Asymptotic expansion for forward-backward SDEs with jumps

References

• D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd ed., Cambridge Studies in Advanced Mathematics, Cambridge University Press, LN, 2009.
   Google Scholar
• G. Barles, R. Buckdahn, and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stoch. Stoch. Rep. 60 (1997), pp. 57–83. doi: 10.1080/17442509708834099
   Google Scholar
• C. Bender and R. Denk, A forward scheme for backward SDEs, Stoch. Process. Appl. 117(12) (2007), pp. 1793–1812. doi: 10.1016/j.spa.2007.03.005
   Web of Science ®Google Scholar
• K. Bichteler, J. Gravereaux, and J. Jacod, Malliavin Calculus for Processes with Jumps, Stochastics Monographs, Gordon and Breach science publishers, LN, 1987.
   Google Scholar
• J.M. Bismut, Conjugate convex functions in optimal stochastic control, J. Math. Anal. Apl. 44 (1973), pp. 384–404. doi: 10.1016/0022-247X(73)90066-8
   Web of Science ®Google Scholar
• B. Bouchard and R. Elie, Discrete-time approximation of decoupled Forward-Backward SDE with Jumps, Stoch. Process. Appl. 118 (2008), pp. 53–75. doi: 10.1016/j.spa.2007.03.010
   Web of Science ®Google Scholar
• B. Bouchard and N. Touzi, Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations, Stoch. Process. Appl. 111(2) (2004), pp. 175–206. doi: 10.1016/j.spa.2004.01.001
   Web of Science ®Google Scholar
• S. Crépey and T.R. Bielecki, Counterparty Risk and Funding, CRC press, Taylor & Francis Group, NY, 2014. with an introductory dialogue by Damiano Brigo.
   Google Scholar
• S. Crepey and A. Matoussi, Reflected and doubly reflected BSDEs with jumps: A priori estimates and comparison, Ann. Appl. Probab. 18(5) (2008), pp. 2041–2069. doi: 10.1214/08-AAP517
   Web of Science ®Google Scholar
• S. Crépey and S. Song, Counterparty Risk and Funding: Immersion and Beyond, Working paper, 2015.
   Google Scholar
• J. Cvitanić and J. Zhang, Contract Theory in Continuous-time Methods, Springer, Berlin, 2013.
   Google Scholar
• L. Delong, Backward Stochastic Differential Equations with Jumps and their Actuarial and Financial Applications, Springer-Verlag, LN, 2013.
   Google Scholar
• L. Delong and P. Imkeller, On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures, Stoch. Process. Appl. 120(9) (2010), pp. 1748–1775. doi: 10.1016/j.spa.2010.05.001
   Web of Science ®Google Scholar
• G. Di Nunno, B. Oksendal, and F. Proske, Malliavin Calculus for Lévy Processes with Applications to Finance, Springer, NY, 2009.
   Google Scholar
• K. Dzhaparidze and E. Valkeila, On the Hellinger type distances for filtered experiments, Probab. Theory Related Fields 85(1) (1990), pp. 105–117. doi: 10.1007/BF01377632
   Web of Science ®Google Scholar
• N. El Karoui and L. Mazliak, (eds.), Backward Stochastic Differential Equations, Addison Wesley Longman Limited, U.S, 1997.
   Google Scholar
• N. El Karoui, S. Peng, and Quenez M.-C., Backward stochastic differential equations in finance, Math. Finance 7(1) (1997), pp. 1–71. doi: 10.1111/1467-9965.00022
   Web of Science ®Google Scholar
• R. Elie, Contrôle stochastique et méthodes numériques en finance mathématique, Ph.D Thesis, University Paris-Dauphine, 2006.
   Google Scholar
• M. Fujii, Momentum-space approach to asymptotic expansion for stochastic filtering, Ann. Inst. Stat. Math. 66 (2014), pp. 93–120. doi: 10.1007/s10463-013-0405-1
   Web of Science ®Google Scholar
• M. Fujii, A polynomial scheme of asymptotic expansion for backward SDEs and option pricing, Quant. Finance 16(3) (2016), pp. 427–445. doi: 10.1080/14697688.2015.1036770
   Web of Science ®Google Scholar
• M. Fujii, Optimal position management for a market-maker with stochastic price impacts, Working paper, CARF-F-360, 2015.
   Google Scholar
• M. Fujii and A. Takahashi, Analytical approximation for non-linear FBSDEs with perturbation scheme, Int. J. Theor. Appl. Finance 15(5) (2012), p. 1250034(24). doi: 10.1142/S0219024912500343
   Google Scholar
• M. Fujii and A. Takahashi, Perturbative expansion technique for non-linear FBSDEs with interacting particle method, Asia-Pacific Financ. Markets 22(3) (2015), pp. 283–304. doi: 10.1007/s10690-015-9201-7
   Web of Science ®Google Scholar
• M. Fujii and A. Takahashi, Solving backward stochastic differential equations with quadratic-growth drivers by connecting the short-term expansions, Stoch. Process. Appl. (2018), in press. Available at https://doi.org/10.1016/j.spa.2018.05.009.
   Google Scholar
• M. Fujii and A. Takahashi, Quadratic-exponential growth BSDEs with jumps and their Malliavin's differentiability, Stoch. Process. Appl. 128(6) (2018), pp. 2083–2130. Available at https://doi.org/10.1016/j.spa.2017.09.002.
   Web of Science ®Google Scholar
• I.I. Gikhman and A.V. Skorohod, Stochastic Differential Equations, Springer, Berlin, 1972.
