Bernoulli

• Bernoulli
• Volume 24, Number 4A (2018), 2569-2609.

Special weak Dirichlet processes and BSDEs driven by a random measure

Abstract

This paper considers a forward BSDE driven by a random measure, when the underlying forward process $X$ is a special semimartingale, or even more generally, a special weak Dirichlet process. Given a solution $(Y,Z,U)$, generally $Y$ appears to be of the type $u(t,X_{t})$ where $u$ is a deterministic function. In this paper, we identify $Z$ and $U$ in terms of $u$ applying stochastic calculus with respect to weak Dirichlet processes.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2569-2609.

Dates
Revised: December 2016
First available in Project Euclid: 26 March 2018

https://projecteuclid.org/euclid.bj/1522051218

Digital Object Identifier
doi:10.3150/17-BEJ937

Mathematical Reviews number (MathSciNet)
MR3779695

Zentralblatt MATH identifier
06853258

Citation

Bandini, Elena; Russo, Francesco. Special weak Dirichlet processes and BSDEs driven by a random measure. Bernoulli 24 (2018), no. 4A, 2569--2609. doi:10.3150/17-BEJ937. https://projecteuclid.org/euclid.bj/1522051218

References

• [1] Bandini, E. Optimal control of Piecewise-Deterministic Markov Processes: A BSDE representation of the value function. In ESAIM: Control, Optimisation and Calculus of Variations. To appear. DOI:10.1051/cocv/2017009.
• [2] Bandini, E. (2015). Existence and uniqueness for BSDEs driven by a general random measure, possibly non quasi-left-continuous. Electron. Commun. Probab. 20 no. 71, 13.
• [3] Bandini, E. and Confortola, F. (2017). Optimal control of semi-Markov processes with a backward stochastic differential equations approach. Math. Control Signals Systems 29 Art. 1, 35.
• [4] Bandini, E. and Fuhrman, M. (2017). Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes. Stochastic Process. Appl. 127 1441–1474.
• [5] Bandini, E. and Russo, F. (2017). Weak Dirichlet processes with jumps. Stochastic Process. Appl. DOI:10.1016/j.spa.2017.04.001.
• [6] Barles, G., Buckdahn, R. and Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastic Rep. 60 57–83.
• [7] Becherer, D. (2006). Bounded solutions to backward SDE’s with jumps for utility optimization and indifference hedging. Ann. Appl. Probab. 16 2027–2054.
• [8] Buckdahn, R. (1993). Backward stochastic differential equations driven by a martingale. Unpublished manuscript.
• [9] Buckdahn, R. and Pardoux, E. (1994). BSDE’s with jumps and associated integral-stochastic differential equations. Preprint.
• [10] Carbone, R., Ferrario, B. and Santacroce, M. (2008). Backward stochastic differential equations driven by càdlàg martingales. Theory Probab. Appl. 52 304–314.
• [11] Ceci, C., Cretarola, A. and Russo, F. (2014). BSDEs under partial information and financial applications. Stochastic Process. Appl. 124 2628–2653.
• [12] Cohen, S. and Elliott, R. (2012). Existence, uniqueness and comparisons for BSDEs in general spaces. Ann. Probab. 40 2264–2297.
• [13] Confortola, F. and Fuhrman, M. (2014). Backward stochastic differential equations associated to Markov jump processes and applications. Stochastic Processes and Their Applications 124 289–316.
• [14] Confortola, F., Fuhrman, M. and Jacod, J. (2016). Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control. Ann. Appl. Probab. 26 1743–1773.
• [15] Davis, M.H.A. (1993). Markov Models and Optimization. Monographs on Statistics and Applied Probability 49. London: Chapman & Hall.
• [16] Delong, L. (2013). Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications. Berlin: Springer.
• [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, 19951996). Pitman Res. Notes Math. Ser. 364 27–36. Longman, Harlow.
• [18] Flandoli, F., Issoglio, E. and Russo, F. (2017). Multidimensional stochastic differential equations with distributional drift. Trans. Amer. Math. Soc. 369 1665–1688.
• [19] Flandoli, F., Russo, F. and Wolf, J. (2003). Some SDEs with distributional drift. I. General calculus. Osaka J. Math. 40 493–542.
• [20] Fuhrman, M. and Tessitore, G. (2003). Generalized directional gradients, backward stochastic differential equations and mild solutions of semilinear parabolic equations. Appl. Math. Optim. 108 263–298.
• [21] Gozzi, F. and Russo, F. (2006). Verification theorems for stochastic optimal control problems via a time dependent Fukushima–Dirichlet decomposition. Stochastic Process. Appl. 116 1530–1562.
• [22] Gozzi, F. and Russo, F. (2006). Weak Dirichlet processes with a stochastic control perspective. Stochastic Process. Appl. 116 1563–1583.
• [23] He, S., Wang, J. and Yan, J. (1992). Semimartingale Theory and Stochastic Calculus. New York: Science Press Bejiing.
• [24] Jacod, J. (1976). Un théorème de représentation pour les martingales discontinues. Z. Wahrsch. Verw. Gebiete 34 225–244.
• [25] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Berlin: Springer.
• [26] Jacod, J. and Protter, P.E. (2003). Probability Essentials. Berlin: Springer.
• [27] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Berlin: Springer.
• [28] Jeanblanc, M., Mania, M., Santacroce, M. and Schweizer, M. (2012). Mean-variance hedging via stochastic control and BSDEs for general semimartingales. Ann. Appl. Probab. 22 2388–2428.
• [29] Kallenberg, O. (1997). Foundations of Modern Probability. Probability and Its Applications (New York). New York: Springer.
• [30] Laachir, I. and Russo, F. (2016). BSDEs càdlàg martingale problems, and orthogonalization under basis risk. SIAM J. Financial Math. 7 308–356.
• [31] Pardoux, É. and Peng, S. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
• [32] Pardoux, É. and Peng, S. (1992). Backward stochastic differential equations and quasilinear parabolic partial differential equations. Lecture Notes in CIS 176 200–217.
• [33] Pardoux, E. and Răşcanu, A. (2014). Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Stochastic Modelling and Applied Probability 69. Cham: Springer.
• [34] Peng, S. (1991). Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stochastics 37 61–74.
• [35] Peng, S. (1992). A generalized dynamic programming principle and Hamilton–Jacobi–Bellman Equation. Stochastics 38 119–134.
• [36] Protter, P.E. (2005). Stochastic Integration and Differential Equations. 2nd ed. Stochastic Modelling and Applied Probability 21. Berlin: Springer.
• [37] Russo, F. and Trutnau, G. (2007). Some parabolic PDEs whose drift is an irregular random noise in space. Ann. Probab. 35 2213–2262.
• [38] Russo, F. and Wurzer, L. (2016). Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time. Stoch. Dyn. Available at DOI:10.1142/S0219493717500307. Preprint HAL INRIA 01023176.
• [39] Tang, S.J. and Li, X.J. (1994). Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32 1447–1475.
• [40] Xia, J. (2000). Backward stochastic differential equation with random measures. Acta Math. Appl. Sin. Engl. Ser. 16 225–234.