Pseudo-differential operators with constant symbols

Svetlana Boyarchenko and Sergei Z. Levendorskiĭ
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Sergei Z. Levendorskiĭ: Rostov State University of Economics, Russia

Chapter 15 in Non-Gaussian Merton-Black-Scholes Theory, 2002, pp 295-364 from World Scientific Publishing Co. Pte. Ltd.

Abstract: AbstractThe following sections are included:IntroductionClasses of functionsGeneral notation and remarksThe space $\mathcal{S}(R^n)$The Fourier transformSpace $\mathcal{S}^\prime(R^n)$ of generalized functions on RnGeneralized functions (distributions)Operations in $\mathcal{S}^\prime(R^n)$The Fourier transform of generalized functionsThe regularization of oscillatory integralsTensor product of generalized functionsGeneralized functions on an open set. The support of a generalized functionPseudo-differential operators with constant symbols on RnDefinition of PDO with constant symbolsThe Sobolev spaces Hs(Rn)Action of PDO in the scale Hs(Rn)Elliptic equationsRestriction to a hyperplaneThe Sobolev embedding theoremAction in the Hölder spacesThe action of PDO in the Sobolev spaces on Rn±Spaces $\mathop {H\limits^{o}}^{s} (R^{n}_\pm)$ and PDO in a half-spaceSpaces Hs(Rn±)The Cauchy-type integral and its decompositionParabolic equationsThe Cauchy problem for the parabolic equationReduction to a family of ODE on R+Parabolic equation as ODE with the operator-valued coefficientParabolic equations and the Hölder-Zygmund spacesThe inhomogeneous parabolic equation and the asymptotics of the solution as t → +∞The Wiener-Hopf equation on a half-line IStatement of the problemThe Wiener-Hopf factorizationMain TheoremsThe asymptotics of the solution near the boundaryParabolic equations on [0,T] × R+The statement of the boundary problemThe Wiener-Hopf factorization with a parameterReduction to ODE with the operator-valued coefficient and the solutionPDO in the Sobolev spaces with exponential weights, in 1DGeneralized functionsThe Sobolev spaces Hs,γ(R)PDO with the symbols holomorphic in a strip, and their action in the scale Hs,γ(R)Action of PDO in the Hölder-Zygmund spaces with exponential weightsThe Sobolev spaces with exponential weights and PDO on a half-lineSpaces $\mathop {H\limits^{o}}^{s,\gamma}(R_{\pm})$Spaces Hs,γ(R±)The Cauchy-type integral and its decompositionParabolic equations in spaces with exponential weightsThe Wiener-Hopf equation on a half-line IIStatement of the problemThe Wiener-Hopf factorization of symbols holomorphic in a stripMain TheoremsThe asymptotics of solutions near the boundary and the free boundary problemsThe asymptotics of solutions at the infinityParabolic equations on R × R+ with exponentially growing data

Keywords: Non-Gaussian Models; Merton-Black-Scholes Theory; Levy Processes; American Options; European Options; Feller Processes (search for similar items in EconPapers)
Date: 2002
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