Abstract.
A common feature that can be consistently found in the works of Professor Kiyosi Itô is a leap from the analysis in distribution family level toward the analysis and synthesis in sample paths level, which has turned analytic descriptions into thoroughly stochastic ones.
Similar content being viewed by others
References
L. Bachelier, Théorie de la spéculation, Ann. Sci. école Norm. Sup., 17 (1900), 21–86.
A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U. S. A., 45 (1959), 208–215.
F. Black and M. Scholes, The pricing of options and corporate liabilities, J. Polit. Economy, 81 (1973), 637–659.
J. L. Doob, Stochastic processes depending on a continuous parameter, Trans. Amer. Math. Soc., 42 (1937), 107–140.
J. L. Doob, Stochastic Processes, John Wiley & Sons, New York, 1953.
E. B. Dynkin, Markov Processes, Moscow, 1963; English translation (in two volumes), Springer-Verlag, Berlin, 1965.
A. Einstein, Über die von der molekularkinematischen Theorie der Wärme geforderte Bewegung von ruhenden Flüssigkeiten suspendierten Teilchen, Ann. Phys., 17 (1905), 549–560.
W. Feller, Zur Theorie der stochastischen Prozesse (Existenz und Eindeutigkeitssätze), Math. Ann., 113 (1936), 113–160.
W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), 468–519.
W. Feller, On second order differential operators, Ann. of Math. (2), 61 (1955), 90–105.
M. Fukushima, Dirichlet Forms and Markov Processes, North-Holland; Kodansha, 1980.
G. A. Hunt, Markov processes and potentials, I, II, III, Illinois J. Math., 1 (1957), 44–93; 1 (1957), 316–369; 2 (1958), 151–213.
N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland; Kodansha, 1980.
K. Itô, On stochastic processes. I. (Infinitely divisible laws of probability), doctoral thesis, Jap. J. Math., 18 (1942), 261–301.
K. Itô, Differential equations determining a Markoff process (in Japanese), J. Pan-Japan Math. Coll. 1077 (1942), 1352–1400; (in English) In: Kiyosi Itô Selected Papers, Springer-Verlag, 1986, pp. 42–75.
K. Itô, A kinematic theory of turbulence, Proc. Imp. Acad. Tokyo, 20 (1944), 120–122.
K. Itô, On stochastic differential equations, Mem. Amer. Math. Soc., 4 (1951), 1–51.
K. Itô, Multiple Wiener integral, J. Math. Soc. Japan, 3 (1951), 157–169.
K. Itô, Stochastic Processes I, II (in Japanese), Iwanami-Shoten, Tokyo, 1957; (English translation by Yuji Ito) Essentials of Stochastic Processes, Transl. Math. Monogr. 231, Amer. Math. Soc., Providence, RI, 2006.
K. Itô, Wiener integral and Feynman integral, In: Proc. Fourth Berkeley Sympos. Math. Statist. and Probability, II, 1960, pp. 227–238.
K. Itô, Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral, In: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, II, 1965, pp. 145–161.
K. Itô, Poisson point processes and their application to Markov processes, Lecture note of Mathematics Department, Kyoto Univ., preprint (1969).
K. Itô, Poisson point processes attached to Markov processes, In: Proc. Sixth Berkeley Sympos. Math. Statist. and Probability, III, 1970, pp. 225–239.
K. Itô, Stochastic differentials, Appl. Math. Optim., 1 (1974), 374–381.
K. Itô, Extensions of stochastic integrals, In: Proc. Int. Symp. Stochastic Differential Equations, Kyoto, 1976, (ed. K. Itô), Kinokuniya, Tokyo, 1978, pp. 95–109.
Kiyosi Itô Selected Papers, (eds. D. W. Stroock and S. R. S. Varadhan), Springer-Verlag, 1986.
K. Itô, Memoirs of My Research on Stochastic Analysis, In: Proc. The Abel Symp. 2005, Stochastic Anal. Appl.–A Symposium in Honor of Kiyosi Itô–, Springer, 2007, pp. 1–5.
K. Itô and H. P. McKean, Jr., Brownian motions on a half line, Illinois J. Math., 7 (1963), 181–231.
K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer-Verlag, 1965; In: Classics Math., Springer-Verlag, 1996.
K. Itô and S. Watanabe, Introduction to stochastic differential equations, In: Proc. Int. Symp. Stochastic Differential Equations, Kyoto, 1976, (ed. K. Itô), Kinokuniya, Tokyo, 1978, pp. 1–30.
S. Kakutani, Two-dimensional Brownian motion and harmonic functions, Proc. Imp. Acad. Tokyo, 20 (1944), 706–714.
A. Kolmogorov, Über die analytischen Methoden in der Wahrscheilichkeitsrechnung, Math. Ann., 104 (1931), 415–458.
A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Erg. der Math., Berlin, 1933.
H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. J., 30 (1967), 209–245.
P. Lévy, Théorie de l'Addition des Variables Aléatoires, Gauthier-Villars, Paris, 1937.
P. Lévy, Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 1948.
B. Maisonneuve, Exit systems, Ann. Probability, 3 (1975), 399–411.
H. P. McKean, Jr., Stochastic Integrals, Academic Press, New York and London, 1969.
R. C. Merton, Theory of rational option pricing, Bell J. Econom. and Management Sci., 4 (1973), 141–183.
R. C. Merton, Continuous-Time Finance, Blackwell, Cambridge, MA, 1990.
P. A. Meyer, Intégrals stochastiques (4 exposés), In: Séminaire de Probabilités I, Lecture Notes in Math. 39, Springer-Verlag, 1967, pp. 72–162.
M. Motoo and S. Watanabe, On a class of additive functionals of Markov processes, J. Math. Kyoto Univ., 4 (1965), 429–469.
D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer-Verlag, 1991.
L. C. G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales, two volumes, John Wiley and Sons Ltd., 1979; 2nd Edition, Cambridge University Press, 2000.
D. Stroock, Markov Processes from K. Itô’s Perspective, Princeton Univ. Press, 2003.
N. Wiener, Differential space, J. Math. Phys., 2 (1923), 131–174.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Toshiyuki Kobayashi
About this article
Cite this article
Fukushima, M. On the works of Kiyosi Itô and stochastic analysis. Jpn. J. Math. 2, 45–53 (2007). https://doi.org/10.1007/s11537-007-0644-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11537-007-0644-0
Keywords and phrases:
- Brownian motion
- Lévy–Itô decomposition
- Itô integral
- Itô formula
- stochastic differential equation
- Wiener–Itô decomposition
- one-dimensional diffusion
- excursions
- stochastic geometry
- stochastic control
- stochastic finance