Narrow Results

Diffusion processes have been widely investigated to understand some essential features of complex networks, and have attracted much attention from physicists, statisticians and computer scientists. In order to understand the evolution of the diffusion process and design the optimal routing strategy according to the maximal entropic diffusion on networks, we propose the information entropy comprehending the structural characteristics and information propagation on the network. Based on the analysis of the diffusion process, we analyze the coupling impact of the structural factor and information propagating factor on the information entropy, where the analytical results fit well with the numerical ones on scale-free complex networks. The information entropy can better characterize the complex behaviors on networks and provides a new way to deepen the understanding of the diffusion process.

We prove an averaging principle which asserts convergence of diffusion processes on domains separated by semi-permeable membranes, when diffusion coefficients tend to infinity while the flux through the membranes remains constant. In the limit, points in each domain are lumped into a single state of a limit Markov chain. The limit chain’s intensities are proportional to the membranes’ permeability and inversely proportional to the domains’ sizes. Analytically, the limit is an example of a singular perturbation in which boundary and transmission conditions play a crucial role. This averaging principle is strongly motivated by recent signaling pathways models of mathematical biology, which are discussed toward the end of the paper.

This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a one-dimensional diffusion with the infinitesimal variance a²(x), drift b(x) and instantaneous discount (killing) rate r(x). The Spectral Theorem for self-adjoint operators in Hilbert space yields the spectral decomposition of the contingent claim value function. Based on the Sturm–Liouville (SL) theory, we classify Feller's natural boundaries into two further subcategories: non-oscillatory and oscillatory/non-oscillatory with cutoff Λ≥0 (this classification is based on the oscillation of solutions of the associated SL equation) and establish additional assumptions (satisfied in nearly all financial applications) that allow us to completely characterize the qualitative nature of the spectrum from the behavior of a, b and r near the boundaries, classify all diffusions satisfying these assumptions into the three spectral categories, and present simplified forms of the spectral expansion for each category. To obtain explicit expressions, we observe that the Liouville transformation reduces the SL equation to the one-dimensional Schrödinger equation with a potential function constructed from a, b and r. If analytical solutions are available for the Schrödinger equation, inverting the Liouville transformation yields analytical solutions for the original SL equation, and the spectral representation for the diffusion process can be constructed explicitly. This produces an explicit spectral decomposition of the contingent claim value function.

In this paper, we study the Kelly criterion in the continuous time framework building on the work of E.O. Thorp and others. The existence of an optimal strategy is proven in a general setting and the corresponding optimal wealth process is found. A simple formula is provided for calculating the optimal portfolio in terms of drift, short term risk-free rate and correlations for a set of generic multi-dimensional diffusion processes satisfying some simple conditions. Properties of the optimal investment strategy are studied. The paper ends with a short discussion of the implications of these ideas for financial markets.

This paper deals with optimal prediction in a regime-switching model driven by a continuous-time Markov chain. We extend existing results for geometric Brownian motion by deriving optimal stopping strategies that depend on the current regime state and prove a number of continuity properties relating to optimal value and boundary functions. Our approach replaces the use of closed form expressions, which are not available in our setting, with PDE arguments that also simplify the approach of [du Toit & Peskir (2009) Selling a stock at the ultimate maximum, Annals of Applied Probability19 (3), 983–1014.] in the classical Brownian case.

The viscous hydrodynamics is investigated via the studying diffusion processes on groups of H^s Sobolev diffeomorphisms of a flat n-dimensional torus (s > ½n + 1). A certain stochastic perturbation of the curve on the above groups, describing the motion of perfect incompressible fluid (or of the diffuse matter), is constructed such that it satisfies a certain stochastic analogue of the geodesic equation and the expectation of its "backward velocity" in the tangent space at identical diffeomorphism (algebra of the group) is a solution of Navier–Stokes (or Burgers, respectively) equations. In particular, the existence of solutions of those equations for lower s is proved. Some other approaches to stochastic presentation of viscous hydrodynamics, using the groups of diffeomorphisms, are also discussed.

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

We study the dynamics of a tagged particle in an environment of infinitely many Brownian particles in continuum. All the particles interact via the gradient of an interaction potential. Our strategy is to construct the infinite particle environment process at first and afterwards the process coupling the motion of the tagged particle and the motion of the environment. First we derive an integration by parts formula with respect to the standard gradient ∇^Γ on configuration spaces Γ for a general class of grand canonical Gibbs measures μ, corresponding to pair potentials ϕ and intensity measures σ = zexp(-ϕ)dx, 0 < z < ∞, having correlation functions fulfilling a Ruelle bound. Combining this with a second integration by parts formula with respect to the gradient ∇^Γ in direction γ ∈ Γ, by Dirichlet form techniques we can construct the environment process and the coupled process, respectively. Our results give the first mathematically rigorous and complete construction of the tagged particle process in continuum with interaction potential. In particular, we can treat interaction potentials which might have a singularity at the origin, nontrivial negative part and infinite range as e.g., the Lennard–Jones potential.

We consider the dynamics of a tagged particle in an infinite particle environment moving according to a stochastic gradient dynamics. For singular interaction potentials this tagged particle dynamics was constructed first in Ref. 7, using closures of pre-Dirichlet forms which were already proposed in Refs. 13 and 24. The environment dynamics and the coupled dynamics of the tagged particle and the environment were constructed separately. Here we continue the analysis of these processes: Proving an essential m-dissipativity result for the generator of the coupled dynamics from Ref. 7, we show that this dynamics does not only contain the environment dynamics (as one component), but is, given the latter, the only possible choice for being the coupled process. Moreover, we identify the uniform motion of the environment as the reversed motion of the tagged particle. (Since the dynamics are constructed as martingale solutions on configuration space, this is not immediate.) Furthermore, we prove ergodicity of the environment dynamics, whenever the underlying reference measure is a pure phase of the system. Finally, we show that these considerations are sufficient to apply Ref. 4 for proving an invariance principle for the tagged particle process. We remark that such an invariance principle was studied before in Ref. 13 for smooth potentials, and shown by abstract Dirichlet form methods in Ref. 24 for singular potentials.

Our results apply for a general class of Ruelle measures corresponding to potentials possibly having infinite range, a non-integrable singularity at 0 and a nontrivial negative part, and fulfill merely a weak differentiability condition on ℝ^d\{0}.

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second-order geometry and Itô integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.

Let G ⊂ ℝ^d be (non-empty) open with euclidean closure . We assume that G is the interior of . Let ρ ∈ L¹(ℝ^d,dx), p > 0 dx-a.e., 0 ≤ α ≤ 1, and ρ_α:= (α1_G + (1 -α) 1_ℝ^d\G) ρ, m_α := ρ_αdx. Let if α = 0, D_α = ℝ^d if 0 < α < 1, D_α = G if α = 1. Let A = (a_ij) be symmetric, and globally uniformly strictly elliptic, ρ satisfying some mild regularity assumption. Non-symmetric conservative diffusion processes which inside obey the generator L given in the following suggestive form

The constructed process reminds one of a multidimensional generalized analogue of the α-skew Brownian motion (see [5], [9]), which corresponds to the case d = 1, , ρ ≡ 1, , a_ij ≡ δij.