Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T09:15:00.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Infinite time interval BSDEs and the convergence of g-martingales

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  09 April 2009

Zengjing Chen  and
Bo Wang
Bo Wang
Affiliation:
Department of Mathematics Shandong UniversityJinan Shandong 250100 P. R.China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we first give a sufficient condition on the coefficients of a class of infinite time interval backward stochastic differential equations (BSDEs) under which the infinite time interval BSDEs have a unique solution for any given square integrable terminal value, and then, using the infinite time interval BSDEs, we study the convergence of g-martingales introduced by Peng via a kind of BSDEs. Finally, we study the applications of g-expectations and g-martingales in both finance and economics.

Keywords

infinite time interval BSDEsg-expectationg-martingaleupcrossing inequality of g-martingaleconvergence of g-martingale

MSC classification

Secondary: 60H10: Stochastic ordinary differential equations 60G48: Generalizations of martingales
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 69 , Issue 2 , October 2000 , pp. 187 - 211
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Chen, Z. and Peng, S., ‘Continuous properties of g-martingales’, to appear in Chinese Ann. Math. (2000).Google Scholar
[2]Chen, Z. and Peng, S., ‘A general downcrossing inequality for g-martingales’, Statist. Probab. Lett. 46 (2000), 169–175.CrossRefGoogle Scholar
[3]Cvitanic, J. and Karatzas, I., ‘Hedging contingent claims with constrained portfolios’, Ann. Appl. Probab. 3 (1993), 652–681.CrossRefGoogle Scholar
[4]Cvitanic, J. and Ma, J., ‘Hedging options for a large investor and forward-backward SDEs’, to appear in Ann. Appl. Probab.Google Scholar
[5]Daling, R., ‘Constructing of Gamma-martingales with prescribed limits, using BSDE’, Ann. Probab. 23 (1995), 1234–1261.Google Scholar
[6]Daling, R. and Pardoux, E., ‘BSDE with random terminal time and applications to semilinear elliptic PDE’, Ann. Probab. 3 (1997), 1135–1159.Google Scholar
[7]Duffie, D. and Epstein, L., ‘Stochastic differential utility’, Econometrica 60 (1992), 353–394.CrossRefGoogle Scholar
[8]El Karoui, N., Peng, S. and Quenez, M-C., ‘BSDEs in finance’, Math. Finance 7 (1997), 1–71.CrossRefGoogle Scholar
[9]El Karoui, N. and Quenez, M-C., ‘Dynamic programming and pricing of contingent claims in an incomplete markets’, SIAM J. Control Optim. 33 (1995), 29–66.CrossRefGoogle Scholar
[10]Epstein, L., ‘A definition of uncertainty aversion’, Rev. Econom. Studies 66 (1999), 579–608.CrossRefGoogle Scholar
[11]Epstein, L. and Wang, T., ‘Intertemporal asset pricing under Knightain uncertainty’, Econometrica 62 (1994), 283–322.CrossRefGoogle Scholar
[12]Epstein, L. and Wang, T., ‘Uncertainty, risk-neutral measures and security price booms and crashes’, J. Econom. Theory 67 (1995), 40–82.CrossRefGoogle Scholar
[13]Gilboa, I. and Schmeidler, D., ‘Maxim expected utility with non-unique prior’, J. Math. Econom. 18 (1989), 141–153.CrossRefGoogle Scholar
[14]Hamadéne, S. and Lepeltier, J. P., ‘Zero-sum stochastic differential games and backward equations’, System Control Lett. 24 (1995), 259–263.CrossRefGoogle Scholar
[15]Hamadéne, S. and Lepeltier, J. P., ‘Backward equations, stochastic control and zero-sum stochastic differential games’, Stochastics 54 (1995), 221–231.Google Scholar
[16]Hu, Y. and Peng, S., ‘Solutions of backward-forward SDE’, Probab. Theory Related Fields 103 (1995), 273–283.CrossRefGoogle Scholar
[17]Kramkov, D., ‘Optional decomposition of supermartingales and hedging contingent claims incomplete security markets’, Probab. Theory Related Fields 105 (1996), 459–479.CrossRefGoogle Scholar
[18]Lepeltier, J. P. and Martin, J. S., ‘BSDEs with continuous coefficients’, Statist. Probab. Lett. 32 (1997), 425–430.CrossRefGoogle Scholar
[19]Pardoux, E., ‘Generalized discontinuous BSDEs’, in: Pitman Res. Notes Math. Ser. 364 (1997), 207–219.Google Scholar
[20]Pardoux, E. and Peng, S., ‘Adapted solution of a backward stochastic differential equation’, System Control Lett. 14 (1990), 55–61.CrossRefGoogle Scholar
[21]Pardoux, E. and Zhang, S., ‘Generalized BSDEs and nonlinear Neumann boundary value problems’, Probab. Theory Related Fields 110 (1998), 535–558.CrossRefGoogle Scholar
[22]Peng, S., ‘A general stochastic maximum principle for optimal control problems’, SIAM J. Control Optim. 28 (1990), 966–979.CrossRefGoogle Scholar
[23]Peng, S., ‘Probabilistic interpretation for systems of quasilinear parabolic partial differential equations’, Stochastics Stochastics Rep. 37 (1991), 61–74.Google Scholar
[24]Peng, S., ‘Monotonic limit theorem of BSDE and its application to Doob-Meyer decomposition theorem’, Probab. Theory Related Fields 113 (1999), 473–499.CrossRefGoogle Scholar
[25]Peng, S., ‘BSDE and related g-expectation’, in: Pitman Res. Notes Math. Ser. 364 (1997), 141–159.Google Scholar
[26]Schmeidler, D., ‘Subjective probability and expected utility without additivity’, Econometrica 57 (1989), 571–587.CrossRefGoogle Scholar
[27]Wasserman, L. and Kadane, J., ‘Bayes' theorem for Choquet capacities’, Ann. Statist. 18 (1990), 1328–1339.CrossRefGoogle Scholar