Volume 9, no. 4Pages 63 - 72

Stochastic Cauchy Problem in Hilbert Spaces: Models, Examples, Solutions

O.S. Starkova
The paper is concerned with the stochastic Cauchy problem for nonlinear first order equation with values in a separable Hilbert space and with multiplicative noise in some other Hilbert space. First, a model of the term structure of interest rate that is a measure of the current bond market is represented. Stochastic behavior of the process describing a temporary bond price structure is caused by the fact that the economic indicators change in time and are not known in advance. We consider methods of calculating the forward curve, which describes the temporal structure of the bond price, and the transition from these methods to the solution of the Cauchy problem of mentioned type. Secondly, we show conditions on initial mappings which are necessary for existence and uniqueness of solution and build examples of mappings satisfying these conditions. We construct weak and mild solutions, show the results of existence and uniqueness of mild solution and the relationship of soft and weak solutions, which implies the existence and uniqueness of a weak solution of the Cauchy problem.
Full text
stochastic Cauchy problem; white noise; Wiener process; weak solution; mild solution; bond price; forward curve.
1. Da Prato G., Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press, 1992. DOI: 10.1017/CBO9780511666223
2. Gawarecki L., Mandrekar V. Stochastic Differential Equations in Infinite Dimensions. Berlin, Heidellberg, Springer-Verlag, 2011. DOI: 10.1007/978-3-642-16194-0
3. Melnikova I.V. Filinkov A.I., Anufrieva U.A. Abstract Stochastic Equations I. Classical and Distributional Solutions. Journal of Mathematical Sciences, 2002, vol. 111, no. 2, pp. 3430-3465. DOI: 10.1023/A:1016006127598
4. Melnikova I.V., Filinkov A.I. Abstract Stochastic Problems with Generators of Regularized Semigroups. Journal of Communications in Applied Analysis, 2009, vol. 13, no. 2, pp. 195-212.
5. Alshanskiy M.A., Melnikova I.V. Regularized and Generalized Solutions of Infinite-Dimensional Stochastic Problems. Sbornik: Mathematics, 2011, vol. 202, no. 11, pp. 1-28. DOI: 10.1070/SM2011v202n11ABEH004199
6. Heath D., Jarrow R., Morton A. Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation. Econometrica, 1992, vol. 60, no. 1, pp. 77-105. DOI: 10.2307/2951677
7. Musiela M. Stochastic PDEs and Term Structure Models. Journees Internationales de France, IGR-AFFI, La Baule, 1993.
8. Filipovic D. Consistency Problems for Heath - Jarrow - Morton Interest Rate Models. Berlin, Heidellberg, Springer, 2001.
9. Balakrishnan A.V. Applied Functional Analysis. Berlin, Heidellberg, Springer, 1976.