. MR 1712239
Freddy Delbaen, Pascale Monat, Walter Schachermayer, Martin Schweizer, and Christophe Stricker, Inégalités de normes avec poids et fermeture d’un espace d’intégrales stochastiques, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 10, 1079–1081 (French, with English and French summaries). MR 1305680
F. Delbaen, P. Monat, W. Schachermayer, M. Schweizer, and C. Stricker, Weighted norm inequalities and hedging in incomplete markets, Finance Stochastics 1 (1997), no. 3, 181–227.
C. Dellacherie and P.-A. Meyer, Probabilities and potential, North-Holland Mathematics Studies, vol. 29, North-Holland, Amsterdam, 1978.
C. Dellacherie and P.-A. Meyer, Probabilities and potential B: Theory of martingales, North-Holland Mathematics Studies, vol. 72, Elsevier Science, New York, 1982.
R. M. Dudley, Real analysis and probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. Revised reprint of the 1989 original. MR 1932358, DOI 10.1017/CBO9780511755347
D. Duffie and P. Protter, From discrete- to continuous-time finance: Weak convergence of the financial gain process, Mathematical Finance 2 (1992), no. 1, 1–15.
N. El Karoui, A. Matoussi, and A. Ngoupeyou, Quadratic exponential semimartingales and application to BSDEs with jumps, arXiv:1603.06191 (2016).
Michel Emery, Stabilité des solutions des équations différentielles stochastiques application aux intégrales multiplicatives stochastiques, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), no. 3, 241–262 (French). MR 464400, DOI 10.1007/BF00534242
M. Emery, Équations différentielles stochastiques lipschitziennes: étude de la stabilité, Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78) Lecture Notes in Math., vol. 721, Springer, Berlin, 1979, pp. 281–293 (French). MR 544801
Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085, DOI 10.1002/9780470316658
Hans Föllmer and Philip Protter, Local martingales and filtration shrinkage, ESAIM Probab. Stat. 15 (2011), no. In honor of Marc Yor, suppl., S25–S38. MR 2817343, DOI 10.1051/ps/2010023
Sheng Wu He, Jia Gang Wang, and Jia An Yan, Semimartingale theory and stochastic calculus, Kexue Chubanshe (Science Press), Beijing; CRC Press, Boca Raton, FL, 1992. MR 1219534
D. N. Hoover, Convergence in distribution and Skorokhod convergence for the general theory of processes, Probab. Theory Related Fields 89 (1991), no. 3, 239–259. MR 1113218, DOI 10.1007/BF01198786
Jean Jacod, Calcul stochastique et problèmes de martingales, Lecture Notes in Mathematics, vol. 714, Springer, Berlin, 1979 (French). MR 542115, DOI 10.1007/BFb0064907
Jean Jacod and Albert N. Shiryaev, Limit theorems for stochastic processes, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288, Springer-Verlag, Berlin, 2003. MR 1943877, DOI 10.1007/978-3-662-05265-5
Jean Jacod, Sylvie Méléard, and Philip Protter, Explicit form and robustness of martingale representations, Ann. Probab. 28 (2000), no. 4, 1747–1780. MR 1813842, DOI 10.1214/aop/1019160506
A. Jakubowski, J. Mémin, and G. Pagès, Convergence en loi des suites d’intégrales stochastiques sur l’espace $\textbf {D}^1$ de Skorokhod, Probab. Theory Related Fields 81 (1989), no. 1, 111–137 (French, with English summary). MR 981569, DOI 10.1007/BF00343739
Y. Kchia, Semimartingales and contemporary issues in quantitative finance, Thesis (Ph.D.)–École Polytechnique.
Younes Kchia and Philip Protter, Progressive filtration expansions via a process, with applications to insider trading, Int. J. Theor. Appl. Finance 18 (2015), no. 4, 1550027, 48. MR 3358108, DOI 10.1142/S0219024915500272
Thomas G. Kurtz and Philip Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab. 19 (1991), no. 3, 1035–1070. MR 1112406
Thomas G. Kurtz and Philip E. Protter, Weak convergence of stochastic integrals and differential equations, Probabilistic models for nonlinear partial differential equations (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1627, Springer, Berlin, 1996, pp. 1–41. MR 1431298, DOI 10.1007/BFb0093176
Dorival Leão and Alberto Ohashi, Weak approximations for Wiener functionals, Ann. Appl. Probab. 23 (2013), no. 4, 1660–1691. MR 3098445, DOI 10.1214/12-aap883
Dorival Leão, Alberto Ohashi, and Alexandre B. Simas, Weak differentiability of Wiener functionals and occupation times, Bull. Sci. Math. 149 (2018), 23–65. MR 3868115, DOI 10.1016/j.bulsci.2018.07.001
Dorival Leão, Alberto Ohashi, and Alexandre B. Simas, A weak version of path-dependent functional Itô calculus, Ann. Probab. 46 (2018), no. 6, 3399–3441. MR 3857859, DOI 10.1214/17-AOP1250
E. Lenglart, D. Lépingle, and M. Pratelli, Présentation unifiée de certaines inégalités de la théorie des martingales, Seminar on Probability, XIV (Paris, 1978/1979) Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 26–52 (French). With an appendix by Lenglart. MR 580107
Rui Lin Long, Martingale spaces and inequalities, Peking University Press, Beijing; Friedr. Vieweg & Sohn, Braunschweig, 1993. MR 1224450, DOI 10.1007/978-3-322-99266-6
Jin Ma, Philip Protter, Jaime San Martín, and Soledad Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab. 12 (2002), no. 1, 302–316. MR 1890066, DOI 10.1214/aoap/1015961165
V. Mackevičius, $s^p$ stabilité des solutions d’équations différentielles stochastiques symétriques avec semi-martingales directrices discontinues, C. R. Acad. Sci., Ser. 1: Math. 302 (1986), no. 19, 689–692.
