Some Remarks on the First-Crossing Area of a Diffusion Process with Jumps over a Constant Barrier

  • Marco Abundo
  • Mario Abundo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8111)


For a given barrier S > 0 and a one-dimensional jump-diffusion process X(t), starting from x < S, we study the probability distribution of the integral \(A_S(x)= \int _0 ^{\tau_S(x)}X(t) \ dt\) determined by X(t) till its first-crossing time τ S (x) over S.


First-crossing time first-crossing area jump-diffusion 


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  1. 1.
    Abundo, M.: On the first-passage area of one-dimensional jump-diffusion process. Methodol. Comput. Appl. Probab. 15, 85–103 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abundo, M., Abundo, M.: On the first-passage area of an emptying Brownian queue. Intern. J. Appl. Math (IJAM) 24(2), 259–266 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Darling, D.A., Siegert, A.J.F.: The first passage problem for a continuous Markov process. Ann. Math. Stat. 24, 624–639 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gihman, I.I., Skorohod, A.V.: Stochastic differential equations. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  5. 5.
    Kearney, M.J., Majumdar, S.N.: On the area under a continuous time Brownian motion till its first-passage time. J. Phys. A: Math. Gen. 38, 4097–4104 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Tuckwell, H.C.: On the first exit time problem for temporally homogeneous Markov processes. Ann. Appl. Probab. 13, 39–48 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marco Abundo
    • 1
  • Mario Abundo
    • 1
  1. 1.Tor Vergata UniversityRomeItaly

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