A general stochastic model for bivariate episodes driven by a gamma sequence

We propose a new stochastic model describing the joint distribution of (X,N), where N is a counting variable while X is the sum of N independent gamma random variables. We present the main properties of this general model, which include marginal and conditional distributions, integral transforms, moments and parameter estimation. We also discuss in more detail a special case where N has a heavy tailed discrete Pareto distribution. An example from finance illustrates the modeling potential of this new mixed bivariate distribution.

A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.

Pathogen Evolution when Transmission and Virulence are Stochastic

Evolutionary processes are inherently stochastic, since we can never know with certainty exactly how many descendants an individual will leave, or what the phenotypes of those descendants will be. Despite this, models of pathogen evolution have nearly all been deterministic, treating values such as transmission and virulence as parameters that can be known ahead of time. We present a broadly applicable analytic approach for modeling pathogen evolution in which vital parameters such as transmission and virulence are treated as random variables, rather than as fixed values. Starting from a general stochastic model of evolution, we derive specific equations for the evolution of transmission and virulence, and then apply these to a particular special case; the SIR model of pathogen dynamics. We show that adding stochasticity introduces new directional components to pathogen evolution. In particular, two kinds of covariation between traits emerge as important: covariance across the population (what is usually measured), and covariance between random variables within an individual. We show that these different kinds of trait covariation can be of opposite sign and contribute to evolution in very different ways. In particular, probability covariation between random variables within an individual is sometimes a better way to capture evolutionarily important tradeoffs than is covariation across a population. We further show that stochasticity can influence pathogen evolution through directional stochastic effects, which results from the inevitable covariance between individual fitness and mean population fitness.

Regulatory mechanisms are revealed by the distribution of transcription initiation times in single microbial cells

Transcription is the dominant point of control of gene expression. Biochemical studies have revealed key molecular components of transcription and their interactions, but the dynamics of transcription initiation in cells is still poorly understood. This state of affairs is being remedied with experiments that observe transcriptional dynamics in single cells using fluorescent reporters. Quantitative information about transcription initiation dynamics can also be extracted from experiments that use electron micrographs of RNA polymerases caught in the act of transcribing a gene (Miller spreads). Inspired by these data we analyze a general stochastic model of transcription initiation and elongation, and compute the distribution of transcription initiation times. We show that different mechanisms of initiation leave distinct signatures in the distribution of initiation times that can be compared to experiments. We analyze published micrographs of RNA polymerases transcribing ribosomal RNA genes inE.coliand compare the observed distributions of inter-polymerase distances with the predictions from previously hypothesized mechanisms for the regulation of these genes. Our analysis demonstrates the potential of measuring the distribution of time intervals between initiation events as a probe for dissecting mechanisms of transcription initiation in live cells.

On the structure of proper Black-Scholes formulae

The thinking behind the original Black-Scholes formula is criticised on the grounds that it holds out the quite unrealistic prospect of risk-free operation, that it can sacrifice asset maximisation to exact meeting of the contract, and that it restricts investment to those stocks on which an option is being sold. An alternative approach is given, based on a model of risk-averse asset maximisation, which, while meeting these criticisms, gives the option price in the familiar form of a discounted and weighted conditional expectation of the seller's liability at maturity. This evaluation is extended to a completely general stochastic model of stock price evolution; consideration is also given to the possibility of seller's ruin.