Estimates of transition densities for Brownian motion on nested fractals

Summary We obtain upper and lower bounds for the transition densities of Brownian motion on nested fractals. Compared with the estimate on the Sierpinski gasket, the results require the introduction of a new exponent,d _J, related to the shortest path metric and chemical exponent on nested fractals. Further, Hölder order of the resolvent densities, sample paths and local times are obtained. The results are obtained using the theory of multi-type branching processes.     

Convergence rates in strong ergodicity for Markov processes

A coupling method is used to obtain the explicit upper and lower bounds for convergence rates in strong ergodicity for Markov processes. For one-dimensional diffusion processes and birth–death processes, these bounds are sharp in the sense that the upper one and the lower one only differ in a constant.     

Bounds for Asian basket options

In this paper we propose pricing bounds for European-style discrete arithmetic Asian basket options in a Black and Scholes framework. We start from methods used for basket options and Asian options. First, we use the general approach for deriving upper and lower bounds for stop-loss premia of sums of non-independent random variables as in Kaas et al. [Upper and lower bounds for sums of random variables, Insurance Math. Econom. 27 (2000) 151–168] or Dhaene et al. [The concept of comonotonicity in actuarial science and finance: theory, Insurance Math. Econom. 31(1) (2002) 3–33]. We generalize the methods in Deelstra et al. [Pricing of arithmetic basket options by conditioning, Insurance Math. Econom. 34 (2004) 55–57] and Vanmaele et al. [Bounds for the price of discrete sampled arithmetic Asian options, J. Comput. Appl. Math. 185(1) (2006) 51–90]. Afterwards we show how to derive an analytical closed-form expression for a lower bound in the non-comonotonic case. Finally, we derive upper bounds for Asian basket options by applying techniques as in Thompson [Fast narrow bounds on the value of Asian options, Working Paper, University of Cambridge, 1999] and Lord [Partially exact and bounded approximations for arithmetic Asian options, J. Comput. Finance 10 (2) (2006) 1–52]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.     

Front propagation in infinite cylinders. II. The sharp reaction zone limit

This paper applies the variational approach developed in part I of this work [22] to a singular limit of reaction–diffusion–advection equations which arise in combustion modeling. We first establish existence, uniqueness, monotonicity, asymptotic decay, and the associated free boundary problem for special traveling wave solutions which are minimizers of the considered variational problem in the singular limit. We then show that the speed of the minimizers of the approximating problems converges to the speed of the minimizer of the singular limit. Also, after an appropriate translation the minimizers of the approximating problems converge strongly on compacts to the minimizer of the singular limit. In addition, we obtain matching upper and lower bounds for the speed of the minimizers in the singular limit in terms of a certain area-type functional for small curvatures of the free boundary. The conclusions of the analysis are illustrated by a number of numerical examples.     

Kernel estimates for a class of Kolmogorov semigroups

We prove short time pointwise upper bounds for the heat kernels of certain Kolmogorov operators. We use Lyapunov function techniques, where the Lyapunov functions depend also on the time variable. Received: 29 November 2007     

Small-time sharp bounds for kernels of convolution semigroups

We study small-time bounds for transition densities of convolution semigroups corresponding to pure jump Lévy processes in R ^d , d ≥ 1, including the processes with jump measures which are exponentially and subexponentially localized at ∞. For a large class of Lévy measures, not necessarily symmetric or absolutely continuous with respect to Lebesgue measure, we find the optimal upper bound in both time and space for the corresponding heat kernels at ∞. In case of Lévy measures that are symmetric and absolutely continuous with densities g such that g(x) ? f(|x|) for non-increasing profile functions f, we also prove the full characterization of the sharp two-sided transition densities bounds of the form

$${p_t}\left( x \right) \asymp h{\left( t \right)^{ - d}} \cdot {1_{\left\{ {\left| x \right| \leqslant \theta h\left( t \right)} \right\}}} + tg\left( x \right) \cdot {1_{\left\{ {\left| x \right| \geqslant \theta h\left( t \right)} \right\}}},t \in \left( {0,{t_0}} \right),{t_0} > 0,x \in {\mathbb{R}^d}.$$

This is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of Lévy measures. Our investigations lead to a surprising dichotomy correspondence of the decay properties at ∞ for transition densities of pure jump Lévy processes. All results are obtained solely by analytic methods, without use of probabilistic arguments.

     

Elementary density bounds for self-similar sets and application

Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition （OSC）, and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.     

Toeplitz and Hankel operators on Bergman-Djrbashian type weighted spaces of harmonic functions on the unit ball

The paper considers weighted spaces of harmonic functions. Having lower and upper bounds for the equivalent kernels of such spaces we consider Toeplitz and Hankel operators with different symbols. The Fredholm criterion for these operators on some compact is also studied.     

