Convergence of numerical methods for stochastic differential equations in mathematical finance

Many stochastic differential equations that occur in financial modelling do not satisfy the standard assumptions made in convergence proofs of numerical schemes that are given in textbooks, i.e., their coefficients and the corresponding derivatives appearing in the proofs are not uniformly bounded and hence, in particular, not globally Lipschitz. Specific examples are the Heston and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia model with rational coefficient functions. Simple examples show that, for example, the Euler-Maruyama scheme may not converge either in the strong or weak sense when the standard assumptions do not hold. Nevertheless, new convergence results have been obtained recently for many such models in financial mathematics. These are reviewed here. Although weak convergence is of traditional importance in financial mathematics with its emphasis on expectations of functionals of the solutions, strong convergence plays a crucial role in Multi Level Monte Carlo methods, so it and also pathwise convergence will be considered along with methods which preserve the positivity of the solutions.Comment: Review Pape

Entropy increase in switching systems

The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them “macroscopic” and “microscopic” complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondo’s paradox

Stable Attracting Sets in Dynamical Systems and in Their One-Step Discretizations

We consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set Λ of arbitrary shape. Under the assumption of uniform asymptotic stability of Λ in the sense of Lyapunov, we show that discretized versions of the dynamical system involving one-step numerical methods have nearby attracting sets Λ(h), which are also uniformly asymptotically stable. Our proof uses the properties of a Lyapunov function which characterizes the stability of Λ

Lyapunov functions for linear nonautonomous dynamical equations on time scales

The existence of a Lyapunov function is established following a method of Yoshizawa for the uniform exponential asymptotic stability of the zero solution of a nonautonomous linear dynamical equation on a time scale with uniformly bounded graininess

Dissipative Quasigeostrophic Motion under Temporally Almost Periodic Forcing

The full nonlinear dissipative quasigeostrophic model is shown to have a unique temporally almost periodic solution when the wind forcing is temporally almost periodic under suitable constraints on the spatial square-integral of the wind forcing and the $\beta$ parameter, Ekman number, viscosity and the domain size. The proof involves the pullback attractor for the associated nonautonomous dynamical system

A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient

The existence of a mean-square continuous strong solution is established for vector-valued Itö stochastic differential equations with a discontinuous drift coefficient, which is an increasing function, and with a Lipschitz continuous diffusion coefficient. A scalar stochastic differential equation with the Heaviside function as its drift coefficient is considered as an example. Upper and lower solutions are used in the proof

Linear-implicit strong schemes for Itô-Galkerin approximations of stochastic PDEs

Linear-implicit versions of strong Taylor numerical schemes for finite dimensional Itô stochastic differential equations (SDEs) are shown to have the same order as the original scheme. The combined truncation and global discretization error of an gamma strong linear-implicit Taylor scheme with time-step delta applied to the N dimensional Itô-Galerkin SDE for a class of parabolic stochastic partial differential equation (SPDE) with a strongly monotone linear operator with eigenvalues lambda 1 <= lambda 2 <= ... in its drift term is then estimated by K(lambda N -½ + 1 + delta gamma) where the constant K depends on the initial value, bounds on the other coefficients in the SPDE and the length of the time interval under consideration. AMS subject classifications: 35R60, 60H15, 65M15, 65U05

In-depth research into rural road crashes

This report was produced under an agreement between Transport SA and the Road Accident Research Unit formed in the late 1990s. Due to various delays in the publication of this report, Transport SA has since become the Department for Transport, Energy and Infrastructure and the Road Accident Research Unit has become the Centre for Automotive Safety Research. The report describes a series of 236 rural road crashes investigated between 1 March 1998 and 29 February 2000 in South Australia. Investigations began with immediate attendance at the scene of the crash. The information collected for each crash included: photographs of the crash scene and vehicles involved, video record of the crash scene and vehicles in selected cases, examination of the road environment, a site plan of the crash scene and vehicle movements in the crash, examination and measurements of the vehicles involved, interviews with crash participants, interviews with witnesses, interviews with police, information on the official police report, information from Coroner’s reports, and injury data for the injured crash participants. The report provides an overall statistical summary of the sample of crashes investigated, followed by a detailed examination of the road infrastructure issues contributing to the crashes. This is done on the basis of crash type, with separate sections concerned with single vehicle crashes, midblock crashes and crashes at intersections. A section is also provided that examines the role of roadside hazards in the crashes.Baldock MRJ, Kloeden CN and McLean A

Periodic solutions of systems with asymptotically even nonlinearities

New conditions of solvability based on a general theorem on the calculation of the index at infinity for vector fields that have degenerate principal linear part as well as degenerate ... next order ... terms are obtained for the 2 Pi-periodic problem for the scalar equation x'' +n2x=g(|x|)+f(t,x)+b(t) with bounded g(u) and f(t,x) -> 0 as |x| -> 0. The result is also applied to the solvability of a two-point boundary value problem and to resonant problems for equations arising in control theory. AMS subject classifications: 47Hll, 47H30

Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a >= 0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields