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Multivalued backward stochastic differential equations with time delayed generators

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Our aim is to study the following new type of multivalued backward stochastic differential equation: $$\left\{ \begin{gathered} - dY\left( t \right) + \partial \phi \left( {Y\left( t \right)} \right)dt \ni F\left( {t,Y\left( t \right),Z\left( t \right),Y_t ,Z_t } \right)dt + Z\left( t \right)dW\left( t \right), 0 \leqslant t \leqslant T, \hfill \\ Y\left( T \right) = \xi , \hfill \\ \end{gathered} \right.$$ where ∂φ is the subdifferential of a convex function and (Y t, Z t):= (Y(t + θ), Z(t + θ))θ∈[−T,0] represent the past values of the solution over the interval [0, t]. Our results are based on the existence theorem from Delong & Imkeller, Ann. Appl. Probab., 2010, concerning backward stochastic differential equations with time delayed generators.
Twórcy
Bibliografia
  • [1] Haïm Brézis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam, 1973.
  • [2] Łukasz Delong, Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management, preprint, 2011 (http://arxiv.org/abs/1005.4417).
  • [3] Łukasz Delong, Peter Imkeller, Backward stochastic differential equations with time delayed generators - results and counterexamples, The Annals of Applied Probability 20 (2010), no. 4, 1512–1536. http://dx.doi.org/10.1214/09-AAP663
  • [4] Łukasz Delong, Peter Imkeller, On Malliavin’s differentiability of BSDE with time delayed generators driven by Brownian motions and Poisson random measures, Stochastic Process. Appl. 120 (2010), no. 9, 1748–1775. http://dx.doi.org/10.1016/j.spa.2010.05.001
  • [5] Gonçalo dos Reis, Anthony Réveillac, Jianing Zhang, FBSDEs with time delayed generators: Lp-solutions, differentiability, representation formulas and path regularity, Stochastic Process. Appl. 121 (2011), no. 9, 2114–2150. http://dx.doi.org/10.1016/j.spa.2011.05.002
  • [6] Nicole El Karoui, Christophe Kapoudjian, Etienne Pardoux, Shige Peng, Marie-Claire Quenez, Reflected solutions of backward SDE’s and related obstacle problems for PDE’s, Ann.Probab. 25 (1997), no. 2, 702–737. http://dx.doi.org/10.1214/aop/1024404416
  • [7] Lucian Maticiuc, Aurel Răşcanu, A stochastic approach to a multivalued Dirichlet-Neumann problem, Stochastic Process. Appl. 120 (2010), no. 6, 777–800. http://dx.doi.org/10.1016/j.spa.2010.02.002
  • [8] Lucian Maticiuc, Aurel Răşcanu, Backward Stochastic Variational Inequalities on Random Interval, accepted for publication in Bernoulli, 2014 (http://arxiv.org/abs/1112.5792).
  • [9] Lucian Maticiuc, Eduard Rotenstein, Numerical schemes for multivalued backward stochastic differential systems, Central European Journal of Mathematics 10 (2012), no. 2, 693–702. http://dx.doi.org/10.2478/s11533-011-0131-y
  • [10] Etienne Pardoux, Shige Peng, Adapted solution of a backward stochastic differential equation, Systems Control Lett. 14 (1990), no. 1, 55–61. http://dx.doi.org/10.1016/0167-6911(90)90082-6
  • [11] Etienne Pardoux, Shige Peng, Backward SDE’s and quasilinear parabolic PDE’s, Stochastic PDE and Their Applications (B.L. Rozovskii, R.B. Sowers eds.), 200–217, LNCIS 176, Springer (1992).
  • [12] Etienne Pardoux, Aurel Răşcanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl. 76 (1998), no. 2, 191–215. http://dx.doi.org/10.1016/S0304-4149(98)00030-1
  • [13] Aurel Răşcanu, Eduard Rotenstein, The Fitzpatrick function-a bridge between convex analysis and multivalued stochastic differential equations, Journal of Convex Analysis 18 (2011), no. 1, 105–138.
  • [14] Qing Zhou, Yong Ren, Reflected backward stochastic differential equations with time delayed generators, Statistics and Probability Letters 82 (2012), no. 5, 979–990. http://dx.doi.org/10.1016/j.spl.2012.02.012
Typ dokumentu
Bibliografia
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