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By Gerard Brunick

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**Additional resources for A Weak Existence Result with Application to the Financial Engineer's Calibration Problem**

**Example text**

29 Lemma. 19, let F0 {Ft0 } where Ft0 FT0 +t and let M be a continuous, real-valued process with M0 = 0. Set M ∆(M, T ) and assume that M is σ G , ∆(X, T ) -measurable. If M T is an (F0 , P1 )-local martingale and M is an (F0 , P2 )-local martingale, then M is an (F0 , P12 )-local martingale. Proof. First we note that if S is an F0 -stopping time, then S ∨ T − T is an 0 F0 -stopping time and FS0 ⊂ FS∨T −T . To see this, notice that T + t is an 46 CHAPTER 3. A CROSS PRODUCT CONSTRUCTION F0 -stopping time, so the first claim follows from the equalities {S ∨ T − T ≤ t} = {S ≤ T + t} ∈ FT0 +t = Ft0 .

27, we may compute µ and σ explicitly. It is clear that µ = 0. s. 19). We now assume that t > T . Write a typical point in R2 as x = (x1 , x2 ). Set A (Yt , YtT ) and define η (x1 , x2 ; v, t) η(x2 , T v) η(x1 − x2 , (t − T )v), for t > T , so η (x1 , x2 ; c, t) is the density of the R2 -valued random variable c Wt , c WT . Recall that η(x, v) was defined as the density of the normal distribution with mean 0 and variance v, and W was defined as a Wiener 29 CHAPTER 2. 10. We then have σ 2 (x, t) = E σt2 Yt = x1 , YtT = x2 = = c1 P σt2 = c1 , At ∈ dx + c2 P σt2 = c2 , At ∈ dx P σt2 = c1 , At ∈ dx + P σt2 = c2 , At ∈ dx c1 η (x1 , x2 ; c1 , t) + c2 η (x1 , x2 ; c2 , t) .

S. of finite variation. s. s. absolutely continuous. Proof. s. of finite variation, and set P0 P ⊗T0 ,G0 P. 0 It then follows from Cor. s. of finite variation. s. of finite variation and setting Pi+1 = Pi ⊗Ti+1 ,Gi+1 P, it again follows from Cor. s. of finite variation. As Pn = P , we have (a). Assertion (b) follows in the same way. 56 CHAPTER 3. 40 Lemma. Let P be a measure on Ω and Π = {(Ti , Gi )}0≤i≤n be an extended partition. Let A be a continuous, Rd -valued process such that ∆(A, Ti ) is σ Gi , ∆(X, Ti ) -measurable for each i ∈ {1, .