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Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. (English) Zbl 1312.60052
The authors study the stability and strong convergence for general quadratic BSDEs and quadratic semimartingales. A quadratic BSDE has the property that the coefficient \(g\) satisfies the quadratic structure equation. Its solution is a quadratic Itō semimartingale where the predictable process which has finite variation also satisfies this structure equation. The authors characterize quadratic semimartingales via an exponential transform where the generator of the BSDE is transformed in a driver with linear-quadratic growth and identify this transformation as quadratic submartingales which are defined by the multiplicative and additive decomposition. Under mild assumptions on the integrabilibility of the exponential terminal value, they derive a general stability result with strong convergence in martingale parts, from \(\mathbb{H}^1\) to BMO-martingales, and convergence in total variation of the finite variation parts. The stability is characterized by an almost sure-convergence regarding to the strong convergence of the martingale parts. When the quadratic semimartingale is bounded, the results are new and direct and within a BSDE-framework in \(\mathbb{H}^1\). For BSDE-like quadratic semimartingales, the authors prove that also the limit process is a BSDE-like semimartingale. Existence results are also established in a more general framework using inf-convolution. In contrast to the approach of M. Kobylanski [Ann. Probab. 28, No. 2, 558–602 (2000; Zbl 1044.60045)], the authors do not assume bounded solutions. These results can be extended to jump processes.

MSC:
60G48 Generalizations of martingales
60G07 General theory of stochastic processes
60G44 Martingales with continuous parameter
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H99 Stochastic analysis
91B26 Auctions, bargaining, bidding and selling, and other market models
91B16 Utility theory
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References:
[1] Ankirchner, S., Imkeller, P. and Dos Reis, G. (2007). Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12 1418-1453 (electronic). · Zbl 1138.60042
[2] Azéma, J., Gundy, R. F. and Yor, M. (1980). Sur l’intégrabilité uniforme des martingales continues. In Seminar on Probability , XIV ( Paris , 1978 / 1979) ( French ). Lecture Notes in Math. 784 53-61. Springer, Berlin. · Zbl 0442.60046
[3] Barlow, M. T. and Protter, P. (1990). On convergence of semimartingales. In Séminaire de Probabilités , XXIV , 1988 / 89. Lecture Notes in Math. 1426 188-193. Springer, Berlin. · Zbl 0703.60041
[4] Barrieu, P. and El Karoui, N. (2004). Optimal derivatives design under dynamic risk measures. In Mathematics of Finance (G. Yin and Q. Zhang, eds.). Contemp. Math. 351 13-25. Amer. Math. Soc., Providence, RI. · Zbl 1070.91019
[5] Barrieu, P. and El Karoui, N. (2009). Pricing, hedging and optimally designing derivatives via minimization of risk measures. In Volume on Indifference Pricing (R. Carmona, ed.) 77-146. Princeton Univ. Press, Princeton. · Zbl 1189.91200
[6] Briand, P. and Hu, Y. (2006). BSDE with quadratic growth and unbounded terminal value. Probab. Theory Related Fields 136 604-618. · Zbl 1109.60052
[7] Briand, P. and Hu, Y. (2008). Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 543-567. · Zbl 1141.60037
[8] Carmona, R., ed. (2009). Indifference Pricing : Theory and Applications . Princeton Univ. Press, Princeton, NJ. · Zbl 1155.91008
[9] Choulli, T. and Schweizer, M. (2011). Stability of Sigma-martingale densities in \(L\log L\) under an equivalent change of measure. NCCR FINRISK Working Paper 676, ETH Zurich.
[10] Delbaen, F., Hu, Y. and Bao, X. (2011). Backward SDEs with superquadratic growth. Probab. Theory Related Fields 150 145-192. · Zbl 1253.60072
[11] Delbaen, F., Hu, Y. and Richou, A. (2011). On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions. Ann. Inst. Henri Poincaré Probab. Stat. 47 559-574. · Zbl 1225.60093
[12] Dellacherie, C. (1979). Inégalités de convexité pour les processus croissants et les sousmartingales. In Séminaire de Probabilités , XIII ( Univ. Strasbourg , Strasbourg , 1977 / 78). Lecture Notes in Math. 721 371-377. Springer, Berlin. · Zbl 0416.60049
[13] Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel . Hermann, Paris. · Zbl 0323.60039
[14] Dos Reis, G. (2011). Some Advances on Quadratic BSDE : Theory-Numerics-Applications . Lap Lambert Academic Publishing, Germany.
