Feynman-Kac Formulae

Abstract

Throughout this chapter, we suppose that E is a Banach space with norm ‖ • ‖, and (Σ, ε) is a measurable space. Further, we suppose that S is a C₀-semigroup of continuous linear operators acting on E and that Q: ε → ℒ(E) is a spectral measure, so that

$$ X = \left( {\Omega ,{{\left\langle {{S_t}} \right\rangle }_{t \geqslant 0}},{{\left\langle {{M_t}} \right\rangle }_{t \geqslant 0}};{{\left\langle {{X_t}} \right\rangle }_{t \geqslant 0}}} \right)$$

is a σ-additive (S, Q)-process. Recall that this means that for each t ≥ 0, M _t : S _t → ℒ(E) is a σ-additive set function defined on a σ-algebra S _t of subsets of Ω containing the collection ε _t {X} of all basic events before time t. To ensure that S_t is not too large, we suppose that it is contained in the completion with respect to the measure M _t of the σ-algebra σ(ε _t {X}) generated by ε _t {X}, that is, the σ-algebra produced by augmenting σ(ε _t {X}) with all subsets of M _t -null sets belonging to σ(ε _t {X}).