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February 2, 2018 | Economy | By admin | 0 Comments

By Gerard Brunick

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12), and set Mt max{Xs : s ∈ [0, t]}. Let N ⊂ R+ be a Lebesgue-null set, and let µ : R2 ×R+ → R and σ : R2 ×R+ → R be functions with µ(Xt , Mt , t) = / N. s. s. 24) dXt = µ(Xt , Mt , t) dt + σ(Xt , Mt , t) dWt , Mt = max{Xs : s ∈ [0, t]}, and L (Xt , Mt ) = L (Xt , Mt ) for all t ∈ R+ . Proof. Take E R2 and let e = [ ee12 ] denote a typical point in E. Set Y ∆(X, 0), set Z (X, M ), and let Φ : E×C0 (R+ ; R) denote the map such that e1 + x(t) Φt (e, x) = , max e2 , e1 + x(s) : s ∈ [0, t] so Z = Φ(Z0 , Y ).

2 Definition. Let {0 = T0 ≤ T1 ≤ . . ≤ Tn < ∞} be an increasing sequence of finite F0 -stopping times, and let {Gi }0≤i≤n be a collection of σ-fields. 3) Hi σ Gi−1 , ∆(X Ti , Ti−1 ) for 1 ≤ i ≤ n + 1. We say that Π (Ti , Gi ) following properties hold: 0≤i≤n is an extended partition if both the (a) Ti − Ti−1 ∈ σ Gi−1 , ∆(X, Ti−1 ) for 1 ≤ i ≤ n, and (b) Gi ⊂ Hi for 0 ≤ i ≤ n. One possible way to interpretation this structure is to think of an extended partition as a filtration-like object in which information is lost at each time Ti−1 , and Gi−1 denotes the information that we keep.

Combining Cor. 30 and Lem. 29 yields the following corollary. 31 Corollary. 19, let M be a continuous, real-valued process. Suppose that M is a local martingale with respect to both (F, P1 ) and (F, P2 ) and that ∆(M, T ) is σ G , ∆(X, T ) -measurable. Then M is an (F, P12 )-local martingale. Before we present the corresponding result for quadratic variation, we give an easy lemma. 32 Lemma. Let M be a uniformly integrable (F0 , P)-martingale, and let S, T , and U be F0 -stopping times with T ≤ U .

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


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