Stochastic differential equations with boundary conditions driven by a Poisson noise

Aureli Alabert (Universitat Autònoma de Barcelona)
Miguel Angel Marmolejo (Universidad del Valle)

Abstract


We consider one-dimensional stochastic differential equations with a boundary condition, driven by a Poisson process. We study existence and uniqueness of solutions and the absolute continuity of the law of the solution. In the case when the coefficients are linear, we give an explicit form of the solution and study the reciprocal process property.

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Pages: 230-254

Publication Date: March 24, 2004

DOI: 10.1214/EJP.v9-157

References

  • Alabert, Aureli; Ferrante, Marco; Nualart, David. Markov field property of stochastic differential equations. Ann. Probab. 23 (1995), no. 3, 1262--1288. MR1349171
  • Alabert, Aureli; Marmolejo, Miguel A. Differential equations with boundary conditions perturbed by a Poisson noise. Stochastic Process. Appl. 91 (2001), no. 2, 255--276. MR1807681
  • S. Bernstein. Sur les liaisons entre les grandeurs aléatoires. In Verh. Internat. Math.-Kongr., Zurich, pages 288--309, 1932.
  • Buckdahn, Rainer; Nualart, David. Skorohod stochastic differential equations with boundary conditions. Stochastics Stochastics Rep. 45 (1993), no. 3-4, 211--235. MR1306932
  • Carlen, Eric A.; Pardoux, Étienne. Differential calculus and integration by parts on Poisson space. Stochastics, algebra and analysis in classical and quantum dynamics (Marseille, 1988), 63--73, Math. Appl., 59, Kluwer Acad. Publ., Dordrecht, 1990. MR1052702
  • Donati-Martin, Catherine. Quasi-linear elliptic stochastic partial differential equation: Markov property. Stochastics Stochastics Rep. 41 (1992), no. 4, 219--240. MR1275584
  • León, Jorge A.; Ruiz de Chávez, J.; Tudor, C. Strong solutions of anticipating stochastic differential equations on the Poisson space. Bol. Soc. Mat. Mexicana (3) 2 (1996), no. 1, 55--63. MR1395911
  • León, Jorge A.; Solé, Josep L.; Vives, Josep. A pathwise approach to backward and forward stochastic differential equations on the Poisson space. Stochastic Anal. Appl. 19 (2001), no. 5, 821--839. MR1857898
  • León, Jorge A.; Tudor, Constantin. Chaos decomposition of stochastic bilinear equations with drift in the first Poisson-Itô chaos. Statist. Probab. Lett. 48 (2000), no. 1, 11--22. MR1767606
  • Nualart, D.; Pardoux, É. Boundary value problems for stochastic differential equations. Ann. Probab. 19 (1991), no. 3, 1118--1144. MR1112409
  • Nualart, David; Vives, Josep. Anticipative calculus for the Poisson process based on the Fock space. Séminaire de Probabilités, XXIV, 1988/89, 154--165, Lecture Notes in Math., 1426, Springer, Berlin, 1990. MR1071538
  • Nualart, David; Vives, Josep. A duality formula on the Poisson space and some applications. Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1993), 205--213, Progr. Probab., 36, Birkhäuser, Basel, 1995. MR1360277
  • Ocone, Daniel; Pardoux, Étienne. Linear stochastic differential equations with boundary conditions. Probab. Theory Related Fields 82 (1989), no. 4, 489--526. MR1002898
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