Probability Distributions and Stochastic Processes

  • Hiroshi Kunita
Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 92)


We introduce some basic facts on probability distributions and stochastic processes. Probability distributions and their characteristic functions are discussed in Sect. 1.1. Criteria for smooth densities of distributions are given by their characteristic functions. In Sect. 1.2, we consider Gaussian, Poisson and infinitely divisible distributions and give criteria for these distributions to have smooth densities. Concerning the density problem of an infinitely divisible distribution, we study the Lévy measure in detail. Regarding that the origin 0 is the center of the Lévy measure, we will give criteria for its smooth density by means of ‘the order condition’ at the center of the Lévy measure.

Then we consider stochastic processes. In Sect. 1.4, we consider Wiener processes, Poisson processes, Poisson random measures and Lévy processes. Among them, Poisson random measures are exposed in detail. Next, in Sects. 1.5 and 1.6, we discuss martingales, semi-martingales and their quadratic variations. These are standard tools for the Itô calculus. In Sect. 1.7, we define Markov processes. The strong Markov property will be discussed. In Sect. 1.8, we study Kolmogorov’s criterion for a random field with multi-dimensional parameter to have a continuous modification.


  1. 20.
    Chentzov, N.N.: Limit theorem for some class of functions, ROC. In: All-Union Conference of Theory Probability and Mathematics Statistics, Erevan (1958) (Selected Transl. Math. Stat. Prob. 9, 11–40 (1970))Google Scholar
  2. 46.
    Ishikawa, Y., Kunita, H., Tsuchiya, M.: Smooth density and its short time estimate for jump process determined by SDE. SPA 128, 3181–3219 (2018)MathSciNetzbMATHGoogle Scholar
  3. 55.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)zbMATHGoogle Scholar
  4. 59.
    Kunita, H.: Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  5. 66.
    Kunita, H., Watanabe, S.: On square integrable martingales. Nagoya Math. J. 30, 209–245 (1967)MathSciNetCrossRefGoogle Scholar
  6. 84.
    Meyer, P.A.: Probability and Potentials. Bleisdell, Waltham (1966)zbMATHGoogle Scholar
  7. 85.
    Meyer, P.A.: Un cours sur integrals stochastiques. In: Meyer, P.A. (ed.) Seminaire Probability X. Lecture Notes in Mathematics, vol. 511, pp. 246–400. Springer, Berlin (1976)Google Scholar
  8. 99.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  9. 114.
    Totoki, H.: A method of construction of measures on function spaces and its applications to stochastic processes. Memories Fac. Sci. Kyushu Univ. Ser. A. Math. 15, 178–190 (1962)MathSciNetzbMATHGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Hiroshi Kunita
    • 1
  1. 1.Kyushu University (emeritus)FukuokaJapan

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