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One barrier reflected backward doubly stochastic differential equations with discontinuous monotone coefficients. (English) Zbl 1261.60056
This paper studies reflected doubly stochastic backward differential equations of the form \[ y_t=\xi + \int_{t}^{T}f(s,y_s,z_s)\,ds+ \int_{t}^{T}g(s,y_s,z_s)dB_s-\int_{t}^{T}z_s dW_s +K_T-K_t, \] \[ y_t \geq S_t,\;K_0=0,\;\int_{0}^{T}(y_t-S_t)dK_t =0, \] where the first stochastic integral is a backward Itō integral and the second one is a standard forward Itō integral. In the above \(S_t\) is given and the equation has to be solved for the triple \((y_t,z_t,K_t)\). \(S_t\) is a given stochastic process such that the unknown process \(y_t\) must always stay above it (called the obstacle) whereas \(K_t\) is a reflecting process. It is the aim of the paper to establish existence and comparison results for the solutions of this equation under the assumption of discontinuous and monotone coefficients.
The proof employs approximation techniques. For example, the generator \(f\) is approximated by an increasing sequence of Lipschitz functions satisfying certain boundedness properties. A sequence of approximation problems is then formulated the solution of which enjoys certain monotonicity and comparison properties that allow us to go to the limit in a meaningful way and to obtain the solution of the original problem. The comparison results proceed in a similar fashion.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Aman, A.; Owo, J.M., Reflected backward doubly stochastic differential equations with discontinuous generator, Random oper. stoch. equ., (2012) · Zbl 1284.60113
[2] Bahlali, K.; Elouaflin, A.; N’zi, M., Backward stochastic differential equations with stochastic monotone coefficients, J. appl. math. stoch. anal., 4, 317-335, (2004) · Zbl 1064.60127
[3] Bahlali, K.; Hassani, M.; Mansouri, B.; Mrhardy, N., One barrier reflected backward doubly differential equations with continuous generator, C. R. math. acad. sci. Paris, 347, 1201-1206, (2009) · Zbl 1176.60041
[4] Boufoussi, B.; Ouknine, Y., On a SDE driven by a fractional Brownian motion and with monotone drift, Electron. comm. probab, 8, 122-134, (2003) · Zbl 1060.60060
[5] Fan, S.; Jiang, L., Uniqueness result for the BSDE whose generator is monotonic in \(y\) and uniformly continuous in \(z\), C. R. acad. sci. Paris, I, 348, 89-92, (2010) · Zbl 1193.60076
[6] Fan, S.; Liu, D., A class of BSDE with integrable parameters, Statist. probab. lett., 80, 2014-2031, (2010) · Zbl 1204.60049
[7] Jia, G., A class of backward stochastic differential equations with discontinuous coefficients, Statist. probab. lett., 78, 231-237, (2007) · Zbl 1134.60041
[8] Lepeltier, J.P.; San Martin, J., Backward stochastic differential equations with continuous coefficients, Statist. probab. lett., 79, 425-430, (1997) · Zbl 0904.60042
[9] Lin, Q., A class of backward doubly stochastic differential equations with non-Lipschitz coefficients, Statist. probab. lett., 79, 20, 2223-2229, (2009) · Zbl 1175.60062
[10] Lin, Q., A generalized existence theorem of backward doubly stochastic differential equations, Acta math. sin. (engl. ser.), 26, 8, 525-1534, (2010) · Zbl 1202.60090
[11] Lin, Q., Backward doubly stochastic differential equations with weak assumptions on the coefficients, Appl. math. comput., 217, 9322-9333, (2011) · Zbl 1220.60035
[12] N’zi, M.; Owo, J.M., Backward doubly stochastic differential equations with discontinuous coefficients, Statist. probab. lett., 79, 920-926, (2009) · Zbl 1168.60353
[13] Pardoux, E.; Peng, S., Adapted solutions of backward stochastic differential equations, Syst. control lett., 14, 55-61, (1990) · Zbl 0692.93064
[14] Pardoux, E.; Peng, S., Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. theory related fields, 98, 209-227, (1994) · Zbl 0792.60050
[15] Saisho, Y., SDE for multidimensional domains with reflecting boundary, Probab. theory related fields, 74, 455-477, (1987) · Zbl 0591.60049
[16] Shi, Y.; Gu, Y.; Liu, K., Comparison theorems of backward doubly stochastic differential equations and applications, Stoch. anal. appl., 23, 97-110, (2005) · Zbl 1067.60046
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