A Weak Existence Result with Application to the Financial by Gerard Brunick

By Gerard Brunick

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5, so we may apply Thm. 11 to conclude that there exists a stochastic basis (Ω, F , F, P) that supports processes W , Y , and Z such that W is a Wiener process, Y satisfies (b) of Thm. 11, Z = Φ(Z0 , Y ), and L (Zt ) = L (Zt ) for all t ∈ R+ . Set (X, N ) Z, and set M = max{Xs : s ∈ [0, t]}. Then Xt = Z01 + Yt where Zti denotes the ith component of Zt . As Y satisfies (b) of Thm. 25) dXt = µ(Xt , Nt , t) dt + σ(Xt , Nt , t) dWt . We also see that Nt = max Z02 , Z01 + Ys : s ∈ [0, t] = max Z02 , Xs : s ∈ [0, t] .

T t ∈ R+ : ∂ If 0 < t < T (ω) and t ∈ / B 1 (ω), then ∂t ATt (ω) exists and AT (ω) and A(ω) ∂ ∂ agree in some neighborhood of t, so ∂t At (ω) exists and agrees with ∂t ATt (ω). 44 CHAPTER 3. A CROSS PRODUCT CONSTRUCTION In particular, if 0 < t < T (ω) and t ∈ / B(ω) ∪ B 1 (ω), then at (ω) = a1t (ω). e. t ∈ 0, T (ω) . 26) 1{AT ∈F V d } f (a1u ) du f (au ) du = 1{AT ∈F V d } 0 0 for all ω, where we use the extended integral of Rem. 7. 26). Similar pathwise arguments show that aT (ω)+t (ω) = a2t (ω) if t > 0, T (ω)+ t ∈ / B(ω), and t ∈ / B 2 (ω).

Tn < tn+1 = ∞}. 5) We take Ti = ti and Gi = σ(Yti ) in Def. 2. For i ∈ {1, . . , n + 1}, we have Hi = σ Gi−1 , ∆(X Ti , XTi−1 ) = σ Yti−1 , Ys − Yti−1 , Cs − Cti−1 : s ∈ [ti−1 , ti ] ∩ R+ = σ Θ(Y ti , ti−1 ), ∆(C ti , ti−1 ) . As Yti = Θti −ti−1 (Y ti , ti−1 ) ∈ Hi , Π = (Gi , ti ) 0≤1≤n is an extended partition. Notice that in this example, we do not have Cti ∈ Hi when i > 0. The goal of this section is to the provide the following theorem. 6 Theorem. Let P be a probability measure on Ω and Π = {(Ti , Gi )}0≤i≤n ⊗Π be an extended partition.

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