Balayage of Excessive Measures

Abstract

Recall [S, (12.1)] that an (F _t ) stopping time, T, is a weak terminal time provided (i) for each t, $t + T \circ {{\theta }_{t}} = T$ a.s. on t < T, and that T is exact if, in addition, (ii) ${{t}_{n}} + T \circ {{\theta }_{{{{t}_{n}}}}} \downarrow T$ a.s. whenever t_n ↓↓ 0. If T is an exact weak terminal time, then according to [S, (55.20)] there exists an (F _t ) stopping time S with S = T a.s. and such that (iii) t + S(θ _t ω) = S(ω) for all t,ω with t < S(ω) and (iv) $S(\omega ) = \downarrow \mathop{{\lim }}\limits_{{t \downarrow 0}} [t + S({{\theta }_{t}}\omega )]$ for every ω. It follows from (iii) and (iv) that t → t + S(θ _t ω) is right continuous and increasing on [0, ∞[ for every ω. For simplicity we define a terminal time S to be an (F_t) stopping time that satisfies (iii), and we say that S is exact if, in addition, it satisfies (iv). In this language, [S, (55.20)] states that an exact weak terminal time is equal a.s. to an exact terminal. Note that if S is a terminal time, then (iii) implies that $t \to t + S \circ {{\theta }_{t}}$ is increasing and one readily checks that ${{S}^{*}}: = \downarrow \mathop{{\lim }}\limits_{{t \downarrow 0}} (t + S \circ {{\theta }_{t}})$ is an exact terminal time called the exact regularization of S. Of course, if B ∈ ε^e then T _B : = inf {t > 0: X _t ∈ B} is an exact terminal time in the above sense, while D _B : = inf {t ≥ 0: X _t ∈ B} is a terminal time. Moreover T _B is the exact regularization of D _B .