Auswahl der wissenschaftlichen Literatur zum Thema „Generalized Feynman-Kac formula“

Let X be a Markov process, which is assumed to be associated with a symmetric Dirichlet form [Formula: see text]. For [Formula: see text], the extended Dirichlet space, we have the classical Fukushima's decomposition: [Formula: see text], where [Formula: see text] is a quasi-continuous version of u, [Formula: see text] the martingale part and [Formula: see text] the zero energy part. In this paper, we investigate two important transformations for X, the Girsanov transform induced by [Formula: see text] and the generalized Feynman–Kac transform induced by [Formula: see text]. For the Girsanov transform, we present necessary and sufficient conditions for which to induce a positive supermartingale and hence to determine another Markov process [Formula: see text]. Moreover, we characterize the symmetric Dirichlet form associated with the Girsanov transformed process [Formula: see text]. For the generalized Feynman–Kac transform, we give a necessary and sufficient condition for the generalized Feynman–Kac semigroup to be strongly continuous.

In this paper we study the solution of the stochastic heat equation where the potential V and the initial condition f are generalized stochastic processes. We construct explicitly the solution and we prove that it belongs to the generalized function space [Formula: see text].

Abstract In this paper, we give a probabilistic representation of the heat equation associated with the quantum K-Gross Laplacian using infinite-dimensional stochastic calculus in two variables. Applying the heat semigroup to the particular case where the operator is the multiplication one, we establish a relation between the classical and the quantum heat semigroup. Finally, using a combination between convolution calculus and the generalized stochastic calculus, we give a generalization of the Feynman–Kac formula.

The aim of this work is to establish and generalize a relationship between fractional partial differential equations (fPDEs) and stochastic differential equations (SDEs) to a wider class of stochastic processes, including fractional Brownian motions {BtH,t≥0} and sub-fractional Brownian motions {ξtH,t≥0} with Hurst parameter H∈(12,1). We start by establishing the connection between a fPDE and SDE via the Feynman–Kac Theorem, which provides a stochastic representation of a general Cauchy problem. In hindsight, we extend this connection by assuming SDEs with fractional- and sub-fractional Brownian motions and prove the generalized Feynman–Kac formulas under a (sub-)fractional Brownian motion. An application of the theorem demonstrates, as a by-product, the solution of a fractional integral, which has relevance in probability theory.

Path integral representations for generalized Schrödinger operators obtained under a class of Bernstein functions of the Laplacian are established. The one-to-one correspondence of Bernstein functions with Lévy subordinators is used, thereby the role of Brownian motion entering the standard Feynman–Kac formula is taken here by subordinate Brownian motion. As specific examples, fractional and relativistic Schrödinger operators with magnetic field and spin are covered. Results on self-adjointness of these operators are obtained under conditions allowing for singular magnetic fields and singular external potentials as well as arbitrary integer and half-integer spin values. This approach also allows to propose a notion of generalized Kato class for which an Lp-Lq bound of the associated generalized Schrödinger semigroup is shown. As a consequence, diamagnetic and energy comparison inequalities are also derived.

<p>In this paper, we considered the parabolic Anderson model with a class of time-independent generalized Gaussian fields on $ \mathbb{R}^d $, which included fractional white noise, Bessel field, massive free field, and other nonstationary Gaussian fields. Under the rough initial conditions, we constructed the Feynman-Kac formula as a solution in the Stratonovich integral by Brownian bridge, and then proved the Hölder continuity of the solution with respect to the time variable. As a comparison, we also studied the Hölder continuity under the regular initial conditions that $ u_0\equiv C $ and $ u_0\in C^\kappa(\mathbb{R}^d) $ with $ \kappa\in(0, 1] $.</p>

The thesis dealing with topics under model uncertainty consists of two main parts. In the first part, we introduce a reduced-form framework in the presence of multiple default times under model uncertainty. In particular, we define a sublinear conditional operator with respect to a family of possibly non-dominated priors for a filtration progressively enlarged by multiple ordered defaults. Moreover, we analyze the properties of this sublinear conditional expectation as a pricing instrument and consider an application to insurance market modeling with non-linear affine intensities. In the second part of this thesis, we prove a Feynman-Kac formula under volatility uncertainty which allows to take into account a discounting factor. In the first part, we generalize the results of a reduced-form framework under model uncertainty for a single default time in order to consider multiple ordered default times. The construction of these default times is based on a generalization of the Cox model under model uncertainty. Within this setting, we progressively enlarge a reference filtration by N ordered default times and define the sublinear expectation with respect to the enlarged filtration and a set of possibly non-dominated probability measures. We derive a weak dynamic programming principle for the operator and use it for the valuation of credit portfolio derivatives under model uncertainty. Moreover, we analyze the properties of the operator as a pricing instrument under model uncertainty. First, we derive some robust superhedging duality results for payment streams, which allow to interpret the operator as a pricing instrument in the context of superhedging. Second, we use the operator to price a contingent claim such that the extended market is still arbitrage-free in the sense of “no arbitrage of the first kind”. Moreover, we provide some conditions which guarantee the existence of a modification of the operator which has quasi-sure càdlàg paths. Finally, we conclude this part by an application to insurance market modeling. For this purpose, we extend the reduced-form framework under model uncertainty for a single default time to include intensities following a non-linear affine process under parameter uncertainty. This allows to introduce a longevity bond under model uncertainty in a way consistent with the classical case under a single prior and to compute its valuation numerically. In the second part, we focus on volatility uncertainty and, more specifically on the G-expectation setting. In this setting, we provide a generalization of a Feynman-Kac formula under volatility uncertainty in presence of a linear term in the PDE due to discounting. We state our result under different hypothesis with respect to the current result in the literature, where the Lipschitz continuity of some functionals is assumed, which is not necessarily satisfied in our setting. Thus, we establish for the first time a relation between non-linear PDEs and G-conditional expectation of a discounted payoff. To do so, we introduce a family of fully non-linear PDEs identified by a regularizing parameter with terminal condition φ at time T > 0, and obtain the G-conditional expectation of a discounted payoff as the limit of the solutions of such a family of PDEs when the regularity parameter goes to zero. Using a stability result, we can prove that such a limit is a viscosity solution of the limit PDE. Therefore, we are able to show that the G-conditional expectation of the discounted payoff is a solution of the PDE. In applications, this permits to calculate such a sublinear expectation in a computationally efficient way.

