. For Mandelbrot’s cascade (Y n ) in an independent and identically distributed (i.i.d.) random environment ξ, we are interested in the a.s. convergence rate of the Mandelbrot’s martingale (W n ) to its limit W, where W n = Y n / E ξ Y n is the normalized partition function. We obtain sufficient conditions under which W−W n has an exponential convergence rate: W − W n = o ( e − n a ) a.s. for some a > 0 explicitly calculated; we also find conditions under which W−W n has a polynomial convergence rate: W − W n = o ( n − α ) a.s. for some α > 0. Similar conclusions hold for Mandelbrot’s cascade in a varying environment.

We consider a model for inferring functional links between genes. We begin with the simple case of two genes whose presence or absence evolves stochastically along a phylogenetic tree. We develop a hidden Markov model where the hidden states of the model correspond to whether or not the genes perform a joint function. In the case that two genes do perform a joint function, the rates of gain or loss of each gene depend on the presence or absence of the other gene. Otherwise, those two genes are assumed to be gained and lost independently. Using simulation, we investigate the conditions under which the package corHMM can infer the hidden state correctly, and we also investigate when the Akaike information criterion (AIC) has the power to reject the simpler model when it is incorrect. We find that we can more accurately determine the dependent and independent rate class regimes when the trees have more tips and when the differences in the two rate classes are larger. We find the accuracy of corHMM is not overly affected by whether or not there are multiple transitions between the rate classes or just a single transition. We show how the two-gene case can be extended to a more general n-gene model with a level-dependent quasi-birth-and-death (LD-QBD) framework. We assume that the level n of the QBD corresponds to the number of genes that are required for some beneficial function, and the phases within each level record the presence/absence of particular genes.

We consider a general class of semi-Markov processes on countable state-space, with a differentiable kernel such that the embedded Markov chain may not be homogeneous. Using an SDE representation of the process, we study the associated flow by investigating meetings and merging related events for solutions, starting with different initial conditions. Some sufficient conditions for almost sure meeting and merging are also obtained. Many examples are considered to clarify the intricacies and implications of this study.

This article proposes and studies a new type of constant elasticity of volatility (CEV) model, titled LVO-CEV, where the name indicates that the excess return is linear in the volatility. The model has either strong or weak solutions, depending on the elasticity parameter, thanks to a connection to radial Ornstein-Uhlenbeck processes. Attainability of lower bounds and the existence of pricing measures are provided. The model allows for closed-form solutions in the context of expected utility theory for investors with hyperbolic absolute risk aversion (HARA) utilities on terminal wealth and consumption. We estimate and implement our model as well as other popular models of indexes and stocks, providing a fair comparison in portfolio management.

In this article, we analyze the strong consistency and r-th (r > 1) mean consistency for the estimators β u, n and g u, n of β and g, respectively, based on m-asymptotic negatively associated errors. Furthermore, the convergence rate for the strong consistency of the estimators is also considered. The results obtained in this article extend the corresponding ones for ρ ∗-mixing random variables and other dependent sequences.

. We consider the cover time of a discrete-time homogenous Markov chain, which is the time needed by the Markov chain to visit all its states. We analyze both the distribution and the moments of the cover time, and we are interested in exact results instead of asymptotic values of the mean cover time, which are generally considered in the literature. We first obtain several general results on the hitting time and the cover time of a subset of the state space, both in terms of distribution and moments. These results are then applied to particular graphs, namely the generalized cycle graph, the complete graph, and the generalized path graph. They lead to recurrence or analytic relations for the distribution and the mean value of their cover times.

In this article, we investigate the estimators for the heteroscadastic partially linear regression model under dependent errors defined by y i = x i β + g ( t i ) + ε i ( 1 ≤ i ≤ n ) , where ε i = σ i e i , σ i 2 = f ( u i ) , the design points ( x i , t i , u i ) are known and nonrandom, β is an unknown parameter to be estimated, the functions g(⋅) and f(⋅) are unknown, which are defined on a closed interval [0,1], and the random errors {e i } are (α, β)-mixing random variables. When the model is heteroscedastic, the unknown parameter β and the unknown function g(⋅) are approximated by the weighted least squares estimators. We derive the asymptotic normality of the weighted least squares estimators under some suitable conditions. Simulation studies are conducted to demonstrate the finite sample performance of the proposed procedure. Finally, we use real data to examine the dependence between oil prices and exchange rates.

. This article focuses on the classic optimal dividend and reinsurance problems. Different from the existing literature, it assumes that the insurance company has two lines of business with a common shock dependence. It can purchase proportional reinsurance to reduce business risk and pay dividends to stay competitive. The goal is to find out the optimal dividend and reinsurance strategies for maximizing the company’s value. Under the diffusion approximation model, we decomposed the problem into several situations and gave the corresponding solutions by using the stochastic control method. Some numerical examples and economic explanations are presented to illustrate the results.

. In Gambler’s Ruin when both players start with the same amount of money, we show the playing time stochastically increases when the games are made more fair. We give two different arguments for this fact that extend results from Peköz and Ross. We then use this to show that the exit time from a symmetric interval for Brownian motion with drift stochastically increases as the drift moves closer to zero; this result is not easily obtainable from the available explicit formulas for the density.

. In this article, we exploit recent results for the Laplace transform of the sojourn time Ω of an entire batch in the M [ X ] / M / 1 processor sharing queue in order to derive the asymptotic behavior of the distribution function of Ω, namely the behavior of ℙ(Ω > x) when x tends to infinity. Specifically, we show that, up to a multiplying factor, ℙ(Ω > x) has the same order of magnitude as ℙ(ω > x) for large x, where ω is the sojourn time of an arbitrary job in the system. From a practical point of view, this means that if a system has to be dimensioned to guarantee the processing time for jobs, then the system can also guarantee processing times for entire batches by introducing a marginal amount of processing capacity.