Conditions for the existence of quasi-stationary distributions for birth–death processes with killing

We consider birth–death processes on the nonnegative integers, where {1,2,…}$\{1,2,\dots \}$ is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 (killing) may occur from any state. Assuming that absorption at 0 is certain we are interested in additional conditions on the transition rates for the existence of a quasi-stationary distribution. Inspired by results of Kolb and Steinsaltz [M. Kolb, D. Steinsaltz, Quasilimiting behavior for one-dimensional diffusions with killing, Ann. Probab. 40 (2012) 162–212] we show that a quasi-stationary distribution exists if the decay rate of the process is positive and exceeds at most finitely many killing rates. If the decay rate is positive and smaller than at most finitely many killing rates then a quasi-stationary distribution exists if and only if the process one obtains by setting all killing rates equal to zero is recurrent.     

Estimates of gradient perturbation series

We give upper and lower bounds of perturbation series for transition densities, corresponding to additive gradient perturbations satisfying certain space–time integrability conditions.     

Some stochastic properties of “semi-magic” and “magic” Markov chains

Gustafson and Styan (Gustafson and Styan, Superstochastic matrices and Magic Markov chains, Linear Algebra Appl. 430 (2009) 2705-2715) examined the mathematical properties of superstochastic matrices, the transition matrices of “magic” Markov chains formed from scaled “magic squares”. This paper explores the main stochastic properties of such chains as well as “semi-magic” chains (with doubly-stochastic transition matrices). Stationary distribution, generalized inverses of Markovian kernels, mean first passage times, variances of the first passage times and expected times to mixing are considered. Some general results are developed, some observations from the chains generated by MATLAB are discussed, some conjectures are presented and some special cases, involving three and four states, are explored in detail.     

Multiple solutions for a class of elliptic equations with jumping nonlinearities

We consider a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using variational methods, we show that the number of solutions tends to infinity as the number of jumped eigenvalues tends to infinity. In order to prove this fact, for every positive integer k we prove that, when a parameter is large enough, there exists a solution which presents k interior peaks. We also describe the asymptotic behaviour and the profile of this solution as the parameter tends to infinity.     

Gradient estimate for Ornstein-Uhlenbeck jump processes

By using absolutely continuous lower bounds of the Lévy measure, explicit gradient estimates are derived for the semigroup of the corresponding Lévy process with a linear drift. A derivative formula is presented for the conditional distribution of the process at time t under the condition that the process jumps before t. Finally, by using bounded perturbations of the Lévy measure, the resulting gradient estimates are extended to linear SDEs driven by Lévy-type processes.     

Plane brownian motion in a region with holes

We consider a fixed family of balls with decreasing radii in the plane. We establish a relationship between a Dirichlet problem in a region without the balls and the solution of a Schroedinger equation in the complete region. Then we find upper bounds for the probability that a brownian motion exits the region without touching these balls. This is used to study harmonic measure and entire functions.     

Some results on the upper convex densities for the self-similar sets

In this paper, by means of a basic result concerning the estimation of the lower bounds of upper convex densities for the self-similar sets, we show that in the Sierpinski gasket, the minimum value of the upper convex densities is achieved at the vertices. In addition, we get new lower bounds of upper convex densities for the famous classical fractals such as the Koch curve, the Sierpinski gasket and the Cartesian product of the middle third Cantor set with itself, etc. One of the main results improves corresponding result in the relevant reference. The method presented in this paper is different from that in the work by Z. Zhou and L. Feng [The minimum of the upper convex density of the product of the Cantor set with itself, Nonlinear Anal. 68 (2008) 3439-3444].     

Off-Diagonal Heat Kernel Lower Bounds Without Poincare

On a manifold with polynomial volume growth satisfying Gaussianupper bounds of the heat kernel, a simple characterization ofthe matching lower bounds is given in terms of a certain Sobolevinequality. The method also works in the case of so-called sub-Gaussianor sub-diffusive heat kernels estimates, which are typical offractals. Together with previously known results, this yieldsa new characterization of the full upper and lower Gaussianor sub-Gaussian heat kernel estimates.     

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the non-linear stochastic heat equation in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.     

Perturbation of Transition Functions and a Feynman-Kac Formula for the Incorporation of Mortality

Markov transition kernels are perturbed by output kernels with a special emphasis on building mortality into structured population models. A Feynman-Kac formula is derived which illustrates the interplay of mortality with a Markov process associated with the unperturbed kernel. partially supported by NSF grants DMS-0314529 and SES-0345945 partially supported by NSF grants DMS-9706787 and DMS-0314529     

三维广义磁流体方程组解的最优衰减率

本文利用Fourier分解法首次建立了三维广义磁流体动力学方程组弱解的时间衰减估计,得到了该方程解关于时间衰减的上下界估计,并且获得了相应的最优代数衰减率.     

Estimates of densities for Lévy processes with lower intensity of large jumps

We obtain general lower estimates of transition densities of jump Lévy processes. We use them for processes with Lévy measures having bounded support, processes with exponentially decaying Lévy measures for large times and for processes with high intensity of small jumps for small times.