Stochastic differential equations with random coefficients

In this paper we establish the existence and uniqueness of a solution for different types of stochastic differential equation with random initial conditions and random coefficients. The stochastic integral is interpreted as a generalized Stratonovich integral, and the techniques used to derive these results are mainly based on the path properties of the Brownian motion, and the definition of the Stratonovich integral.

Large deviations for stochastic Volterra equations

This paper is devoted to prove a large-deviation principle for solutions to multidimensional stochastic Volterra equations.

Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H 1/2

We consider the Cauchy problem for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H>¿. We prove an existence and uniqueness result for this problem, when the coefficients are sufficiently regular. Furthermore, if the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of all orders, we prove that the law of the solution admits a smooth density with respect to Lebesgue measure on R.

Stochastic models and numerical algorithms for a class of regulatory gene networks.

Regulatory gene networks contain generic modules, like those involving feedback loops, which are essential for the regulation of many biological functions (Guido et al. in Nature 439:856-860, 2006). We consider a class of self-regulated genes which are the building blocks of many regulatory gene networks, and study the steady-state distribution of the associated Gillespie algorithm by providing efficient numerical algorithms. We also study a regulatory gene network of interest in gene therapy, using mean-field models with time delays. Convergence of the related time-nonhomogeneous Markov chain is established for a class of linear catalytic networks with feedback loops.

Essays in equilibrium asset pricing

Introduction This dissertation consists of three essays in equilibrium asset pricing. The first chapter studies the asset pricing implications of a general equilibrium model in which real investment is reversible at a cost. Firms face higher costs in contracting than in expanding their capital stock and decide to invest when their productive capital is scarce relative to the overall capital of the economy. Positive shocks to the capital of the firm increase the size of the firm and reduce the value of growth options. As a result, the firm is burdened with more unproductive capital and its value lowers with respect to the accumulated capital. The optimal consumption policy alters the optimal allocation of resources and affects firm's value, generating mean-reverting dynamics for the M/B ratios. The model (1) captures convergence of price-to-book ratios -negative for growth stocks and positive for value stocks - (firm migration), (2) generates deviations from the classic CAPM in line with the cross-sectional variation in expected stock returns and (3) generates a non-monotone relationship between Tobin's q and conditional volatility consistent with the empirical evidence. The second chapter proposes a standard portfolio-choice problem with transaction costs and mean reversion in expected returns. In the presence of transactions costs, no matter how small, arbitrage activity does not necessarily render equal all riskless rates of return. When two such rates follow stochastic processes, it is not optimal immediately to arbitrage out any discrepancy that arises between them. The reason is that immediate arbitrage would induce a definite expenditure of transactions costs whereas, without arbitrage intervention, there exists some, perhaps sufficient, probability that these two interest rates will come back together without any costs having been incurred. Hence, one can surmise that at equilibrium the financial market will permit the coexistence of two riskless rates that are not equal to each other. For analogous reasons, randomly fluctuating expected rates of return on risky assets will be allowed to differ even after correction for risk, leading to important violations of the Capital Asset Pricing Model. The combination of randomness in expected rates of return and proportional transactions costs is a serious blow to existing frictionless pricing models. Finally, in the last chapter I propose a two-countries two-goods general equilibrium economy with uncertainty about the fundamentals' growth rates to study the joint behavior of equity volatilities and correlation at the business cycle frequency. I assume that dividend growth rates jump from one state to other, while countries' switches are possibly correlated. The model is solved in closed-form and the analytical expressions for stock prices are reported. When calibrated to the empirical data of United States and United Kingdom, the results show that, given the existing degree of synchronization across these business cycles, the model captures quite well the historical patterns of stock return volatilities. Moreover, I can explain the time behavior of the correlation, but exclusively under the assumption of a global business cycle.

An allele-specific, stochastic gene expression process controls the expression of multiple Ly49 family genes and generates a diverse, MHC-specific NK cell receptor repertoire.

Mouse NK cells express MHC class I-specific inhibitory Ly49 receptors. Since these receptors display distinct ligand specificities and are clonally distributed, their expression generates a diverse NK cell receptor repertoire specific for MHC class I molecules. We have previously found that the Dd (or Dk)-specific Ly49A receptor is usually expressed from a single allele. However, a small fraction of short-term NK cell clones expressed both Ly49A alleles, suggesting that the two Ly49A alleles are independently and randomly expressed. Here we show that the genes for two additional Ly49 receptors (Ly49C and Ly49G2) are also expressed in a (predominantly) mono-allelic fashion. Since single NK cells can co-express multiple Ly49 receptors, we also investigated whether mono-allelic expression from within the tightly linked Ly49 gene cluster is coordinate or independent. Our clonal analysis suggests that the expression of alleles of distinct Ly49 genes is not coordinate. Thus Ly49 alleles are apparently independently and randomly chosen for stable expression, a process that directly restricts the number of Ly49 receptors expressed per single NK cell. We propose that the Ly49 receptor repertoire specific for MHC class I is generated by an allele-specific, stochastic gene expression process that acts on the entire Ly49 gene cluster.

