Feynman–kac Formulas for Black–scholes-Type Operators

We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to the computation of expectations of suitable nonnegative local martingales. Moreover, we characterize the distribution function of the time to explosion as the minimal solution to a certain Cauchy problem for an appropriate parabolic differential equation; this leads to alternative characterizations of Feller's criterion for explosions. We discuss in detail several examples for which To the Memory of Marc Yor. for their insightful remarks that improved the manuscript. We thank Alex Mijatović for raising the issues of strict positivity and full support and for prompting us to think about them (Sects. 4.2 and 4.3, respectively). We are grateful to the referees for their meticulous reading of the paper and their many and incisive suggestions. J.R. acknowledges generous support from the Oxford-Man Institute of Quantitative Finance, University of Oxford, where a major part of this work was completed. The problem discussed here was suggested to us by Marc Yor, who also gave us generous expert advice on several of its aspects. We dedicate the paper to his memory with deep sorrow at his passing. I.

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“…The proof of Proposition 5.2 resembles the arguments in [30]. For similar results, see Section 3.5 in [39], Theorem 5.6.1 in [3,21,27,48].…”

Section: Connections With Parabolic Partial Differential Equationsmentioning

confidence: 72%

Distribution of the time to explosion for one-dimensional diffusions

Karatzas

Ruf

2015

Probab. Theory Relat. Fields

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“…The relationship between CEV stochastic equations and PDEs was established by Janson and Tysk [23] who proved a Feynman-Kac type theorem: precisely, they showed that, if the payoff function ϕ is continuous and polynomially bounded (i.e. |ϕ(S)| ≤ C(1 + S m ) for some C and m), then the function C in (4.39) is the unique, polynomially bounded classical solution…”

Section: Numerical Testsmentioning

confidence: 99%

Analytical approximation of the transition density in a local volatility model

Pagliarani

Pascucci

2011

Open Mathematics

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“…For a more recent discussion of the well-posedness of the problem, see also [20]. In the numerical experiments here, we will use (6.4) as in [10] even if it is not needed.…”

Section: Multi-asset American Option Pricingmentioning

confidence: 99%

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

2014

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A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications. Computing, http://dx.doi.org/10.1007/s10915-014-9935-9 Journal of ScientificAccess to the published version may require subscription. Abstract Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations (PDEs) because they are flexible with respect to geometry, they can provide high order convergence, they allow for local refinement, and they are easy to implement in higher dimensions. For global RBF methods, one of the major disadvantages is the computational cost associated with the dense linear systems that arise. Therefore, research is currently directed towards localized RBF approximations such as the RBF partition of unity collocation method (RBF-PUM) proposed here. The objective of this paper is to establish that RBF-PUM is viable for parabolic PDEs of convection-diffusion type. The stability and accuracy of RBF-PUM is investigated partly theoretically and partly numerically. Numerical experiments show that high-order algebraic convergence can be achieved for convection-diffusion problems. Numerical comparisons with finite difference and pseudospectral methods have been performed, showing that RBF-PUM is competitive with respect to accuracy, and in some cases also with respect to computational time. As an application, RBF-PUM is employed for a two-dimensional American option pricing problem. It is shown that using a node layout that captures the solution features improves the accuracy significantly compared with a uniform node distribution.

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Feynman–kac Formulas for Black–scholes-Type Operators

Cited by 61 publications

References 14 publications

Distribution of the time to explosion for one-dimensional diffusions

Distribution of the time to explosion for one-dimensional diffusions

Analytical approximation of the transition density in a local volatility model

A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications

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