Information on jump sizes and hedging

We study a hedging problem in a market where traders have various levels of information. The exclusive information available only to informed traders is modelled by a diffusion process rather than discrete arrivals of new information. The asset price follows a jump–diffusion process and an information process affects jump sizes of the asset price. We find the local risk minimization hedging strategy of informed traders. Numerical examples as well as their comparison with the Black–Scholes strategy are provided via Monte Carlo.     

On differentiability with respect to the initial data of the solution to an SDE with a Lévy noise and discontinuous coefficients

We construct a stochastic flow generated by an stochastic differential equation with its drift being a function of bounded variation and its noise being a stable process with exponent from (1,2). It is proved that the flow is non-coalescing and Sobolev differentiable with respect to the initial data. The representation for the derivative is given.     

On semimartingale diffusions and stochastic differential equations

Given a regular diffusion X on the real axis which is a semimartingale we describe the semimartingale decomposition of X. We then give necessary and sufficient conditions in terms of the scale and the speed measure for X being a solution of an Ito type stochastic differential equation driven by a Wiener process and with classical drift or a drift term involving the local time of X. A regular diffusion is also characterized as unique solution of a certain martingale problem. Finally we discuss an example related to skew Brownian motion     

A Method to Explore Protein Side Chain Conformational Variability Using Experimental Data

Experimentally measured values of molecular properties or observables of biomolecules such as proteins are generally averages over time and space, which do not contain su?cient information to determine the underlying conformational distribution of the molecules in solution. The relationship between experimentally measured NMR ³J‐coupling values and the corresponding dihedral angle values is a particularly complicated case due to its nonlinear, multiple‐valued nature. Molecular dynamics (MD) simulations at constant temperature can generate Boltzmann ensembles of molecular structures that are free from a priori assumptions about the nature of the underlying conformational distribution. They suffer, however, from limited sampling with respect to time and conformational space. Moreover, the quality of the obtained structures is dependent on the choice of force ?eld and solvation model. A recently proposed method that uses time‐averaging with local‐elevation (LE) biasing of the conformational search provides an elegant means of overcoming these three problems. Using a set of side chain ³J‐coupling values for the FK506 binding protein (FKBP), we ?rst investigate the uncertainty in the angle values predicted theoretically. We then propose a simple MD‐based technique to detect inconsistencies within an experimental data set and identify degrees of freedom for which conformational averaging takes place or for which force ?eld parameters may be de?cient. Finally, we show that LE MD is the best method for producing ensembles of structures that, on average, ?t the experimental data.     

A stability property for locally uniformly rotund renorming

It is proved that C(K,E) (the space of all continuous functions on a Hausdorff compact space K taking values in a Banach space E) admits an equivalent locally uniformly rotund norm if C(K) and E do so. Moreover, if the equivalent LUR norms on C(K) and E are lower semicontinuous with respect to some weak topologies, the LUR norm on C(K,E) can be chosen to be lower semicontinuous with respect to an appropriate weak topology. As a consequence we prove that if X and Y are two Hausdorff compacta and C(X), C(Y) admit equivalent (pointwise lower semicontinuous) LUR norms, then so does C(X×Y).     

A local duality principle for the Baire classes of functions

A local dual of a Banach space X is a closed subspace of X^∗ that satisfies the properties that the principle of local reflexivity assigns to X as a subspace of X∗∗. We show that, for every ordinal 1?α?ω₁, the spaces Bα[0,1] of bounded Baire functions of class α are local dual spaces of the space M[0,1] of all Borel measures. As a consequence, we derive that each annihilator Bα^⊥[0,1] is the kernel of a norm-one projection.     

Multi‐dimensional Lotka–Volterra systems for carcinogenesis mutations

In the paper we consider three classes of models describing carcinogenesis mutations. Every considered model is described by the system of (n+1) equations, and in each class three models are studied: the first is expressed as a system of ordinary differential equations (ODEs), the second—as a system of reaction–diffusion equations (RDEs) with the same kinetics as the first one and with the Neumann boundary conditions, while the third is also described by the system of RDEs but with the Dirichlet boundary conditions. The models are formulated on the basis of the Lotka–Volterra systems (food chains and competition systems) and in the case of RDEs the linear diffusion is considered. The differences between studied classes of models are expressed by the kinetic functions, namely by the form of kinetic function for the last variable, which reflects the dynamics of malignant cells (that is the last stage of mutations). In the first class the models are described by the typical food chain with favourable unbounded environment for the last stage, in the second one—the last equation expresses competition between the pre‐malignant and malignant cells and the environment is also unbounded, while for the third one—it is expressed by predation term but the environment is unfavourable. The properties of the systems in each class are studied and compared. It occurs that the behaviour of solutions to the systems of ODEs and RDEs with the Neumann boundary conditions is similar in each class; i.e. it does not depend on diffusion coefficients, but strongly depends on the class of models. On the other hand, in the case of the Dirichlet boundary conditions this behaviour is related to the magnitude of diffusion coefficients. For sufficiently large diffusion coefficients it is similar independently of the class of models, i.e. the trivial solution that is unstable for zero diffusion gains stability. Copyright © 2009 John Wiley & Sons, Ltd.     

LOCAL TIME ANALYSIS OF ADDITIVE LEVY PROCESSES WITH DIFFERENT L(E)VY EXPONENTS

Let X1 XN be independent, classical Levy processes on R^d with Levy exponents ψ1,…, ψN, respectively. The corresponding additive Levy process is defined as the following N-parameter random field on R^d, X（t） △= X1（t1） ＋ ... ＋ XN（tN）, At∈N. Under mild regularity conditions on the ψi＇s, we derive estimate for the local and uniform moduli of continuity of local times of X = {X（t）; t ∈R^N}.