Kac Formula Research Articles

Well-posedness of scalar BSDEs with sub-quadratic generators and related PDEs

We first establish the existence of an unbounded solution to a backward stochastic differential equation (BSDE) with generator g allowing a general growth in the state variable y and a sub-quadratic growth in the state variable z, when the terminal condition satisfies a sub-exponential moment integrability condition, which is weaker than the usual exp(μL)-integrability and stronger than Lp(p>1)-integrability. Then, we prove the uniqueness and comparison theorem for the unbounded solutions of the preceding BSDEs under some additional assumptions and establish a general stability result for the unbounded solutions. Finally, we derive the nonlinear Feynman–Kac formula in this context.

Finite time-horizon optimal investment and consumption with time-varying subsistence consumption constraints

In this paper we consider a general optimal consumption and portfolio selection problem of a finitely-lived agent whose consumption rate process is subject to time-varying subsistence consumption constraints. That is, her consumption rate should be greater than or equal to some convex, non-decreasing and continuous function of time t. Using martingale duality approach and Feynman–Kac formula, we derive the partial differential equation of the Cauchy problem satisfied by the dual value function. We use the integral transform method for solving this Cauchy problem to obtain the general optimal policies in an explicit form. With constant relative risk aversion and constant absolute risk aversion utility functions we illustrate some numerical results of the optimal policies.

Representation formula for viscosity solution to a PDE problem involving Pucci’s extremal operator

We provide a representation formula for viscosity solutions to an elliptic Dirichlet problem involving Pucci’s extremal operators. This is done through a dynamic programming principle derived from Denis et al. (2010). The formula can be seen as a nonlinear extension of the Feynman–Kac formula.

Spectral Rigidity of Random Schrödinger Operators via Feynman–Kac Formulas

We develop a technique for proving number rigidity (in the sense of Ghosh and Peres in Duke Math J 166(10):1789–1858, 2017) of the spectrum of general random Schrodinger operators (RSOs). Our method makes use of Feynman–Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman–Kac formulas for RSOs with multiplicative noise (Gaudreau Lamarre in Semigroups for one-dimensional Schrodinger operators with multiplicative Gaussian noise, Preprint arXiv:1902.05047v3 , 2019; Gaudreau Lamarre and Shkolnikov in Ann Inst Henri Poincare Probab Stat 55(3):1402–1438, 2019; Gorin and Shkolnikov in Ann Probab 46(4):2287–2344, 2018) by Gorin, Shkolnikov, and the first-named author, we use this method to prove number rigidity for a class of one-dimensional continuous RSOs of the form $$-\frac{1}{2}\Delta +V+\xi $$ , where V is a deterministic potential and $$\xi $$ is a stationary Gaussian noise. Our results require only very mild assumptions on the domain on which the operator is defined, the boundary conditions on that domain, the regularity of the potential V, and the singularity of the noise $$\xi $$ .

Household Lifetime Strategies under a Self-Contagious Market

In this paper, we consider the optimal strategies in asset allocation, consumption, and life insurance for a household with an exogenous stochastic income under a self-contagious market which is modeled by bivariate self-exciting Hawkes jump processes. By using the Hawkes process, jump intensities of the risky asset depend on the history path of that asset. In addition to the financial risk, the household is also subject to an uncertain lifetime and a fixed retirement date. A lump-sum payment will be paid as a heritage, if the wage earner dies before the retirement date. Under the dynamic programming principle, explicit solutions of the optimal controls are obtained when asset prices follow special jump distributions. For more general cases, we apply the Feynman–Kac formula and develop an iterative numerical scheme to derive the optimal strategies. We also prove the existence and uniqueness of the solution to the fixed point equation and the convergence of an iterative numerical algorithm. Numerical examples are presented to show the effect of jump intensities on the optimal controls.

Optimal Investment Policy for Insurers under the Constant Elasticity of Variance Model with a Correlated Random Risk Process

This paper investigates the optimal portfolio choice problem for a large insurer with negative exponential utility over terminal wealth under the constant elasticity of variance (CEV) model. The surplus process is assumed to follow a diffusion approximation model with the Brownian motion in which is correlated with that driving the price of the risky asset. We first derive the corresponding Hamilton–Jacobi–Bellman (HJB) equation and then obtain explicit solutions to the value function as well as the optimal control by applying a variable change technique and the Feynman–Kac formula. Finally, we discuss the economic implications of the optimal policy.