   Google Scholar
• E. Gobet, J.P. Lemor, and X. Warin, A regression-based Monte Carlo method to solve backward stochastic differential equations, Ann. Appl. Probab. 15(3) (2005), pp. 2172–2202. doi: 10.1214/105051605000000412
   Web of Science ®Google Scholar
• S. Hamadène and M. Jeanblanc, On the stating and stopping problem: Application in reversible investments, Math. Oper. Res. 32(1) (2007), pp. 182–192. doi: 10.1287/moor.1060.0228
   Web of Science ®Google Scholar
• S. He, J. Wang, and J. Yan, Semimartingale Theory and Stochastic Calculus, CRC Press, London, UK, 1992.
   Google Scholar
• P. Henry-Labordère, Cutting CVA's complexity, Risk Mag. July Issue (2012), pp. 67–73.
   Google Scholar
• S. Igari, Introduction of Real Analysis, Iwanami publisher, Japanese, 1996.
   Google Scholar
• N. Kazamaki, A sufficient condition for the uniform integrability of exponential martingales, Math. Rep. Toyama Univ. 2 (1979), pp. 1–11. MR-0542374.
   Google Scholar
• T. Kruse and T. Popier, BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration, Stochastics 88(4) (2015), pp. 491–539. 2016. doi: 10.1080/17442508.2015.1090990
   Google Scholar
• A. Lejay, E. Mordecki, and S. Torres, Numerical approximation of backward stochastic differential equations with jumps, HAL: inria-00357992, 2014.
   Google Scholar
• G. Leoni, A First Course in Sobolev Spaces, Graduate Studies in Mathematics, vol. 105, American Mathematical Society, US, 2009.
   Google Scholar
• A.E. Lim, Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market, Math. Oper. Res. 29( 1) (2004), pp. 132–161. doi: 10.1287/moor.1030.0065
   Web of Science ®Google Scholar
• R.Sh. Liptser and A.N. Shiryayev, Theory of Martingales, Mathematics and its Applications, Kluwer Academic Publishers, LN, 1989.
   Google Scholar
• J. Ma and J. Yong, Forward-backward Stochastic Differential Equations and their Applications, Springer, Berlin, 2000.
   Google Scholar
• J. Ma and J. Zhang, Representation theorems for backward stochastic differential equations, Ann. Appl. Probab. 12(4) (2002), pp. 1390–1418. doi: 10.1214/aoap/1037125868
   Web of Science ®Google Scholar
• M. Mania and R. Tevzadze, Backward stochastic PDE and imperfect hedging, Int. J. Theor. Appl. Finance 6(7) (2003), pp. 663–692. doi: 10.1142/S0219024903002122
   Google Scholar
• M-A. Morlais, A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem, Stoch. Process. Appl. 120 (2010), pp. 1966–1995. doi: 10.1016/j.spa.2010.05.011
   Web of Science ®Google Scholar
• E. Pardoux and S. Peng, Adapted solution of a backward stochastic differential equation, Syst. Control Lett. 14 (1990), pp. 55–61. doi: 10.1016/0167-6911(90)90082-6
   Web of Science ®Google Scholar
• E. Pardoux and S. Peng, Backward stochastic differential equations and quasilinear parabolic partial differential equations, Lecture Notes in Control and Inform. Sci., vol. 176, Springer, New York, 1992, pp. 200–217.
   Google Scholar
• E. Pardoux and A. Rascanu, Stochastic Differential Equations, Backward SDEs, Partial Differential Equations, Springer International Publishing, Switzerland, 2014.
   Google Scholar
• E. Petrou, Malliavin calculus in Lévy spaces and applications to finance, Electric J. Probab. 13(27) (2008), pp. 852–879. doi: 10.1214/EJP.v13-502
   Web of Science ®Google Scholar
• H. Pham, Stochastic control under progressive enlargement of filtrations and applications to multiple defaults risk management, Stoch. Process. Appl. 120 (2010), pp. 1795–1820. doi: 10.1016/j.spa.2010.05.003
   Web of Science ®Google Scholar
• P.E. Protter, Stochastic Integration and Differential Equations, Springer, NY, 2005.
   Google Scholar
• M.C. Quenez and A. Sulem, BSDEs with jumps, optimization and applications to dynamic risk measures, Stoch. Process. Appl. 123(8) (2013), pp. 3328–3357. doi: 10.1016/j.spa.2013.02.016
   Web of Science ®Google Scholar
• M. Royer, Backward stochastic differential equations with jumps and related non-linear expectations, Stoch. Process. Appl. 116 (2006), pp. 1358–1376. doi: 10.1016/j.spa.2006.02.009
   Web of Science ®Google Scholar
• A. Takahashi, Asymptotic expansion approach in finance, in Large Deviations and Asymptotic Methods in Finance, P. Fritz, J. Gatheral, A. Gulisashvili, A. Jacquier and J. Teichmann, editors, Springer, New York, 2015, pp. 345–411.
   Google Scholar
• A. Takahashi and T. Yamada, A weak approximation with asymptotic expansion and multidimensional Malliavin weights, Ann. Appl. Probab. 26(2) (2016), pp. 818–856.
   Google Scholar
• N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDEs, Springer, New York, 2013.
   Google Scholar
• J. Zhang, A numerical scheme for bsdes, Ann. Appl. Probab. 14(1) (2004), pp. 459–488. doi: 10.1214/aoap/1075828058
   Web of Science ®Google Scholar