V. Mackevičius, $s^p$ stability of solutions of symmetric stochastic differential equations with discontinuous driving semimartingales, Ann. Inst. Henri Poincaré, B 23 (1987), no. 4, 575–592.
Dilip Madan, Martijn Pistorius, and Mitja Stadje, Convergence of $\textrm {BS}\Delta \textrm {Es}$ driven by random walks to BSDEs: the case of (in)finite activity jumps with general driver, Stochastic Process. Appl. 126 (2016), no. 5, 1553–1584. MR 3473105, DOI 10.1016/j.spa.2015.11.013
Jean Mémin, Espaces de semi martingales et changement de probabilité, Z. Wahrsch. Verw. Gebiete 52 (1980), no. 1, 9–39 (French). MR 568256, DOI 10.1007/BF00534184
Jean Mémin, Stability of Doob-Meyer decomposition under extended convergence, Acta Math. Appl. Sin. Engl. Ser. 19 (2003), no. 2, 177–190. MR 2011481, DOI 10.1007/s10255-003-0094-2
P.-A. Meyer, Sur le lemme de la Vallée Poussin et un théorème de Bismut, Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977) Lecture Notes in Math., vol. 649, Springer, Berlin, 1978, pp. 770–774 (French). MR 520045
Markus Mocha and Nicholas Westray, Quadratic semimartingale BSDEs under an exponential moments condition, Séminaire de Probabilités XLIV, Lecture Notes in Math., vol. 2046, Springer, Heidelberg, 2012, pp. 105–139. MR 2933935, DOI 10.1007/978-3-642-27461-9_{5}
P. Monat and C. Stricker, Fermeture de $G_T(\Theta )$ et de $L^2(\scr F_0)+G_T(\Theta )$, Séminaire de Probabilités, XXVIII, Lecture Notes in Math., vol. 1583, Springer, Berlin, 1994, pp. 189–194 (French). MR 1329114, DOI 10.1007/BFb0073847
Pascale Monat and Christophe Stricker, Föllmer-Schweizer decomposition and mean-variance hedging for general claims, Ann. Probab. 23 (1995), no. 2, 605–628. MR 1334163
Itrel Monroe, On embedding right continuous martingales in Brownian motion, Ann. Math. Statist. 43 (1972), 1293–1311. MR 343354, DOI 10.1214/aoms/1177692480
A. Ngoupeyou, Optimisation des portefeuilles d’actifs soumis au risque de défaut, 2010. Thesis (Ph.D.)–Université Évry-Val-d’Essonne,.
Antonis Papapantoleon, Dylan Possamaï, and Alexandros Saplaouras, Existence and uniqueness results for BSDE with jumps: the whole nine yards, Electron. J. Probab. 23 (2018), Paper No. 121, 69. MR 3896858, DOI 10.1214/18-EJP240
A. Papapantoleon, D. Possamaï, and A. Saplaouras, Stability results for martingale representations: The general case, arXiv:1806.01172 (2018).
K. R. Parthasarathy, Probability measures on metric spaces, AMS Chelsea Publishing, Providence, RI, 2005. Reprint of the 1967 original. MR 2169627, DOI 10.1090/chel/352
Dylan Possamaï and Xiaolu Tan, Weak approximation of second-order BSDEs, Ann. Appl. Probab. 25 (2015), no. 5, 2535–2562. MR 3375883, DOI 10.1214/14-AAP1055
Philip Protter, ${\cal H}^{p}$ stability of solutions of stochastic differential equations, Z. Wahrsch. Verw. Gebiete 44 (1978), no. 4, 337–352. MR 509206, DOI 10.1007/BF01013196
Philip Protter, Approximations of solutions of stochastic differential equations driven by semimartingales, Ann. Probab. 13 (1985), no. 3, 716–743. MR 799419
Philip E. Protter, Stochastic integration and differential equations, Stochastic Modelling and Applied Probability, vol. 21, Springer-Verlag, Berlin, 2005. Second edition. Version 2.1; Corrected third printing. MR 2273672, DOI 10.1007/978-3-662-10061-5
A. Saplaouras, Backward stochastic differential equations with jumps are stable, 2017. Thesis (Ph.D.)–Technische Universität Berlin.
Martin Schweizer, Variance-optimal hedging in discrete time, Math. Oper. Res. 20 (1995), no. 1, 1–32. MR 1320445, DOI 10.1287/moor.20.1.1
L. Słomiński, Stability of strong solutions of stochastic differential equations, Stochastic Process. Appl. 31 (1989), no. 2, 173–202.
Boris Tsirelson, Within and beyond the reach of Brownian innovation, Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), 1998, pp. 311–320. MR 1648165
Marc Yor, Sous-espaces denses dans $L^{1}$ ou $H^{1}$ et représentation des martingales, Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977) Lecture Notes in Math., vol. 649, Springer, Berlin, 1978, pp. 265–309 (French). Avec un appendice de l’auteur et J. de Sam Lazaro. MR 520008