The size of the primes obstructing the existence of rational points

AbstractThe sequence of prime numbers p for which a variety over ℚ has no p-adic point plays a fundamental role in arithmetic geometry. This sequence is deterministic, however, we prove that if we choose a typical variety from a family then the sequence has random behaviour. We furthermore prove that this behaviour is modelled by a random walk in Brownian motion. This has several consequences, one of them being the description of the finer properties of the distribution of the primes in this sequence via the Feynman–Kac formula.

Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula

This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies

We obtain an explicit formula for the full expansion of the spectral action on Robertson–Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman–Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson–Walker spacetimes along an Apollonian sphere packing. Using Mellin transforms, we show that the asymptotic expansion of the spectral action contains the same terms as in the case of a single Robertson–Walker spacetime, but with zeta-regularized coefficients, given by values at integers of the zeta function of the fractal string of the radii of the sphere packing, as well as additional log-periodic correction terms arising from the poles (off the real line) of this zeta function.

Markovian integral equations

Nous analysons des équations intégrales Markoviennes multidimensionnelles qui sont formulées avec un processus de Markov progressif et non homogène dans le temps qui a des probabilités de transition Borel-mesurables. Dans les cas d’un processus de trajectoire d’une diffusion dépendante de trajectoire, les solutions à ces équations intégrales mènent au concept de solutions «mild » d’équations aux dérivées partielles dépendant de trajectoire. Notre objectif est d’établir unicité, stabilité, existence et non-extensibilité des solutions parmi une certaine classe de fonctions. En exigeant la continuité de Feller du processus de Markov, nous donnons des conditions faibles sous lesquelles les solutions deviennent continues. En outre, nous fournissons une formule multidimensionnelle de Feynmanc–Kac et un résultat unidimensionnel d’existence et d’unicité globales.

Finance Without Brownian Motions: An Introduction To Simplified Stochastic Calculus

The paper introduces a simple way of recording and manipulating general stochastic processes without explicit reference to a probability measure. In the new calculus, operations traditionally presented in a measure-specific way are instead captured by tracing the behaviour of jumps (also when no jumps are physically present). The calculus is fail-safe in that, under minimal assumptions, all informal calculations yield mathematically well-defined stochastic processes. The calculus is also intuitive as it allows the user to pretend all jumps are of compound Poisson type. The new calculus is very effective when it comes to computing drifts and expected values that possibly involve a change of measure. Such drift calculations yield, for example, partial integro-differential equations, Hamilton–Jacobi–Bellman equations, Feynman–Kac formulae, or exponential moments needed in numerous applications. We provide several illustrations of the new technique, among them a novel result on the Margrabe option to exchange one defaultable asset for another.

Integral equation characterization of the Feynman–Kac formula for a regime-switching diffusion

In this paper, we provide an integral equation characterization of the solution to a Cauchy problem associated to the Feynman–Kac formula for a regime-switching diffusion. We give a sufficient condition to guarantee the uniqueness of solutions to the integral equation and provide an example in the context of option pricing under the Ornstein–Uhlenbeck regime-switching model.

Precise high moment asymptotics for parabolic Anderson model with log-correlated Gaussian field

In this paper, we consider the continuous parabolic Anderson model (PAM) driven by a time-independent log-correlated Gaussian field (LGF). We obtain an asymptotic result of Eexp{12∑j,k=1N∫0t∫0tγ(Bj(s)−Bk(r))drds}(N→∞) which is composed of the independent Brownian motions {Bj(s)} and the function γ approximating to a logarithmic potential at 0, such as the covariances of massive free field and Bessel field. Based on the asymptotic result, we get the precise high moment asymptotics for Feynman–Kac formula of the PAM with LGF.

Heat conservation for generalized Dirac Laplacians on manifolds with boundary

We consider a notion of conservation for the heat semigroup associated with a generalized Dirac Laplacian acting on sections of a vector bundle over a noncompact manifold with a (possibly noncompact) boundary under mixed boundary conditions. Assuming that the geometry of the underlying manifold is controlled in a suitable way and imposing uniform lower bounds on the zero-order piece (Weitzenbock potential) of the Dirac Laplacian, and on the endomorphism defining the mixed boundary condition, we show that the corresponding conservation principle holds. A key ingredient in the proof is a domination property for the heat semigroup which follows from an extension to this setting of a Feynman–Kac formula recently proved by the author de Lima (Pac J Math 292(1):177–201, 2018) in the context of differential forms. When applied to the Hodge Laplacian acting on differential forms satisfying absolute boundary conditions, this extends previous results by Vesentini (Ann Math Pura Appl 182:1–19, 2003) and Masamune (Atti Accad Naz Lincei Rend Lincei Mat Appl 18(4):351–358, 2007) in the boundaryless case. Along the way, we also prove a vanishing result for $$L^2$$ harmonic sections in the broader context of generalized (not necessarily Dirac) Laplacians. These results are further illustrated with applications to the Dirac Laplacian acting on spinors and to the Jacobi operator acting on sections of the normal bundle of a free boundary minimal immersion.

