Stochastic Differential Equations

  • Vincenzo Capasso
  • David Bakstein
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


Let \((W_{t})_{t\in \mathbb{R}_{+}}\) be a Wiener process on the probability space \((\varOmega,\mathcal{F},P)\), equipped with its natural filtration \((\mathcal{F}_{t})_{t\in \mathbb{R}_{+}}\), \(\mathcal{F}_{t} =\sigma (W_{s},0 \leq s \leq t)\). Furthermore, let a(t, x), b(t, x) be deterministic measurable functions in \([t_{0},T] \times \mathbb{R}\) for some \(t_{0} \in \mathbb{R}_{+}.\) Finally, consider a real-valued random variable u 0; we will denote by \(\mathcal{F}_{u^{0}}\) the σ-algebra generated by u 0, and we assume that \(\mathcal{F}_{u^{0}}\) is independent of \((\mathcal{F}_{t})\) for t ∈ (t 0, +). We will denote by \(\mathcal{F}_{u^{0},t}\) the σ-algebra generated by the union of \(\mathcal{F}_{u^{0}}\) and \(\mathcal{F}_{t}\) for t ∈ (t 0, +). 


  1. Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)zbMATHGoogle Scholar
  2. Baldi, P.: Equazioni differenziali stocastiche. UMI, Bologna (1984)Google Scholar
  3. Breiman, L.: Probability. Addison-Wesley, Reading, MA (1968)zbMATHGoogle Scholar
  4. Champagnat, N., Ferriére, R., Méléard, S.: Unifying evolutionary dynamics: From individula stochastic processes to macroscopic models. Theor. Pop. Biol. 69, 297–321 (2006)CrossRefzbMATHGoogle Scholar
  5. Chan, K.C., Karolyi, G.A., Longstaff, F.A., Sanders, A.B.: An empirical comparison of alternative models of the short-term interest rate. J. Fin. 47, 1209–1227 (1992)CrossRefGoogle Scholar
  6. Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1971)zbMATHGoogle Scholar
  7. Friedman, A.: Stochastic Differential Equations and Applications. Academic, London (1975). Two volumes bounded as one, Dover, Mineola, NY (2004)Google Scholar
  8. Gihman, I.I., Skorohod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  9. Gihman, I.I., Skorohod, A.V.: The Theory of Random Processes. Springer, Berlin (1974)Google Scholar
  10. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg (1999)Google Scholar
  11. Lapeyre, B., Pardoux, E., Sentis, R.: Introduction to Monte-Carlo Methods for Transport and Diffusion Equations. Oxford University Press, Oxford (2003)zbMATHGoogle Scholar
  12. Lipster, R., Shiryaev, A.N.: Statistics of Random Processes, I: General Theory. Springer, Heidelberg (1977)Google Scholar
  13. Lipster, R., Shiryaev, A.N.: Statistics of Random Processes, II: Applications, 2nd edn. Springer, Heidelberg (2010)Google Scholar
  14. Mao, X., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics. Stoch. Proc. Appl. 97, 95–110 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  15. Nowman, K.B.: Gaussian estimation of single-factor continuous time models of the term structure of interest rate. J. Fin. 52, 1695–1706 (1997)CrossRefGoogle Scholar
  16. Øksendal, B.: Stochastic Differential Equations. Springer, Berlin (1998)CrossRefGoogle Scholar
  17. Pascucci, A.: Calcolo Stocastico per la Finanza. Springer Italia, Milano (2008)CrossRefzbMATHGoogle Scholar
  18. Risken, H.: The Fokker–Planck Equation. Methods of Solution and Applications, 2nd edn. Springer, Heidelberg (1989)Google Scholar
  19. Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, vol. 1. Wiley, New York (1994)zbMATHGoogle Scholar
  20. Schuss, Z.: Theory and Applications of Stochastic Processes: An Analytical Approach. Springer, New York (2010)CrossRefGoogle Scholar
  21. Sobczyk, K.: Stochastic Differential Equations: With Applications to Physics and Engineering. Kluwer, Dordrecht (1991)CrossRefzbMATHGoogle Scholar
  22. Taira, K.: Diffusion Processes and Partial Differential Equations. Academic, New York (1988)zbMATHGoogle Scholar
  23. Ventcel’, A.D.: A Course in the Theory of Stochastic Processes. Nauka, Moscow (1975) (in Russian). Second Edition 1996Google Scholar
  24. Wu, F., Mao, X., Chen, K.: A highly sensitive mean-reverting process in finance and the Euler–Maruyama approximations. J. Math. Anal. Appl. 348, 540–554 (2008)CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Vincenzo Capasso
    • 1
  • David Bakstein
    • 1
  1. 1.ADAMSS (Interdisciplinary Centre for Advanced Applied Mathematical and Statistical Sciences)Università degli Studi di MilanoMilanItaly

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