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S. of finite variation. s. s. s. absolutely continuous. 42 CHAPTER 3. A CROSS PRODUCT CONSTRUCTION Proof. Set FV d {x ∈ C0 (R+ ; Rd ) : x ∈ BV [0, t]; Rd for all t}, and AC d {x ∈ C0 (R+ ; Rd ) : x ∈ AC [0, t]; Rd for all t}. , Cor. 9 and Cor. 11). Fixing any ω = (e, x) ∈ Ω, we see that ∇ x, T (ω) ∈ BV [0, t]; Rd and ∆ x, T (ω) ∈ BV [0, (t−T (ω))∨0]; Rd implies that x ∈ BV [0, t]; Rd , so AT ∈ F V d ∩ {∆(A, T ) ∈ F V d } ⊂ A ∈ F V d . As P2 [∆(A, T ) ∈ F V d ] = 1, 1 is a version of P2 [∆(A, T ) ∈ F V d | G ], and we may apply the properties of P12 listed in Cor.

In particular, if A ∈ A , then Q[Ac ] = 1 − Q[A] so Ac ∈ A . Similarly, if An ∈ A and the An are disjoint, then Q ∪n An = n Q[An ] so ∪n An ∈ A . We have now shown that A is a λ-system. Set B F ∈ F : F = A ∩ B for some A ∈ FT0 and B = {∆(X, T ) ∈ C } , and take F ∈ B. Then Q[F ] = Q[A ∩ B] = 1A Q(C) ∈ FT0 by the previous lemma, so B ⊂ A . 3), so F ⊂ A by the π-λ theorem, and Q[A] is an FT0 -measurable for all A ∈ F . As Qω is a probability measure for fixed ω by construction, Q is an FT0 -measurable probability kernel on (Ω, F ) Now define the measure Q[F ] EP Q[F ] for F ∈ F .

In the first lemma, we take an initial point ω = (e, x) ∈ Ω, and we cut the path x at time t, keeping the initial segment from 0 to t and discarding the rest. We then randomly draw a path from C0 (R+ ; Rd ) according to some measure Q and append this path to the initial segment of x. Recall that C0 (R+ ; Rd ) denotes the set of continuous functions from R+ to Rd that start at 0. 36 CHAPTER 3. 9 Lemma. Fix some ω = (e , x ) ∈ Ω, t ≥ 0, and let Q be a probability measure on C0 (R+ ; Rd ). Then there exists a unique measure on Ω, denoted δω ⊕t Q, such that δω ⊕t Q A ∩ {∆(X, t) ∈ B } = 1A (ω ) Q[B ] for all A ∈ Ft0 and B ∈ C0 , where C0 denotes the Borel σ-field on C0 (R+ ; Rd ).

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A Weak Existence Result with Application to the Financial Engineer's Calibration Problem by Gerard Brunick


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