The Heston stochastic volatility model has a boundary trace at zero volatility

Bénédicte Alziary () and Peter Takáč
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Bénédicte Alziary: IMT - Institut de Mathématiques de Toulouse UMR5219 - UT Capitole - Université Toulouse Capitole - UT - Université de Toulouse - INSA Toulouse - Institut National des Sciences Appliquées - Toulouse - INSA - Institut National des Sciences Appliquées - UT - Université de Toulouse - UT2J - Université Toulouse - Jean Jaurès - UT - Université de Toulouse - UT3 - Université Toulouse III - Paul Sabatier - UT - Université de Toulouse - CNRS - Centre National de la Recherche Scientifique
Peter Takáč: Universität Rostock

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Abstract: We establish boundary regularity results in Holder spaces for the degenerate parabolic problem obtained from the Heston stochastic volatility model in Mathematical Finance set up in the spatial domain (upper half-plane) H = R x (0, infinity) subset of R-2. Starting with nonsmooth initial data u0 is an element of H, we take advantage of smoothing properties of the parabolic semigroup e(-tA) : H -> H, t is an element of R+, generated by the Heston model, to derive the smoothness of the solution u(t) = e(-tA)u(o) for all t > 0. The existence and uniqueness of a weak solution is obtained in a Hilbert space H = L-2(H; pi) with very weak growth restrictions at infinity and on the boundary partial derivative H = R x {0} subset of R-2 of the half-plane H. We investigate the influence of the boundary behavior of the initial data u(o) is an element of H on the boundary behavior of u(t) for t > 0.

Keywords: Degenerate parabolic equation; Weighted Sobolev space; Holomorphic semigroup; Parabolic smoothing effect; Dynamic boundary conditions; Heston’s stochastic volatility model (search for similar items in EconPapers)
Date: 2023
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Published in RACSAM. Real Academia de Ciencias. Serie A, Matemáticas - Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas, 2023, 117, pp.52. ⟨10.1007/s13398-022-01374-7⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-04888235

DOI: 10.1007/s13398-022-01374-7

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