Stochastic Calculus of Variations in Mathematical Finance - download pdf or read online

By Paul Malliavin, Anton Thalmaier

ISBN-10: 3540434313

ISBN-13: 9783540434313

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Then we have N (Ip (Fp )) = p Ip (Fp ); N (φ) L2 ≤ φ D22 , √ and φ D22 ≤ 2 N (φ) L2 , if E[φ] = 0. 51) Furthermore N (φψ) = ψN (φ) + φN (ψ) + 2 1 0 Dτ φ Dτ ψ dτ . 52) Consider a finite linearly independent system h1 , . . , hq in L2 ([0, 1]), to which we assign the q×q covariance matrix Γ defined by Γij = (hi |hj )L2 ([0,1]) . Letting Φ(W ) = F (W (h1 ), . . , W (hq )) be a generalized cylindrical function, then N Φ = Ψ, Ψ(W ) = G(W (h1 ), . . , W (hq )), 2N {h. } = Γij i,j ∂2 ∂ − ξi j i j ∂ξ ∂ξ ∂ξ where G = N {h.

Tp−1 ) := Fp (τ, t1 , . . , tp−1 ); then Ip (Fp ) ∈ D12 (W ), Dτ (Ip (Fp )) = Ip−1 (Fpτ ) . 46) Expanding a given φ ∈ L2 in terms of a normalized series of iterated integrals ∞ φ − E[φ] = p! s. 47) is finite and then φ 2 D12 = (p + 1) Fp p≥0 2 L2 ([0,1]) . 47) 20 1 Gaussian Stochastic Calculus of Variations Proof. 46) by recursion on the integer p. 45) along with the fact that Dτ Z = 0. Assuming the formula for all integers p < p, we denote Z(λ) = Ip−1 Fpλ 1[0,λ[ (ts ) ; 1≤s≤p−1 then 1 Ip (Fp ) = Z(τ ) dW (τ ) .

We start from the point of view of intertwining partial differential operators as in Sect. 1. If f satisfies Dirichlet boundary conditions on ∂D, then the tangential component of the differential df of f , denoted by (df )ρ , vanishes on ∂D. 4) by the backward heat semigroup ∂t ϕt + ∆1 ϕt = 0, ϕT = ϕ with boundary condition ϕt |∂D = 0 . 38) Then ωt := dϕt satisfies the parabolic system ∂t ωt + L ωt = 0, ωT = dϕ, (ωt )ρ = 0 . 39) 40 2 Computation of Greeks and Integration by Parts Formulae The Ocone–Karatzas formula takes the form T ϕ − Et0 ,S0 [ϕ] = ωtk dW (t) , t0 where ωtk = ωt , Ak (S(t)) .

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Stochastic Calculus of Variations in Mathematical Finance by Paul Malliavin, Anton Thalmaier


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