Viability Theorem for SPDE's including HJM framework

J. Math. Sci. Univ. Tokyo
Vol. 11 (2004), No. 3, Page 313--324.

NAKAYAMA, Toshiyuki
Viability Theorem for SPDE's including HJM framework
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]

A viability theorem is proven for the mild solution of the stochastic differential equation in a Hilbert space of the form: $$ \begin{cases} dX^x(t)=AX^x(t)dt+b(X^x(t))dt +\sigma(X^x(t))dB(t), X^x(0)=x. \end{cases} $$ It is driven by a Hilbert space-valued Wiener process $B$, with the infinitesimal generator $A$ of a ($C_0$)-semigroup. This equation contains the stochastic partial differential equation within HJM framework in mathematical finance. Especially a viability theorem for \lq\lq finite dimensional manifold" is proved, which is important for \lq\lq consistency problems" in mathematical finance.

Mathematics Subject Classification (1991): 60G17, 60H15.
Mathematical Reviews Number: MR2097528

Received: 2004-04-06