Estimating functions for jump–diffusions

Asymptotic theory for approximate martingale estimating functions is generalised to diffusions with finite-activity jumps, when the sampling frequency and terminal sampling time go to infinity. Rate-optimality and efficiency are of particular concern. Under mild assumptions, it is shown that estimators of drift, diffusion, and jump parameters are consistent and asymptotically normal, as well as rate-optimal for the drift and jump parameters. Additional conditions are derived, which ensure rate-optimality for the diffusion parameter as well as efficiency for all parameters. The findings indicate a potentially fruitful direction for the further development of estimation for jump–diffusions.     

A Feynman–Kac formula for stochastic Dirichlet problems

A representation formula for solutions of stochastic partial differential equations with Dirichlet boundary conditions is proved. The scope of our setting is wide enough to cover the general situation when the backward characteristics that appear in the usual formulation are not even defined in the Itô sense.     

First order Feynman–Kac formula

We study the parabolic integral kernel for the weighted Laplacian with a potential. For manifolds with a pole we deduce formulas and estimates for the derivatives of the Feynman–Kac kernels and their logarithms, these are in terms of a ‘Gaussian’ term and the semi-classical bridge.     

Magnetic Energies and Feynman–Kac–Itô Formulas for Symmetric Markov Processes

Given a (conservative) symmetric Markov process on a metric space we consider related bilinear forms that generalize the energy form for a particle in an electromagnetic field. We obtain one bilinear form by semigroup approximation and another, closed one, by using a Feynman–Kac–Itô formula. If the given process is Feller, its energy measures have densities and its jump measure has a kernel, then the two forms agree on a core and the second is a closed extension of the first. In this case we provide the explicit form of the associated Hamiltonian.     

Explicit formulas for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift and their applications

We give an explicit formula for the twisted Koecher–Maaß series of the Duke–Imamoglu–Ikeda lift. As an application we prove a certain algebraicity result for the values of twisted Rankin–Selberg series at integers of half-integral weight modular forms, which was not treated by Shimura (J Math Soc Japan 33:649–672, 1981).     

A maximum principle for Markov regime-switching forward–backward stochastic differential games and applications

In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward–backward stochastic differential equations with jumps. First, we prove a sufficient maximum principle for nonzero-sum stochastic differential games problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for nonzero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a problem of robust utility maximization under a relative entropy penalty and to find optimal investment of an insurance firm under model uncertainty.     

Asymptotic Properties of Generalized Feynman–Kac Functionals

Let ( ) be a regular Dirichlet form on L2(X;m) and {Px}x X the Hunt process generated by ( ). Let be a signed 'smooth measure' associated with ( ) and At the continuous additive functional corresponding to the measure . Under some conditions on ( ) and , we shall prove that

Linear variance bounds for particle approximations of time-homogeneous Feynman–Kac formulae

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance for particle approximations of time-homogeneous Feynman–Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman–Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state space: (1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and (2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.     

Generalized Feynman–Kac Semigroups,Associated Quadratic Forms and Asymptotic Properties

In this paper, we study the Feynman–Kac semigroup T _t f(x)=E _x[f(X _t)exp(N _t)],where X _t is a symmetric Levy process and N _t is a continuous additive functional of zero energy which is not necessarily of bounded variation. We identify the corresponding quadratic form and obtain large time asymptotics of the semigroup. The Dirichlet form theory plays an important role in the whole paper.     

Derivation of Feynman–Kac and Bloch–Torrey Equations in a Trapping Medium

Methodology and Computing in Applied Probability - We derive rigorously the fractional counterpart of the Feynman–Kac equation for a transport problem with trapping events characterized by...     

A Large Deviation Principle for Symmetric Markov Processes with Feynman–Kac Functional

We establish a large deviation principle for the occupation distribution of a symmetric Markov process with Feynman–Kac functional. As an application, we show the L ^p -independence of the spectral bounds of a Feynman–Kac semigroup. In particular, we consider one-dimensional diffusion processes and show that if no boundaries are natural in Feller’s boundary classification, the L ^p -independence holds, and if one of the boundaries is natural, the L ^p -independence holds if and only if the L ²-spectral bound is non-positive.     

On the Generalized Feynman–Kac Transformation for?Nearly Symmetric Markov Processes

Suppose that X is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on L ²(E;m). For $u\in D(\mathcal{E})$ , we have Fukushima??s decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$ . In this paper, we investigate the strong continuity of the generalized Feynman?CKac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$ . Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$ . Denote by J ₁ the dissymmetric part of the jumping measure J of $(\mathcal{E},D(\mathcal{E}))$ . Under the assumption that J ₁ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant ?? ₀??0 such that $\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}$ for every t>0. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on L ²(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J ₁ is finite.     

Some extensions for Ramanujan’s circular summation formulas and applications

The Ramanujan Journal - In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we...     

Gibbs–butzer derivatives and their applications 1

A series of results on Gibbs derivatives and their applications made by the group of modern harmonic analysts and approximation theorists in Nanjing University is presented in this paper. It is a survey of these results concerned with certain locally compact groups obtained by that group. Moreover, some new results are also included.     

The Probabilistic Feynman–Kac Formula for an Infinite-Dimensional Schrödinger Equation with Exponential and Singular Potentials

Using the probabilistic Feynman–Kac formula, the existence of solutions of the Schrödinger equation on an infinite dimensional space E is proven. This theorem is valid for a large class of potentials with exponential growth at infinity as well as for singular potentials. The solution of the Schrödinger equation is local with respect to time and space variables. The space E can be a Hilbert space or other more general infinite dimensional spaces, like Banach and locally convex spaces (continuous functions, test functions, distributions). The specific choice of the infinite dimensional space corresponds to the smoothness of the fields to which the Schrödinger equation refers. The results also express an infinite-dimensional Heisenberg uncertainty principle: increasing of the field smoothness implies increasing of divergence of the momentum part of the quantum field Hamiltonian.     

Recurrence and transience for jump–diffusion processes

The purpose of this work is to obtain sufficient conditions for transience and recurrence of multidimensional jump–diffusion processes, which are driven by Brownian motion and Poisson random measure. The approach adopted here is to construct appropriate Lyapounov functions