We introduce renormalized integrals which generalize conventional measure theoretic integrals. One approximates the integration domain by measure spaces and defines the integral as the limit of integrals over the approximating spaces. This concept is implicitly present in many mathematical contexts such as Cauchy's principal value, the determinant of operators on a Hilbert space and the Fourier transform of an L^p function. We use renormalized integrals to define a path integral on manifolds by approximation via geodesic polygons. The main part of the paper is dedicated to the proof of a path integral formula for the heat kernel of any self-adjoint generalized Laplace operator acting on sections of a vector bundle over a compact Riemannian manifold.

Cette thèse se consacre à étudier les symétries de Lie d'une classe particulière d'équations différentielles partielles (EDP), désignée sous le nom d'équation de Kolmogorov rétrograde. Cette équation joue un rôle essentiel dans le cadre des modèles financiers, notamment en lien avec le modèle de Longstaff-Schwartz, qui est largement utilisé pour la valorisation des options et des produits dérivés.Dans un contexte plus générale, notre étude s'oriente vers l'analyse des symétries de Lie de l'équation de Kolmogorov rétrograde, en introduisant un terme non linéaire. Cette généralisation est significative, car l'équation ainsi modifiée est liée à une équation différentielle stochastique rétrograde et progressive (EDSRP) via la formule de Feynman-Kac généralisée (non linéaire). Nous nous intéressons également à l'exploration des symétries de cette équation stochastique, ainsi qu'à la manière dont les symétries de l'EDP sont connectées à celles de l'EDSRP.Enfin, nous proposons un recalcul des symétries de l'équation différentielle stochastique rétrograde (EDSR) et de l'EDSRP, en adoptant une nouvelle approche. Cette approche se distingue par le fait que le groupe de symétries qui opère sur le temps dépend lui-même du processus Y, qui constitue la solution de l'EDSR. Cette dépendance ouvre de nouvelles perspectives sur l'interaction entre les symétries temporelles et les solutions des équations
This thesis is dedicated to studying the Lie symmetries of a particular class of partialdifferential equations (PDEs), known as the backward Kolmogorov equation. This equa-tion plays a crucial role in financial modeling, particularly in relation to the Longstaff-Schwartz model, which is widely used for pricing options and derivatives.In a broader context, our study focuses on analyzing the Lie symmetries of thebackward Kolmogorov equation by introducing a nonlinear term. This generalization issignificant, as the modified equation is linked to a forward backward stochastic differ-ential equation (FBSDE) through the generalized (nonlinear) Feynman-Kac formula.We also examine the symmetries of this stochastic equation and how the symmetriesof the PDE are connected to those of the BSDE.Finally, we propose a recalculation of the symmetries of the BSDE and FBSDE,adopting a new approach. This approach is distinguished by the fact that the symme-try group acting on time itself depends also on the process Y , which is the solutionof the BSDE. This dependence opens up new perspectives on the interaction betweentemporal symmetries and the solutions of the equations

Abstract These two papers describe the development of a variant of diffusion QMC with importance sampling and its applications to a few simple systems. The method is based on the generalized FeynmanKac path integral formalisma and is termed the “full generalized Feynman-Kac” or FGFK method. Its major advantage, relative to other diffusion-based methods, is the elimination of branching or multiplication of walkers. Its major disadvantage is a greater complexity. The basic implementation is very nearly the same as that of standard diffusion QMC with importance sampling, and it incorporates diffusion with drift in the same way. The uses of trial functions, fixed nodes, extrapolation to zero time-step size, and other procedures except for branching are identical. However, the evaluation of mean quantities can be carried out in terms of ‘lj;2 sampling to obtain energies and other properties based on the FGFK formula. The applications described include ground and excited states of the harmonic oscillator and the helium atom and the ground state of the hydrogen molecule. For these the energies as well as a dozen other observables such as the averages of interelectron distances were determined. The results were found to be in good agreement with accepted values. A combination of FGFK with the released-node method was investigated for the case of the one-dimensional harmonic oscillator and found to be successful for that case.