Doubly reflected BSDEs with call protection and their approximation
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 613-641.

We study the numerical approximation of doubly reflected backward stochastic differential equations with intermittent upper barrier (RIBSDEs). These denote reflected BSDEs in which the upper barrier is only active on certain random time intervals. From the point of view of financial interpretation, RIBSDEs arise as pricing equations of game options with constrained callability. In a Markovian set-up we prove a convergence rate for a time-discretization scheme by simulation to an RIBSDE. We also characterize the solution of an RIBSDE as the largest viscosity subsolution of a related system of variational inequalities, and we establish the convergence of a deterministic numerical scheme for that problem. Due to the potentially very high dimension of the system of variational inequalities, this approach is not always practical. We thus subsequently prove a convergence rate for a time-discretisation scheme by simulation to an RIBSDE.

DOI : 10.1051/ps/2013047
Classification : 93E20, 65C99, 60H30
Mots clés : reflected BSDEs, variational inequalities, discrete-time approximation, game option, Call protection
@article{PS_2014__18__613_0,
     author = {Chassagneux, Jean-Fran\c{c}ois and Cr\'epey, St\'ephane},
     title = {Doubly reflected {BSDEs} with call protection and their approximation},
     journal = {ESAIM: Probability and Statistics},
     pages = {613--641},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2013047},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps/2013047/}
}
TY  - JOUR
AU  - Chassagneux, Jean-François
AU  - Crépey, Stéphane
TI  - Doubly reflected BSDEs with call protection and their approximation
JO  - ESAIM: Probability and Statistics
PY  - 2014
SP  - 613
EP  - 641
VL  - 18
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/ps/2013047/
DO  - 10.1051/ps/2013047
LA  - en
ID  - PS_2014__18__613_0
ER  - 
%0 Journal Article
%A Chassagneux, Jean-François
%A Crépey, Stéphane
%T Doubly reflected BSDEs with call protection and their approximation
%J ESAIM: Probability and Statistics
%D 2014
%P 613-641
%V 18
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/ps/2013047/
%R 10.1051/ps/2013047
%G en
%F PS_2014__18__613_0
Chassagneux, Jean-François; Crépey, Stéphane. Doubly reflected BSDEs with call protection and their approximation. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 613-641. doi : 10.1051/ps/2013047. http://archive.numdam.org/articles/10.1051/ps/2013047/

[1] V. Bally and G. Pagès, Error analysis of the quantization algorithm for obstacle problems. Stoch. Process. Appl. 106 (2003) 1-40. | MR | Zbl

[2] G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations. Asymptot. Anal. 4 (1991) 271-283. | MR | Zbl

[3] B. Bouchard and J.-F. Chassagneux, Discrete time approximation for continuously and discretely reflected BSDEs. Stoch. Process. Appl. 118 (2008) 2269-2293. | MR | Zbl

[4] B. Bouchard and S. Menozzi, Strong Approximations of BSDEs in a domain. Bernoulli 15 (2009) 1117-1147. | MR | Zbl

[5] J.-F. Chassagneux, Processus réfléchis en finance et probabilité numérique. Ph.D. thesis Université Paris Diderot - Paris (2008) 7.

[6] J.-F. Chassagneux, Discrete time approximation of doubly reflected BSDEs. Adv. Appl. Probab. 41 (2009) 101-130. | MR | Zbl

[7] M. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (1992). | MR | Zbl

[8] S. Crépey, Financial Modeling: A Backward Stochastic Differential Equations Perspective. Springer Finance Textbooks. Springer (2013). | MR | Zbl

[9] S. Crépey and A. Matoussi, Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison principle. Ann. Appl. Probab. 18 (2008) 2041-2069. | MR | Zbl

[10] S. Crépey and A. Rahal, Pricing Convertible Bonds with Call Protection. J. Comput. Finance 15 (2011/12) 37-75.

[11] J. Cvitanić and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games. Ann. Probab. 24 (1996) 2024-2056. | MR | Zbl

[12] E.B. Dynkin, Game variant of a problem on optimal stopping. Soviet Math. Dokl. 10 (1969) 270-274. | Zbl

[13] N. El Karoui, E. Kapoudjian, C. Pardoux and S. Peng, and M.-C. Quenez, Reflected solutions of backward SDE's, and related obstacle problems for PDE's. Ann. Probab. 25 (1997) 702-737. | MR | Zbl

[14] N. El Karoui, S. Peng and M.-C. Quenez, Backward stochastic differential equations in finance. Math. Finance 7 (1997) 1-71. | MR | Zbl

[15] E. Gobet and A. Makhlouf L2-time regularity of BSDEs with irregular terminal functions. Stoch. Process. Appl. 120 (2010) 1105-1132. | MR | Zbl

[16] S. Hamadène, Reflected BSDEs with Discontinuous Barrier and Application. Stoch. Stoch. Reports 74 (2002) 571-596. | MR | Zbl

[17] S. Hamadène and M. Hassani, BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game. Electr. J. Probab. 11 (2006) 121-145. | MR | Zbl

[18] S. Hamadène and M. Hassani, BSDEs with two reflecting barriers: the general result. Probab. Theory Relat. Fields 132 (2005) 237-264. | MR | Zbl

[19] S. Hamadène, M. Hassani and Y. Ouknine, BSDEs with general discontinuous reflecting barriers without Mokobodski's condition. Bull. Sci. Math. 134 (2010) 874-899. | MR | Zbl

[20] Y. Kifer, Game options. Fin. Stoch. 4 (2000) 443-463. | MR | Zbl

[21] P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations. Springer (2000). | MR | Zbl

[22] J.-P. Lepeltier and M. Xu, Reflected backward stochastic differential equations with two RCLL barriers. ESAIM: PS 4 (2007) 3-22. | Numdam | MR | Zbl

[23] D. Nualart, The Malliavin Calculus and Related Topics, 2nd Edition. Springer (2006). | MR | Zbl

Cité par Sources :