### STOCHASTIC INTEGRAL OF PROCESSES TAKING VALUES OF GENERALIZED OPERATORS

CHOI, BYOUNG JIN;CHOI, JIN PIL;JI, UN CIG

• 투고 : 2015.11.15
• 심사 : 2015.12.17
• 발행 : 2016.01.30
• 7 3

#### 초록

In this paper, we study the stochastic integral of processes taking values of generalized operators based on a triple E ⊂ H ⊂ E, where H is a Hilbert space, E is a countable Hilbert space and E is the strong dual space of E. For our purpose, we study E-valued Wiener processes and then introduce the stochastic integral of L(E, F)-valued process with respect to an E-valued Wiener process, where F is the strong dual space of another countable Hilbert space F.

#### 키워드

countable Hilbert space;Q-Wiener process;generalized operator;stochastic integral

#### 참고문헌

1. G. Da Prato and J. Zabczyk, Stochastic Equations in infinite Dimensions, Cambridge University Press, 1992.
2. S. Albeverio and B. Rüdiger, Stochastic integrals and the Lévy-Itô decomposition theorem on separable Banach space, Stoch. Anal. and Appl. 23 (2005), 217-253. https://doi.org/10.1081/SAP-200026429
3. D. Applebaum, Matingale-valued measure, Ornstein-Uhlenbeck process with jump and op-erator self-decomposabliity in Hilbert space, Lecture Notes in Math. 1874 (2006), 173-198.
4. R.F. Curtain and P.L. Falb, Ito's lemma in infinite dimensions, J. Math. Anal. Appl. 31 (1970), 434-448. https://doi.org/10.1016/0022-247X(70)90037-5
5. E. Dettweiler, Banach space valued processes with independent increments and stochastic integration, in "Probability in Banach spaces IV", Lecture Notes in Math. 990 (1982), 54-83.
6. I.M. Gelfand and N.Y. Vilenkin, Generalized Functions, Vol. 4, Academic Press, 1964.
7. K. Itô, Stochastic integral, Proc. Imp. Acad. Tokyo 20 (1944), 519-524. https://doi.org/10.3792/pia/1195572786
8. N. Obata, White Noise Calculus and Fock Space, Lecture Notes in Math. 1577 (1994).
9. H. Kunita, Stochastic integrals based on martingales taking values in Hilbert space, Nagoya Math. J. 38 (1970), 41-52. https://doi.org/10.1017/S0027763000013507
10. P. Lu and C. Gao, Existence-and-uniqueness and mean-square boundedness of the solution to stochastic control systems, J. Appl. Math. & Informatics 31 (2013), 513-522. https://doi.org/10.14317/jami.2013.513
11. B.I. Mamporia, Wiener processes and stochastic integrals in a Banach space, Probab. Math. Statist. 7 (1986), 59-75.
12. J.M.A.M. van Neerven and L. Weis, Stochastic integration of functions with values in a Banach space, Studia Math. 166 (2005), 131-170. https://doi.org/10.4064/sm166-2-2
13. S.L. Yadava, Stochastic evolution equations in locally convex space, Proc. Indian Acad. Sci. Math. Sci. 95 (1986), 79-96. https://doi.org/10.1007/BF02881072