Björn Böttcher

APPROXIMATION OF FELLER PROCESSES BY MARKOV CHAINS WITH LÉVY INCREMENTS

We consider Feller processes whose generators have the test functions as an operator core. In this case, the generator is a pseudo differential operator with negative definite symbol q(x, ξ). If |q(x, ξ)| < c(1 + |ξ|2), the corresponding Feller process can be approximated by Markov chains whose steps are increments of Levy processes. This approximation can easily be used for a simulation of the sample path of a Feller process. Further, we provide conditions in terms of the symbol for the transition operators of the Markov chains to be Feller. This gives rise to a sequence of Feller processes approximating the given Feller process.

Lévy Matters III

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Convergence rate of numerical solutions to SFDEs with jumps

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An overshoot approach to recurrence and transience of Markov processes

Abstract In this paper, we are interested in numerical solutions of stochastic functional differential equations with jumps. Under a global Lipschitz condition, we show that the p th-moment convergence of Euler–Maruyama numerical solutions to stochastic functional differential equations with jumps has order 1 / p for any p ≥ 2 . This is significantly different from the case of stochastic functional differential equations without jumps, where the order is 1 / 2 for any p ≥ 2 . It is therefore best to use the mean-square convergence for stochastic functional differential equations with jumps. Moreover, under a local Lipschitz condition, we reveal that the order of mean-square convergence is close to 1 / 2 , provided that local Lipschitz constants, valid on balls of radius j , do not grow faster than log j .

The Moderator Effect That Wasn't There: Statistical Problems in Ambivalence Research

We develop criteria for recurrence and transience of one-dimensional Markov processes which have jumps and oscillate between +∞ and −∞. The conditions are based on a Markov chain which only consists of jumps (overshoots) of the process into complementary parts of the state space. In particular we show that a stable-like process with generator −(−∆) such that α(x) = α for x < −R and α(x) = β for x > R for some R > 0 and α, β ∈ (0, 2) is transient if and only if α+ β < 2, otherwise it is recurrent. As a special case this yields a new proof for the recurrence, point recurrence and transience of symmetric α-stable processes.

The Euler Scheme for Feller Processes

Ambivalence researchers often collapse separate measures of positivity and negativity into a single numerical index of ambivalence and refer to it as objective, operative, or potential ambivalence. The authors argue that this univariate approach to ambivalence models undermines the validity of subsequent statistical analyses because it confounds the effects of the index and its components. To remedy this situation, they demonstrate how the assumptions underlying the indices derived from the conflicting reactions model and similarity-intensity model can be tested using a multivariate approach to ambivalence models. On the basis of computer simulations and reanalyses of published moderator effects, the authors show that the frequently reported moderating influence of ambivalence on attitude effects may be a statistical artifact resulting from unmodeled correlations of positivity and negativity with attitude and the dependent variable. On the basis of extensive power analyses, they conclude that it may be extremely difficult to detect moderator effects of ambivalence in observational data. Therefore, they encourage ambivalence researchers to take an experimental approach to study design and a multivariate approach to data analysis.

Feynman formulas and path integrals for some evolution semigroups related to τ-quantization

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented. In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.

Brownian motion : an introduction to stochastic processes

This note is devoted to Feynman formulas (i.e., representations of semigroups by limits of n-fold iterated integrals as n → ∞) and their connections with phase space Feynman path integrals. Some pseudodifferential operators corresponding to different types of quantization of a quadratic Hamiltonian function are considered. Lagrangian and Hamiltonian Feynman formulas for semigroups generated by these operators are obtained. Further, a construction of Hamiltonian (phase space) Feynman path integrals is introduced. Due to this construction, the Hamiltonian Feynman formulas obtained here and in our previous papers do coincide with Hamiltonian Feynman path integrals. This connects phase space Feynman path integrals with some integrals with respect to probability measures. These connections enable us to make a contribution to the theory of phase space Feynman path integrals, to prove the existence of some of these integrals, and to study their properties by means of stochastic analysis. The Feynman path integrals thus obtained are different for different types of quantization. This makes it possible to distinguish the process of quantization in the language of Feynman path integrals.

Feller Processes: The Next Generation in Modeling. Brownian Motion, Lévy Processes and Beyond

Stochastic processes occur everywhere in sciences and engineering, and need to be understood by applied mathematicians, engineers and scientists alike. This is a first course introducing the reader gently to the subject. Brownian motions are a stochastic process, central to many applications and easy to treat.

Feller evolution systems: Generators and approximation

We present a simple construction method for Feller processes and a framework for the generation of sample paths of Feller processes. The construction is based on state space dependent mixing of Lévy processes. Brownian Motion is one of the most frequently used continuous time Markov processes in applications. In recent years also Lévy processes, of which Brownian Motion is a special case, have become increasingly popular. Lévy processes are spatially homogeneous, but empirical data often suggest the use of spatially inhomogeneous processes. Thus it seems necessary to go to the next level of generalization: Feller processes. These include Lévy processes and in particular Brownian motion as special cases but allow spatial inhomogeneities. Many properties of Feller processes are known, but proving the very existence is, in general, very technical. Moreover, an applicable framework for the generation of sample paths of a Feller process was missing. We explain, with practitioners in mind, how to overcome both of these obstacles. In particular our simulation technique allows to apply Monte Carlo methods to Feller processes.