Stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with Dirichlet boundary conditions

Feehan, Paul M. N.; Pop, Camelia A.

doi:10.1090/s0002-9947-2014-06043-1

Cited by 11 publications

(5 citation statements)

References 45 publications

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“…Note that Feehan and Pop did prove regularity results in the elliptic case, see [9]. They also announce results for the parabolic case in [8].…”

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confidence: 92%

“…Note that Feehan and Pop did prove regularity results in the elliptic case, see [9]. They also announce results for the parabolic case in [8].The aim of this paper is to give a precise analytical characterization of the American option price function for a large class of payoffs which includes the standard put and call options. In particular, we give a variational formulation of the American pricing problem using the weighted Sobolev spaces and the bilinear form introduced in [5].…”

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confidence: 99%

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Variational Formulation of American Option Prices in the Heston Model

Lamberton¹,

Terenzi²

2019

SIAM J. Finan. Math.

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We give an analytical characterization of the price function of an American option in Heston-type models. Our approach is based on variational inequalities and extends recent results of Feehan (2011, 2016) and . We study the existence and uniqueness of a weak solution of the associated degenerate parabolic obstacle problem. Then, we use suitable estimates on the joint distribution of the log-price process and the volatility process in order to characterize the analytical weak solution as the solution to the optimal stopping problem. We also rely on semi-group techniques and on the affine property of the model.It should be emphasized that a clear analytic characterization of the price function allows not only to formally justify the theoretical convergence of some classical pricing algorithms but also to investigate the regularity properties of the price function (see [13] for the case of the Black and Scholes models).Concerning the existing literature, E. Ekstrom and J. Tysk in [7] give a rigorous and complete analysis of these issues in the case of European options, proving that, under some regularity assumptions on the payoff functions, the price function is the unique classical solution of the associated PDE with a certain boundary behaviour for vanishing values of the volatility. However, the payoff functions they consider do not include the case of standard put and call options.Recently, P. Daskalopoulos and P. Feehan studied the existence, the uniqueness, and some regularity properties of the solution of this kind of degenerate PDE and obstacle problems in the elliptic case, introducing suitable weighted Sobolev spaces which clarify the behaviour of the solution near the degenerate boundary (see [5,6]). In another paper ([8]) P. Feehan and C. Pop addressed the issue of the probabilistic representation of the solution, but we do not know if their assumptions on the solution of the parabolic obstacle problem are satisfied in the case of standard American options. Note that Feehan and Pop did prove regularity results in the elliptic case, see [9]. They also announce results for the parabolic case in [8].The aim of this paper is to give a precise analytical characterization of the American option price function for a large class of payoffs which includes the standard put and call options. In particular, we give a variational formulation of the American pricing problem using the weighted Sobolev spaces and the bilinear form introduced in [5]. The paper is organized as follows. In Section 2, we introduce our notations and we state our main results. Then, in Section 3, we study the existence and uniqueness of the solution of the associated variational inequality, extending the results obtained in [5] in the elliptic case. The proof relies, as in [5], on the classical penalization technique introduced by Bensoussan and Lions [4] with some technical devices due to the degenerate nature of the problem. We also establish a Comparison Theorem. Finally, in section 4, we prove that the solution of the variational ineq...

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“…Note that Feehan and Pop did prove regularity results in the elliptic case, see [9]. They also announce results for the parabolic case in [8].…”

mentioning

confidence: 92%

mentioning

confidence: 99%

Variational Formulation of American Option Prices in the Heston Model

Lamberton¹,

Terenzi²

2019

SIAM J. Finan. Math.

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“…On the other hand, for the problems dealing with multivalued terms modeled by Clarke's subdifferential we refer to the papers of Averna et al [2], Denkowski et al [13][14][15][16], Filippakis et al [18,19], Gasiński [20,21], Gasiński et al [22], Gasiński and Papageorgiou [24,25], Kalita and Kowalski [27], Papageorgiou et al [41,42], Zeng et al [48]. Finally, for the problems dealing with obstacle problems we refer to the papers of Caffarelli et al [4], Caffarelli et al [5], Choe [10], Choe and Lewis [11], Feehan and Pop [17], Oberman [40].…”

Section: Introductionmentioning

confidence: 99%

Qualitative analysis of solutions of obstacle elliptic inclusion problem with fractional Laplacian

Zeng

Cen

Atangana

et al. 2021

Z. Angew. Math. Phys.

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In this paper, we study an elliptic obstacle problem with a generalized fractional Laplacian and a multivalued operator which is described by a generalized gradient. Under quite general assumptions on the data, we employ a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded multivalued pseudomonotone mapping to prove that the set of weak solutions to the problem is nonempty, bounded and closed. Then, we introduce a sequence of penalized problems without obstacle constraints. Finally, we prove that the Kuratowski upper limit of the sets of solutions to penalized problems is nonempty and is contained in the set of solutions to original elliptic obstacle problem, i.e., ∅ ≠ wlim sup n→∞ Sn = slim sup n→∞ Sn ⊂ S.

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“…In the case of Laplacian operator L = ∆, [9] (Sections 4.4 and 4.7) shows that the Feynman-Kac functional v of (2) solves (1). With L being a second order differential operator, and thus almost surely continuous X ∼ L, the relation between the Feynman-Kac functional and the Dirichlet problem is discussed in [4,15,16,18,21,19] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Exit Problems as the Generalized Solutions of Dirichlet Problems

Han¹,

Song²,

Wang³

2019

SIAM J. Control Optim.

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This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to α-stable processes, which help verify the solvability of the original HJB equation.

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Stochastic representation of solutions to degenerate elliptic and parabolic boundary value and obstacle problems with Dirichlet boundary conditions

Cited by 11 publications

References 45 publications

Variational Formulation of American Option Prices in the Heston Model

Variational Formulation of American Option Prices in the Heston Model

Qualitative analysis of solutions of obstacle elliptic inclusion problem with fractional Laplacian

Exit Problems as the Generalized Solutions of Dirichlet Problems

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