Edge of spiked beta ensembles, stochastic Airy semigroups and reflected Brownian motions

Nous accedons a l’extremite du spectre des ensembles beta gaussiens avec perturbation de rang un par l’entremise de grandes puissances des matrices tridiagonales qui y sont associees. Pour les valeurs traditionnelles $\beta =1,2,4$, ceci correspond a l’etude des fluctuations des valeurs propres maximales des matrices aleatoires GOE/GUE/GSE assujetties a une perturbation additive de rang un. En dimensions infinies, nos resultats nous menent vers une famille de semi-groupes de type Feynman–Kac qui, dans un cas extreme, correspond au stochastic Airy semigroup introduit par Gorin et Shkolnikov (Ann. Probab. 46 (2018) 2287–2344). De plus, nos resultats ont pour corollaire des formules de Feynman–Kac pour les spiked stochastic Airy operators introduits par Bloemendal et Virag (Probab. Theory Related Fields 156 (2013) 795–825). Ces formules sont exprimees a l’aide de certaines fonctionnelles du mouvement brownien reflechi et de ses temps locaux. Ce faisant, les operateurs en question peuvent etre etudies a l’aide du calcul stochastique. Nous obtenons un premier resultat dans cette lignee en demontrant une nouvelle identite decrivant la distribution du mouvement brownien reflechi ayant ete conditionne sur son temps local a zero. La principale innovation de notre demonstration consiste en la preuve d’un nouveau resultat sur l’approximation forte du mouvement brownien reflechi et de son temps local par une marche aleatoire non negative en utilisant la methode de reflexion de Skorokhod.

A closed-form pricing formula for variance swaps under MRG–Vasicek model

In this paper, the pricing problems of variance swaps with discrete sampling times are studied, where the volatility of underlying assets follows a mean-reverting Gaussian (MRG in short) process, and the instantaneous interest rate is described by classical Vasicek model. By using measure transformation, Feynman–Kac formula and Fourier transform algorithm, a closed-form analytic pricing formula for variance swaps with the actual-return realized variance is presented.

A Feynman–Kac result via Markov BSDEs with generalised drivers

In this paper, we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman–Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE. Due to the irregularity of the driver, the $Y$-component of a couple $(Y,Z)$ solving the BSDE is not necessarily a semimartingale but a weak Dirichlet process.

Stochastic representation of solution to nonlocal-in-time diffusion

The aim of this paper is to derive a stochastic representation of the solution to a nonlocal-in-time evolution equation (with a historical initial condition), which serves a bridge between normal diffusion and anomalous diffusion. We first derive the Feynman–Kac formula by reformulating the original model into an auxiliary Caputo-type evolution equation with a specific forcing term subject to certain smoothness and compatibility conditions. After that, we confirm that the stochastic formula also provides the solution in the weak sense even though the problem data is nonsmooth. Finally, numerical experiments are presented to illustrate the theoretical results and the application of the stochastic formula.

A Feynman–Kac formula approach for computing expectations and threshold crossing probabilities of non-smooth stochastic dynamical systems

Abstract We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman–Kac formula, where the underlying stochastic processes satisfy variational inequalities modeling elasto-plastic and obstacle oscillators. We then focus on solving them numerically The main challenge in solving these equations is the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. We illustrate how to use our approach to compute expectations and other statistical quantities, such as the asymptotic growth rate of variance in asymptotic formulae for threshold crossing probabilities.

Simple nonlinear models with rigorous extreme events and heavy tails

Extreme events and the heavy tail distributions driven by them are ubiquitous in various scientific, engineering and financial research. They are typically associated with stochastic instability caused by hidden unresolved processes. Previous studies have shown that such instability can be modeled by a stochastic damping in conditional Gaussian models. However, these results are mostly obtained through numerical experiments, while a rigorous understanding of the underlying mechanism is sorely lacking. This paper contributes to this issue by establishing a theoretical framework, in which the tail density of conditional Gaussian models can be rigorously determined. In rough words, we show that if the stochastic damping takes negative values, the tail is polynomial; if the stochastic damping is nonnegative but takes value zero at a point, the tail is between exponential and Gaussian. The proof is established by constructing a novel, product-type Lyapunov function, where a Feynman–Kac formula is applied. The same framework also leads to a non-asymptotic large deviation bound for long-time averaging processes.