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Stochastics
An International Journal of Probability and Stochastic Processes
Volume 94, 2022 - Issue 8
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Research Article

Solving some stochastic partial differential equations driven by Lévy noise using two SDEs*

Pages 1265-1298 | Received 28 Apr 2021, Accepted 12 Jan 2022, Published online: 01 Feb 2022

References

  • H. Allouba and W. Zheng, Brownian-time processes: The PDE connection and the half-derivative generator, Annals of Probability 29(4) (2001), pp. 1780–1795.
  • T. Amaba, D. Taguchi, and G. Yûki, Convergence implications via dual flow method, Markov Process. Relat. Fields 25(3) (2019), pp. 533–568.
  • S. Asmussen and J. Rosiński, Approximations of small jumps of Lévy processes with a view towards simulation, J. Appl. Probab. 38(2) (2001), pp. 482–493.
  • M.T. Barlow and M. Yor, Semi-martingale inequalities via the Garsia–Rodemich–Rumsey lemma, and applications to local times, J. Funct. Anal. 49(2) (1982), pp. 198–229.
  • N. Bruti-Liberati and E. Platen, Strong approximations of stochastic differential equations with jumps, J. Comput. Appl. Math. 205(2) (2007), pp. 982–1001.
  • S. Dereich, Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction, Ann. Appl. Probab. 21(1) (2011), pp. 283–311.
  • N. El Karoui and M. Mrad, An exact connection between two solvable SDEs and a non linear utility stochastic PDEs, SIAM J. Financ. Math. 4(1) (2013), pp. 737–783.
  • T. Fujiwara and H. Kunita, Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group, J. Math. Kyoto Univ. 25(1) (1985), pp. 71–106.
  • A.M. Garsia, E. Rodemich, and H. Rumsey Jr., A real variable lemma and the continuity of paths of some Gaussian processes, Indiana Univ. Math. J. 20(6) (1970), pp. 565–578.
  • M.B. Giles, Multilevel Monte Carlo path simulation, Oper. Res. 56 (2008), pp. 607–617.
  • E. Gobet and M. Mrad, Strong approximation of stochastic processes at random times and application to their exact simulation, Stochastics 89(6–7) (2017), pp. 883–895.
  • E. Gobet and M. Mrad, Convergence rate of strong approximations of compound random maps, application to SPDEs, Discrete Contin. Dyn. Syst. B 23(10) (2018), p. 4455.
  • S. Heinrich, Multilevel monte carlo methods, in International Conference on Large-Scale Scientific Computing, Lecture Notes in Computer Science vol. 2179, Springer, Berlin, Heidelberg, 2001. pp. 58–67.
  • A. Kohatsu-Higa, S. Ortiz-Latorre, and P. Tankov, Optimal simulation schemes for Lévy driven stochastic differential equations, Math. Comput. 83(289) (2014), pp. 2293–2324.
  • A. Kohatsu-Higa and P. Tankov, Jump-adapted discretization schemes for Lévy-driven SDEs, Stoch. Process. Appl. 120(11) (2010), pp. 2258–2285.
  • H. Kunita, Stochastic Flows and Jump-Diffusions, Springer, 2019.
  • H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics Vol. 24, Cambridge University Press, 1997.
  • H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, in Real and Stochastic Analysis, Springer, 2004, pp. 305–373.
  • A. Matoussi and M. Mrad, Dynamic Utility and related nonlinear SPDEs driven by Levy Noise. International Journal of Theoretical and Applied Finance. https://doi.org/10.1142/S0219024922500042.
  • E. Mordecki, A. Szepessy, R. Tempone, and G.E. Zouraris, Adaptive weak approximation of diffusions with jumps, SIAM J. Numer. Anal. 46(4) (2008), pp. 1732–1768.
  • M. Musiela and T. Zariphopoulou, Investment and valuation under backward and forward dynamic exponential utilities in a stochastic factor model, in Advances in Mathematical Finance, Birkhäuser, Boston, 2007, pp. 303–334.
  • M. Musiela and T. Zariphopoulou, Stochastic partial differential equations and portfolio choice, in Contemporary Quantitative Finance, Springer, 2010, pp. 195–216.
  • D. Nualart, Malliavin Calculus and Related Topics, 2nd ed., Springer, Berlin, 2006.
  • B. Øksendal, T. Zhang, others, The Itô-Ventzell formula and forward stochastic differential equations driven by Poisson random measures, Osaka J. Math. 44(1) (2007), pp. 207–230.
  • P. Protter and D. Talay, The Euler scheme for Lévy driven stochastic differential equations, Ann. Probab. 25(1) (1997), pp. 393–423.
  • D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd ed., Comprehensive Studies in Mathematics, Springer, Berlin, 1999.
  • C. Rhee and P.W. Glynn, A new approach to unbiased estimation for SDEs, in Proceedings of the 2012 Winter Simulation Conference, C. Laroque, J. Himmelspach, R. Pasupathy, O. Rose, and A. M. Uhrmacher, eds., pp. 495–503.
  • S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process, Stoch. Process. Appl. 103(2) (2003), pp. 311–349.

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