We show the existence and uniqueness of a continuous viscosity solution of a system of partial differential equations (PDEs for short) without assuming the usual monotonicity conditions on the driver function as in Hamadène and Morlais’s article [Applied Mathematics & Optimization67 (2013), 163–196]. Our method strongly relies on the link between PDEs and reflected backward stochastic differential equations with interconnected obstacles for which we already know that the solution exists and is unique for general drivers.
J.-F.Chassagneux, R.Elie and I.Kharroubi, A note on existence and uniqueness for solutions of multidimensional reflected bsdes, Electronic Communications in Probability16 (2011), 120–128. doi:10.1214/ECP.v16-1614.
2.
B.Djehiche, S.Hamadene and M.A.Morlais, Viscosity solutions of systems of variational inequalities with interconnected bilateral obstacles, Funkcialaj Ekvacioj58 (2015), 135–175. doi:10.1619/fesi.58.135.
3.
D.Duffie and L.Epstein, Stochastic differential utility, Econometrica60 (1992), 353–394. doi:10.2307/2951600.
4.
B.El Asri and S.Hamadène, The finite horizon optimal multi-modes switching problem: The viscosity solution approach, Applied Mathematics & Optimization60(2) (2009), 213–235. doi:10.1007/s00245-009-9071-3.
5.
N.El Karoui, C.Kapoudjian, E.Pardoux, S.Peng and M.C.Quenez, Reflected solutions of backward SDE’s, and related obstacle problems for PDE’s, The Annals of Probability (1997), 702–737.
6.
N.El Karoui, S.Peng and M.C.Quenez, Backward stochastic differential equations in finance, Mathematical finance7(1) (1997), 1–71.
7.
S.Hamadène and M.-A.Morlais, Viscosity solutions of systems of PDEs with interconnected obstacles and switching problem, Applied Mathematics & Optimization67(2) (2013), 163–196. doi:10.1007/s00245-012-9184-y.
8.
S.Hamadène and J.Zhang, Switching problem and related system of reflected backward sdes, Stochastic Processes and their applications120(4) (2010), 403–426. doi:10.1016/j.spa.2010.01.003.
9.
Y.Hu and S.Tang, Multi-dimensional bsde with oblique reflection and optimal switching, Probability Theory and Related Fields147(1) (2010), 89–121. doi:10.1007/s00440-009-0202-1.
10.
E.Pardoux and S.Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters14(1) (1990), 55–61. doi:10.1016/0167-6911(90)90082-6.
11.
D.Revuz and M.Yor, Continuous Martingales and Brownian Motion, Vol. 293, Springer Science & Business Media, 2013.