[15] El-Karoui, N. and Hamadène, S. (2003). BSDEs and risk-sensitive control, zero-sum and nonzero-sum game problems of stochastic functional differential equations. Stochastic Process. Appl. 107 145-169. · Zbl 1075.60534
[16] El Karoui, N., Hamadène, S. and Matoussi, A. (2009). BSDEs and applications. In Volume on Indifference Pricing (R. Carmona, ed.) 267-320. Princeton Univ. Press, Princeton. · Zbl 1168.60030
[17] El Karoui, N. and Huang, S. J. (1997). A general result of existence and uniqueness of backward stochastic differential equations. In Backward Stochastic Differential Equations ( Paris , 1995 - 1996) (N. El Karoui and L. Mazliak, eds.). Pitman Res. Notes Math. Ser. 364 27-36. Longman, Harlow. · Zbl 0887.60064
[18] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1-71. · Zbl 0884.90035
[19] Fleming, W. H. and Sheu, S. J. (2000). Risk-sensitive control and an optimal investment model. Math. Finance 10 197-213. · Zbl 1039.93069
[20] Föllmer, H. and Schied, A. (2004). Stochastic Finance : An Introduction in Discrete Time , extended ed. de Gruyter Studies in Mathematics 27 . de Gruyter, Berlin. · Zbl 1126.91028
[21] Frittelli, M. (2000). The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10 39-52. · Zbl 1013.60026
[22] Harremoës, P. (2008). Some new maximal inequalities. Statist. Probab. Lett. 78 2776-2780. · Zbl 1156.60014
[23] Hiriart-Urruty, J. B. and Lemaréchal, C. (2004). Fundamentals of Convex Analysis . Springer, Berlin. · Zbl 0998.49001
[24] Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Probab. 15 1691-1712. · Zbl 1083.60048
[25] Hu, Y. and Schweizer, M. (2011). Some new BSDE results for an infinite-horizon stochastic control problem. In Advanced Mathematical Methods for Finance 367-395. Springer, Heidelberg. · Zbl 1230.91202
[26] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558-602. · Zbl 1044.60045
[27] Lenglart, E., Lépingle, D. and Pratelli, M. (1980). Présentation unifiée de certaines inégalités de la théorie des martingales. In Seminar on Probability , XIV ( Paris , 1978 / 1979) ( French ). Lecture Notes in Math. 784 26-52. Springer, Berlin. · Zbl 0427.60042
[28] Lepeltier, J. P. and San Martin, J. (1997). Backward stochastic differential equations with continuous coefficient. Statist. Probab. Lett. 32 425-430. · Zbl 0904.60042
[29] Lepeltier, J. P. and San Martín, J. (1998). Existence for BSDE with superlinear-quadratic coefficient. Stochastics Stochastics Rep. 63 227-240. · Zbl 0910.60046
[30] Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15 2113-2143. · Zbl 1134.91449
[31] Mania, M. and Tevzadze, R. (2006). An exponential martingale equation. Electron. Commun. Probab. 11 206-216 (electronic). · Zbl 1112.60035
[32] Mocha, M. and Westray, N. (2011). Quadratic semimartingale BSDEs under and exponential moments condition. Working paper. Available at . 1101.2582v1 · Zbl 1254.60061
[33] Morlais, M.-A. (2009). Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem. Finance Stoch. 13 121-150. · Zbl 1199.91188
[34] Morlais, M. A. (2010). A new existence result for BSDEs with jumps and application to the utility maximization problem. Stochastic Process. Appl. 120 1966-1995. · Zbl 1232.91631
[35] Neveu, J. (1972). Martingales à Temps Discret . Dunod, Paris. · Zbl 0235.60010
[36] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55-61. · Zbl 0692.93064
[37] Protter, P. E. (2005). Stochastic Integration and Differential Equations , 2nd ed. Stochastic Modelling and Applied Probability 21 . Springer, Berlin.
[38] Rouge, R. and El Karoui, N. (2000). Pricing via utility maximization and entropy. Math. Finance 10 259-276. · Zbl 1052.91512
[39] Tevzadze, R. (2008). Solvability of backward stochastic differential equations with quadratic growth. Stochastic Process. Appl. 118 503-515. · Zbl 1175.60064
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