## 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
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.