Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing
The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and u...
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| Vydáno v: | Mathematics (Basel) Ročník 9; číslo 13; s. 1463 |
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| Jazyk: | angličtina |
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MDPI AG
01.07.2021
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| ISSN: | 2227-7390, 2227-7390 |
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| Abstract | The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function. |
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| AbstractList | The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function. |
| Author | Udeani, Cyril Izuchukwu Ševčovič, Daniel |
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| Cites_doi | 10.1016/0304-405X(76)90022-2 10.1016/j.crma.2019.03.001 10.1137/19M1290061 10.1023/A:1009703431535 10.1007/978-0-387-70914-7 10.1007/s10690-016-9213-y 10.1515/9781400883882 10.1007/BF02450422 10.1007/s007800050046 10.1007/s13160-020-00414-2 10.1080/1350486X.2019.1585267 10.1088/1469-7688/1/3/303 10.1007/s00780-005-0153-z 10.1080/13504869500000005 10.1137/S0036139996308534 10.1515/9783110571561 10.1007/s007800050035 10.1007/BFb0089647 10.2139/ssrn.300527 10.1155/JAM.2005.235 10.1007/s10440-012-9687-1 10.1111/j.1540-6261.1985.tb02383.x 10.1287/mnsc.48.8.1086.166 10.1111/1467-9965.00036 10.1016/j.camwa.2018.05.012 |
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| SubjectTerms | bessel potential spaces Differential equations Food science hölder continuity lévy measure Mathematical functions option pricing partial integro-differential equation Pricing Put & call options Stochastic models Stochastic processes strong kernel Uniqueness Variables |
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