The Annals of Applied Probability

Differentiability of quadratic BSDEs generated by continuous martingales

Peter Imkeller, Anthony Réveillac, and Anja Richter

Full-text: Open access

Abstract

In this paper we consider a class of BSDEs with drivers of quadratic growth, on a stochastic basis generated by continuous local martingales. We first derive the Markov property of a forward–backward system (FBSDE) if the generating martingale is a strong Markov process. Then we establish the differentiability of a FBSDE with respect to the initial value of its forward component. This enables us to obtain the main result of this article, namely a representation formula for the control component of its solution. The latter is relevant in the context of securitization of random liabilities arising from exogenous risk, which are optimally hedged by investment in a given financial market with respect to exponential preferences. In a purely stochastic formulation, the control process of the backward component of the FBSDE steers the system into the random liability and describes its optimal derivative hedge by investment in the capital market, the dynamics of which is given by the forward component. The representation formula of the main result describes this delta hedge in terms of the derivative of the BSDEs solution process on the one hand and the correlation structure of the internal uncertainty captured by the forward process and the external uncertainty responsible for the market incompleteness on the other hand. The formula extends the scope of validity of the results obtained by several authors in the Brownian setting. It is designed to extend a genuinely stochastic representation of the optimal replication in cross hedging insurance derivatives from the classical Black–Scholes model to incomplete markets on general stochastic bases. In this setting, Malliavin’s calculus which is required in the Brownian framework, is replaced by new tools based on techniques related to a calculus of quadratic covariations of basis martingales.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 1 (2012), 285-336.

Dates
First available in Project Euclid: 7 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1328623701

Digital Object Identifier
doi:10.1214/11-AAP769

Mathematical Reviews number (MathSciNet)
MR2932548

Zentralblatt MATH identifier
1254.60058

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 60J25: Continuous-time Markov processes on general state spaces

Keywords
Forward–backward stochastic differential equation driven by continuous martingale quadratic growth Markov property BMO martingale utility indifference hedging and pricing sensitivity analysis stochastic calculus of variations delta hedge

Citation

Imkeller, Peter; Réveillac, Anthony; Richter, Anja. Differentiability of quadratic BSDEs generated by continuous martingales. Ann. Appl. Probab. 22 (2012), no. 1, 285--336. doi:10.1214/11-AAP769. https://projecteuclid.org/euclid.aoap/1328623701


Export citation

References

  • [1] Ankirchner, S. and Heyne, G. (2009). Cross hedging with stochastic correlation. Finance Stoch. To appear.
  • [2] 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).
  • [3] Ankirchner, S., Imkeller, P. and Dos Reis, G. (2010). Pricing and hedging of derivatives based on nontradable underlyings. Math. Finance 20 289–312.
  • [4] Bally, V., Pardoux, E. and Stoica, L. (2005). Backward stochastic differential equations associated to a symmetric Markov process. Potential Anal. 22 17–60.
  • [5] Briand, P. and Confortola, F. (2008). BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces. Stochastic Process. Appl. 118 818–838.
  • [6] Çinlar, E. and Jacod, J. (1981). Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981). Progress in Probability 1 159–242. Birkhäuser, Boston, MA.
  • [7] Çinlar, E., Jacod, J., Protter, P. and Sharpe, M. J. (1980). Semimartingales and Markov processes. Z. Wahrsch. Verw. Gebiete 54 161–219.
  • [8] Dellacherie, C. and Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies 29. North-Holland, Amsterdam.
  • [9] Doléans-Dade, C. (1976). On the existence and unicity of solutions of stochastic integral equations. Z. Wahrsch. Verw. Gebiete 36 93–101.
  • [10] 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, 19951996). Pitman Research Notes in Mathematics Series 364 27–36. Longman, Harlow.
  • [11] El Karoui, N., Peng, S. and Quenez, M. C. (1997). Backward stochastic differential equations in finance. Math. Finance 7 1–71.
  • [12] Frei, C. and Schweizer, M. (2009). Exponential utility indifference valuation in a general semimartingale model. In Optimality and Risk—Modern Trends in Mathematical Finance (F. Delbaen, M. Rásonyi and C. Stricker, eds.) 49–86. Springer, Berlin.
  • [13] Fukushima, M., Ōshima, Y. and Takeda, M. (1994). Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics 19. de Gruyter, Berlin.
  • [14] Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Probab. 15 1691–1712.
  • [15] Kazamaki, N. (1994). Continuous Exponential Martingales and BMO. Lecture Notes in Math. 1579. Springer, Berlin.
  • [16] Ma, J. and Zhang, J. (2002). Representation theorems for backward stochastic differential equations. Ann. Appl. Probab. 12 1390–1418.
  • [17] Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15 2113–2143.
  • [18] Morlais, M.-A. (2009). Quadratic BSDEs driven by a continuous martingale and applications to the utility maximization problem. Finance Stoch. 13 121–150.
  • [19] N’Zi, M., Ouknine, Y. and Sulem, A. (2006). Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations. Stochastic Process. Appl. 116 1319–1339.
  • [20] Protter, P. (1977). Markov solutions of stochastic differential equations. Z. Wahrsch. Verw. Gebiete 41 39–58.
  • [21] Protter, P. E. (1977). On the existence, uniqueness, convergence and explosions of solutions of systems of stochastic integral equations. Ann. Probab. 5 243–261.
  • [22] Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd ed. Stochastic Modelling and Applied Probability 21. Springer, Berlin.