Search

We discuss the geometry induced by pairs of diffusion operators on two states spaces related by a map from one space to the other. This geometry led us to an intrinsic point of view on filtering. This will be explained plainly by examples, in local coordinates and in the metric setting. This article draws largely from the books (˵Elworthy et al., On the geometry of diffusion operators and stochastic flows, Lecture Notes in Mathematics 1720, Springer, 1999″, ˵Elworthy et al., The Geometry of Filtering. To appear in Frontiers in Mathematics Series, Birkhauser″) and aims to have a comprehensive account of the geometry for a general audience. [ABSTRACT FROM AUTHOR]

We quantize a multidimensional SDE (in the Stratonovich sense) by solving the related system of ODE΄s in which the d-dimensional Brownian motion has been replaced by the components of functional stationary quantizers. We make a connection with rough path theory to show that the solutions of the quantized solutions of the ODE converge toward the solution of the SDE. On our way to this result we provide convergence rates of optimal quantizations toward the Brownian motion for ]> -Hölder distance, q > 2, in Lp(ℙ). [ABSTRACT FROM AUTHOR]

Agent-based models are rigorous tools for simulating the interactions of individual entities, such as organisms or molecules within cells and assessing their effects on the dynamic behavior of the system as a whole. In context with bioprocess and biosystems engineering there are several interesting and important applications. This contribution aims at introducing this strategy with the aid of two examples characterized by striking distinctions in the scale of the individual entities and the mode of their interactions. In the first example a structured-segregated model is applied to travel along the lifelines of single cells in the environment of a three-dimensional turbulent field of a stirred bioreactor. The modeling approach is based on an Euler-Lagrange formulation of the system. The strategy permits one to account for the heterogeneity present in real reactors in both the fluid and cellular phases, respectively. The individual response of the cells to local variations in the extracellular concentrations is pictured by a dynamically structured model of the key reactions of the central metabolism. The approach permits analysis of the lifelines of individual cells in space and time. The second application of the individual modeling approach deals with dynamic modeling of signal transduction pathways in individual cells. Usually signal transduction networks are portrayed as being wired together in a spatially defined manner. Living circuitry, however, is placed in highly malleable internal architecture. Creating a homogenous bag of molecules, a well-mixed system, the dynamic behavior of which is modeled with a set of ordinary differential equations is normally not valid. The dynamics of the MAP kinase and a steroid hormone pathway serve as examples to illustrate how single molecule tracking can be linked with the stochasticity of biochemical reactions, where diffusion and reaction occur in a probabilistic manner. The problem of hindered diffusion caused by macromolecular crowding is also taken into account. [ABSTRACT FROM AUTHOR]

In this communication we have proposed a simple model for the dynamics of the target leverage ratio of a firm and performed a theoretical investigation of its time path. The gap between a firm's current target leverage ratio and the long-term target ratio is found to diminish exponentially, and the decay rate is determined by the volatility of the leverage ratio and the long-term target ratio only. The time-varying target leverage ratios can be readily incorporated into the dynamics of the leverage ratios of individual firms, and the default probabilities of individual firms can be generated to assess the default risks of the firms. [ABSTRACT FROM AUTHOR]

This paper discusses the use of the Automatic Differentiation approach in evaluating derivatives of functions represented by computer programs. We then considered a Cournot oligopoly modeled by a system of stochastic differential equations. The setting is that of a set of self-interested firms striving to adjust their productions in the direction of higher profits subject to mistakes or random shocks. The stochastic differential equations are solved by a numerical method and the profits are calculated using a Monte Carlo simulation. Then, Automatic Differentiation is used to propagate sensitivities along each path in an automated fashion. Numerical results have confirmed the intuition one may have that noisy environments can lead to important profit differences between firms as well as higher sensitivities as opposed to less noisy ones. [ABSTRACT FROM AUTHOR]