2011
DOI: 10.1007/s00780-011-0166-8
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Singular risk-neutral valuation equations

Abstract: Many risk-neutral pricing problems proposed in the finance literature do not admit closed-form expressions and have to be dealt with by solving the corresponding partial integro-differential equation. Often, these PIDEs have singular diffusion matrices and coefficients that are not Lipschitz-continuous up to the boundary. In addition, in general, boundary conditions are not specified. In this paper, we prove existence and uniqueness of (continuous) viscosity solutions for linear PIDEs with all the above featur… Show more

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Cited by 12 publications
(18 citation statements)
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“…It may be interesting to note that, in the context of Hamilton-Jacobi equations, the idea of studying a parabolic equation by solving a resolvent equation in the viscosity sense appears already in [7], Section VI.3, where it is applied to a model problem. The methodology is also important for related problems in finance (for example [26], [3], [18], [1], [5] and many others).…”
Section: Introductionmentioning
confidence: 99%
“…It may be interesting to note that, in the context of Hamilton-Jacobi equations, the idea of studying a parabolic equation by solving a resolvent equation in the viscosity sense appears already in [7], Section VI.3, where it is applied to a model problem. The methodology is also important for related problems in finance (for example [26], [3], [18], [1], [5] and many others).…”
Section: Introductionmentioning
confidence: 99%
“…Remark 4.4. Already the results in the case without uncertainty in Costantini et al (2012) show that these results do not generalize to R ≥0 , because then the Lipschitz property on compact subsets which is crucially used in the proof will no longer be satisfied. Moreover, Example 1 in Amadori (2007) shows that the condition b 0 ≥ā 1 /2 > 0 is indeed necessary for uniqueness.…”
Section: The Kolmogorov Equationmentioning
confidence: 99%
“…As mentioned in the Introduction, we will obtain existence and uniqueness of the viscosity solution to (3.28) from a general result for valuation equations for contingent claims written on jump-diffusion underlyings proved in [CPD12]. Since the state space of (3.28) is unbounded, as usual in the literature uniqueness will hold in the class of functions with a prescribed growth rate.…”
Section: The Valuation Equationmentioning
confidence: 98%
“…The equation is not uniformly parabolic because the second order coefficient is not bounded away from zero, therefore it is not obvious that a classical solution exists. However we are able to prove existence and uniqueness of the viscosity solution by applying a result of Costantini, Papi and D'Ippoliti [CPD12].…”
Section: Introductionmentioning
confidence: 97%