Singular risk-neutral valuation equations

Costantini, Cristina; Papi, Marco; D'Ippoliti, Fernanda

doi:10.1007/s00780-011-0166-8

Cited by 12 publications

(18 citation statements)

References 28 publications

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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%

Viscosity methods giving uniqueness for martingale problems

Costantini

Kurtz

2015

Electron. J. Probab.

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Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes. We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as solutions of constrained martingale problems. In the case of reflecting diffusions in $D\subset {\bf R}^d$, our assumptions allow $ D$ to be nonsmooth and the direction of reflection to be degenerate. Two examples are presented: A diffusion with degenerate oblique direction of reflection and a class of jump diffusion processes with infinite variation jump component and possibly degenerate diffusion matrix

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Section: Introductionmentioning

confidence: 99%

Viscosity methods giving uniqueness for martingale problems

Costantini

Kurtz

2015

Electron. J. Probab.

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“…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%

Affine processes under parameter uncertainty

Fadina

Neufeld

Schmidt

2019

Probab Uncertain Quant Risk

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We develop a one-dimensional notion of affine processes under parameter uncertainty, which we call non-linear affine processes. This is done as follows: given a set Θ of parameters for the process, we construct a corresponding non-linear expectation on the path space of continuous processes. By a general dynamic programming principle we link this non-linear expectation to a variational form of the Kolmogorov equation, where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in Θ. This non-linear affine process yields a tractable model for Knightian uncertainty, especially for modelling interest rates under ambiguity.We then develop an appropriate Itô-formula, the respective term-structure equations and study the non-linear versions of the Vasiček and the Cox-Ingersoll-Ross (CIR) model. Thereafter we introduce the non-linear Vasiček-CIR model. This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence the approach solves this modelling issue arising with negative interest rates.

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“…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%

Risk Neutral Valuation of Inflation-Linked Interest Rate Derivatives

Antonacci¹,

Costantini²,

D'Ippoliti³

et al. 2019

Preprint

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We propose a model for the joint evolution of European inflation, the European Central Bank official interest rate and the short-term interest rate, in a stochastic, continuous time setting.We derive the valuation equation for a contingent claim depending potentially on all three factors. This valuation equation reduces to a finite number of Cauchy problems for a degenerate parabolic PDE with non-local terms. We show that the price of the contingent claim is the only viscosity solution of the valuation equation.We also provide an efficient numerical scheme to compute the price and implement it in an example.

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Singular risk-neutral valuation equations

Cited by 12 publications

References 28 publications

Viscosity methods giving uniqueness for martingale problems

Viscosity methods giving uniqueness for martingale problems

Affine processes under parameter uncertainty

Risk Neutral Valuation of Inflation-Linked Interest Rate Derivatives

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