Stochastic resonance in a dipole

We show that the dipole, a system usually proposed to model relaxation phenomena, exhibits a maximum in the signal-to-noise ratio at a nonzero noise level, thus indicating the appearance of stochastic resonance. The phenomenon occurs in two different situations, i.e., when the minimum of the potential of the dipole remains fixed in time and when it switches periodically between two equilibrium points. We have also found that the signal-to-noise ratio has a maximum for a certain value of the amplitude of the oscillating field.

Stochastic inversion of tracer test and electrical geophysical data to estimate hydraulic conductivities.

Quantifying the spatial configuration of hydraulic conductivity (K) in heterogeneous geological environments is essential for accurate predictions of contaminant transport, but is difficult because of the inherent limitations in resolution and coverage associated with traditional hydrological measurements. To address this issue, we consider crosshole and surface-based electrical resistivity geophysical measurements, collected in time during a saline tracer experiment. We use a Bayesian Markov-chain-Monte-Carlo (McMC) methodology to jointly invert the dynamic resistivity data, together with borehole tracer concentration data, to generate multiple posterior realizations of K that are consistent with all available information. We do this within a coupled inversion framework, whereby the geophysical and hydrological forward models are linked through an uncertain relationship between electrical resistivity and concentration. To minimize computational expense, a facies-based subsurface parameterization is developed. The Bayesian-McMC methodology allows us to explore the potential benefits of including the geophysical data into the inverse problem by examining their effect on our ability to identify fast flowpaths in the subsurface, and their impact on hydrological prediction uncertainty. Using a complex, geostatistically generated, two-dimensional numerical example representative of a fluvial environment, we demonstrate that flow model calibration is improved and prediction error is decreased when the electrical resistivity data are included. The worth of the geophysical data is found to be greatest for long spatial correlation lengths of subsurface heterogeneity with respect to wellbore separation, where flow and transport are largely controlled by highly connected flowpaths.

First-passage and escape problems in the Feller process

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single-neuron firing to volatility of financial assets. While general properties of the process have long been well known, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.

Entropic stochastic resonance

We present a novel scheme for the appearance of stochastic resonance when the dynamics of a Brownian particle takes place in a confined medium. The presence of uneven boundaries, giving rise to an entropic contribution to the potential, may upon application of a periodic driving force result in an increase of the spectral amplification at an optimum value of the ambient noise level. The entropic stochastic resonance, characteristic of small-scale systems, may constitute a useful mechanism for the manipulation and control of single molecules and nanodevices.

A hybrid algorithm combining metaheuristic with Monte Carlo simulation for solving the Stochastic Flow Shop problem

In this paper, a hybrid simulation-based algorithm is proposed for the StochasticFlow Shop Problem. The main idea of the methodology is to transform the stochastic problem into a deterministic problem and then apply simulation to the latter. In order to achieve this goal, we rely on Monte Carlo Simulation and an adapted version of a deterministic heuristic. This approach aims to provide flexibility and simplicity due to the fact that it is not constrained by any previous assumption and relies in well-tested heuristics.

A hybrid algorithm combining metaheuristic with Monte Carlo simulation for solving the Stochastic Flow Shop problem

In this paper, a hybrid simulation-based algorithm is proposed for the StochasticFlow Shop Problem. The main idea of the methodology is to transform the stochastic problem into a deterministic problem and then apply simulation to the latter. In order to achieve this goal, we rely on Monte Carlo Simulation and an adapted version of a deterministic heuristic. This approach aims to provide flexibility and simplicity due to the fact that it is not constrained by any previous assumption and relies in well-tested heuristics.

Asymmetric stochastic switching driven by intrinsic molecular noise

Low-copy-number molecules are involved in many functions in cells. The intrinsic fluctuations of these numbers can enable stochastic switching between multiple steady states, inducing phenotypic variability. Herein we present a theoretical and computational study based on Master Equations and Fokker-Planck and Langevin descriptions of stochastic switching for a genetic circuit of autoactivation. We show that in this circuit the intrinsic fluctuations arising from low-copy numbers, which are inherently state-dependent, drive asymmetric switching. These theoretical results are consistent with experimental data that have been reported for the bistable system of the gallactose signaling network in yeast. Our study unravels that intrinsic fluctuations, while not required to describe bistability, are fundamental to understand stochastic switching and the dynamical relative stability of multiple states.