The Wiener process, the stochastic integral over the Wiener process, and stochastic differential equations

  • R. S. Liptser
  • A. N. Shiryayev
Part of the Applications of Mathematics book series (SMAP, volume 5)

Abstract

Let (Ω, , P) be a probability space and β = (β t ), t ≥ 0, be a Brownian motion process (in the sense of the definition given in Section 1.4). Denote t β = σ{ω: β s , st}. Then, according to (1.30) and (1.31), (P-a.s.)
$$ M({\beta _t}|\mathcal{F}_s^\beta ) = {\beta _s},{\text{ t}} \geqslant {\text{s}} $$
(4.1)
,
$$ M[{({\beta _t} - {\beta _s})^2}|\mathcal{F}_s^\beta ] = t - s,{\text{ t}} \geqslant s $$
(4.2)
.

Keywords

Weak Solution Stochastic Differential Equation Strong Solution Simple Function Wiener Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes and references

  1. [46]
    Doob J. L., Probability Processes. Russian transi., IL. Moscow, 1956.Google Scholar
  2. [20]
    Wiener N., Differential space. J. Math. and Phys. 58 (1923), 131–174.Google Scholar
  3. [59]
    Ito K., Stochastic integrals. Proc. Imp. Acad. Tokyo 20 (1944), 519–524.MathSciNetMATHCrossRefGoogle Scholar
  4. [34]
    Gikhman 1. I., Skorokhod A. V., Introduction to Random Processes Theory. “Nauka,” Moscow, 1965.Google Scholar
  5. [36]
    Gikhman I. I., Skorokhod A. V., Stochastic Differential Equations. “Naukova dumka,” Kiev, 1968 (Ukranian).Google Scholar
  6. [52]
    Yershov M. P., On representations of Ito processes. Teoria Verojatn. i Primenen. XVII, 1 (1972), 167–172.Google Scholar
  7. [47]
    Dynkin Ye. B., Markov Processes. Fizmatgiz, Moscow, 1963.Google Scholar
  8. [60]
    Ito K., On one formula on stochastic differentials. Matematika. Sbornik perevodov inostr. statei. 3: 5 (1959), 131–141.Google Scholar
  9. [144]
    Skorokhod A. V., The investigation of a random processes theory. Izd-vo Kievsk. univ-ta, 1961.Google Scholar
  10. [53]
    Yershov M. P., On absolute continuity of measures corresponding to diffusion type processes. Teoria Verojatn. i Primenen. XVII, 1 (1972), 173–178.Google Scholar
  11. [166]
    Shiryayev A. N., Stochastic equations of nonlinear filtering of jump Markov processes. Problemy peredachi informatsii. II, 3 (1966), 3–22.Google Scholar
  12. [111]
    Liptser R. S., Shiryayev A. N., Nonlinear filtering of diffusion type Markov processes. Trudy matem. in-ta im. V. A. Steklova AN SSSR 104 (1968), 135–180.Google Scholar
  13. [174]
    Yamada T., Watanabe Sh., On the uniqueness of solution of stochastic differential equations. J. Math. Kyoto Univ. 11, 1 (1971), 155–167.MathSciNetMATHGoogle Scholar
  14. [62]
    Ito K., Nisio M., On stationary solutions of stochastic differential equations. J. Math. Kyoto Univ. 4, 1 (1964), 1–79.MathSciNetMATHGoogle Scholar
  15. [74]
    Kallianpur G., Striebel C., Estimation of stochastic systems: Arbitrary system process with additive white noise observation errors. AMS 39 (1968), 785–801.MathSciNetMATHGoogle Scholar
  16. [213]
    Tsyrelson B. S., An example of the stochastic equation having no strong solution. Teoria Verojatn. i Primenen. XX, 2 (1975), 427–430.Google Scholar

Copyright information

© Springer Science+Business Media New York 1977

Authors and Affiliations

  • R. S. Liptser
    • 1
  • A. N. Shiryayev
    • 2
  1. 1.Institute for Problems of Control TheoryMoscowUSSR
  2. 2.Institute of Control SciencesMoscowUSSR

Personalised recommendations