Tomasz Zastawniak

Fundamentals of Finslerian Diffusion with Applications

Introduction. 1. Finsler Spaces. 2. Introduction to Stochastic Calculus on Manifolds. 3. Stochastic Development on Finsler Spaces. 4. Volterra-Hamilton Systems of Finsler Type. 5. Finslerian Diffusion and Curvature. 6. Diffusion on the Tangent and Indicatrix Bundles. A. Diffusion and Laplacian on the Base Space. B. Two-Dimensional Constant Berwald Spaces. Bibliography. Index.

A Relativistic Version of Nelson's Stochastic Mechanics

Markov diffusions corresponding to solutions of the Klein-Gordon equation are introduced in a similar way as in Nelsons stochastic mechanics. A relativistic version of the Nelson-Newton law for Markov diffusions is established. Stochastic generalizations of the proper time and of the relativistic action functional are proposed and a stochastic version of the relativistic principle of least action is stated. The formalism can be regarded as a quantization procedure leading from classical special relativity to the Klein-Gordon equation.

Utility maximizing entropy and the second law of thermodynamics

Expected utility maximization problems in mathematical finance lead to a generalization of the classical definition of entropy. It is demonstrated that a necessary and sufficient condition for the second law of thermodynamics to operate is that any one of the generalized entropies should tend to its minimum value of zero.

Diffusions on Finsler Manifolds

Let M be a finite-dimensional Finsler manifold with arc element ds = L(x 1,… , x n, dx 1,… , dx n), L being a non-negative smooth homogeneous function of degree one in dx i. The Finsler metric tensor, where will be assumed to be positive definite for all x ∈ M and 0 ≠y ∈TM x. (For some examples, y 1…y n ≠ 0, is required.) Let M be equipped with the Cartan connection being the nonlinear, the horizontal, and the vertical connection coefficients, which can be expressed as where as above, with and. Here and throughout the paper we use the standard summation convention. The horizontal and vertical covariant derivatives of a Finsler vector field A i(x, y) will be denoted by These formulae can be extended to any Finsler tensor fields in the usual way. The Cartan connection is metrical, i.e., (1.1)

Stochastic Calculus on Finsler Manifolds and an Application in Biology

Since the pioneering works by Ito [15,16,17,18] the theory of Brownian motion and stochastic development on Riemannian manifolds has become a classical branch of stochastic calculus (see, for example [10,11,13]) with numerous applications in other areas. In the present article we extend the theory of Brownian motion and stochastic development to the case of Finsler manifolds.

The nonexistence of the path-space measure for the Dirac equation in four space-time dimensions

It is proved that the path‐space measure for the Dirac equation in four space‐time dimensions does not exist. The origin of the nonexistence of the measure turns out to be the dependence of the solution on the first derivative of the initial condition.

Noise Induced Transitions in a Stochastic Volterra-Hamilton Open System

A Volterra-Hamilton system describing the evolution of a dimorphic clone in the presence of inner developmental noise is considered as an open system in interaction with a fluctuating environment, subject to optimum growth conditions.In the case of constant environment considered previously by Antonelli and Křivan the system is confined to an invariant set of a stationary diffusion process, which provides a model of growth canalization. Different invariant sets can be identified with different clonal types of a given species. The probability distribution of the diffusion over an invariant set accounts for the variability within the corresponding clonal type.In this paper, the external noise in a non-constant environment is shown to trigger transitions between invariant sets as it interacts with the inner developmental noise. Such transitions from one clonal type to another, which do not involve any genetic alterations, are known in biology as plastic responses to the environment.This is an entirely different mechanism than genetic mutations, which can disturb the equilibrium of the system. If after such a mutation the system settles down in a new stationary state with its own invariant sets and probability distribution, then one or more new genetically altered species will emerge.

Diffusion on the Tangent and Indicatrix Bundles of a Finsler Manifold

The theory of diffusion processes on Riemannian manifolds, which goes back to the pioneering articles [7], [8], [9], [10] by Ito, has now become a classical branch of stochastic calculus (see, for example, [4], [5], [6]). Recently, the theory has been extended by the present authors [1], [2] to the case of diffusions on Finsler manifolds, the extension being motivated by certain models in developmental and population biology involving systems in a noisy environment. The goal of this article is to represent and study Finslerian diffusions as processes on the slit tangent bundle TM and the indicatrix bundle I M of a Finsler manifold M.

Density-dependent host/parasite systems of Rothschild type and Finslerian diffusion

Host/parasite interactions are considered in which the host has hormonal control over parasite reproductivity. A previously studied model is modified using social interactions arising from evolution for smaller size and simpler morphology for parasites and higher reproductivity for hosts (i.e., progenesis a la S.J. Gould). The evolutionary process necessarily takes place in a noisy environment typical of r-selective regimes. Emergent density-dependent effects of progenesis require Finsler diffusion theory to obtain the stochastically perturbed model equations. An application is made to the myxomatosis epizootic of the European wild rabbit. Qualitative information about this disease is obtained from the Feynman-Kac formula for the forward Cauchy problem in a Finsler space of Wagner class.

Linear Vector Optimization and European Option Pricing Under Proportional Transaction Costs

A method for pricing and superhedging European options under proportional transaction costs based on linear vector optimisation and geometric duality developed by Lohne & Rudloff (2014) is compared to a special case of the algorithms for American type derivatives due to Roux & Zastawniak (2014). An equivalence between these two approaches is established by means of a general result linking the support function of the upper image of a linear vector optimisation problem with the lower image of the dual linear optimisation problem.