JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2026, Vol. 61 ›› Issue (4): 123-132.doi: 10.6040/j.issn.1671-9352.0.2024.122

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Combination solution method for stochastic differential equations with Poisson jumps

DUAN Yingpeng, HU Lin*   

  1. School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, Jiangxi, China
  • Online:2026-04-20 Published:2026-04-08

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Abstract: For stochastic differential equations containing only one random term(Poisson term), the Poisson-type Itô formula is given. The existence and uniqueness conditions for the solution of stochastic differential equations containing only Poisson terms are obtained. The boundedness and convergence of the compensated θ method on such equations are proved. For general stochastic differential equations with Poisson jumps, a new combined solution method is established. This combined solution method can help obtain analytical and numerical solutions for some stochastic differential equations with Poisson jumps, and can also be used to correct numerical solutions and improve the convergence of numerical solutions.

Key words: stochastic differential equation, Poisson jump, combination solution method, numerical solution

CLC Number: 

  • O211.63
[1] PADGETT W J, TSOKOS C P. The origins and applications of stochastic integral equations[J]. International Journal of Systems Science, 1971, 2(2):135-148.
[2] ITÔ K. Stochastic integral[J]. Proceedings of the Imperial Academy, 1944, 20(8):519-524.
[3] ITÔ K. On stochastic differential equations[M]. New York: American Mathematical Society, 1951:289-302.
[4] KOU S G. A jump-diffusion model for option pricing[J]. Management science, 2002, 48(8):1086-1101.
[5] SOBCZYK K. Stochastic differential equations:with applications to physics and engineering[M]. Berlin: Springer, 2013:15-35.
[6] SITU R. Theory of stochastic differential equations with jumps and applications:mathematical and analytical techniques with applications to engineering[M]. New York: Springer, 2005:552-563.
[7] HIGHAM D J, KLOEDEN P E. Numerical methods for nonlinear stochastic differential equations with jumps[J]. Numerische Mathematik, 2005, 101(1):101-119.
[8] HIGHAM D J, KLOEDEN P E. Convergence and stability of implicit methods for jump-diffusion systems[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2006, 3(2):125-140.
[9] RONGHUA L, HONGBING M, YONGHONG D. Convergence of numerical solutions to stochastic delay differential equations with jumps[J]. Applied Mathematics and Computation, 2006, 172(1):584-602.
[10] CHALMERS G D, HIGHAM D J. Asymptotic stability of a jump-diffusion equation and its numerical approximation[J]. SIAM Journal on Scientific Computing, 2009, 31(2):1141-1155.
[11] WEI M. Convergence of numerical solutions for variable delay differential equations driven by Poisson random jump measure[J]. Applied Mathematics and Computation, 2009, 212(2):409-417.
[12] LI Qiyong, GAN Siqing. Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps[J]. Journal of Applied Mathematics and Computing, 2011, 37(1):541-557.
[13] MARUYAMA G. Continuous Markov processes and stochastic equations[J]. Rendiconti del Circolo Matematico di Palermo, 1955, 4:48-90.
[14] MILSTEIN G. Approximate integration of stochastic differential equations[J]. Theory of Probability & Its Applications, 1975, 19(3):557-562.
[15] RÜEMELIN W. Numerical treatment of stochastic differential equations[J]. SIAM Journal on Numerical Analysis, 1982, 19(3):604-613.
[16] HU Lin, GAN Siqing. Numerical analysis of the balanced implicit methods for stochastic pantograph equations with jumps[J]. Applied Mathematics and Computation, 2013, 223:281-297.
[17] SCHEUTZOW M. A stochastic Gronwall lemma[J]. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2013, 16(2):1350019.
[18] PLATEN E, BRUTI-LIBERATI N. Numerical solution of stochastic differential equations with jumps in finance[M]. Berlin:Springer, 2010:11-15.
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