Quantile clocks are defined as convolutions of subordinators L, with quantile functions of positive random variables. We show that quantile clocks can be chosen to be strictly increasing and continuous and discuss their practical modeling advantages as business activity times in models for asset prices. We show that the marginal distributions of a quantile clock, at each fixed time, equate with the marginal distribution of a single subordinator. Moreover, we show that there are many quantile clocks where one can specify L, such that their marginal distributions have a desired law in the class of generalized s-self decomposable distributions, and in particular the class of self-decomposable distributions. The development of these results involves elements of distribution theory for specific classes of infinitely divisible random variables and also decompositions of a gamma subordinator, that is of independent interest. As applications, we construct many price models that have continuous trajectories, exhibit volatility clustering and have marginal distributions that are equivalent to those of quite general exponential Lévy price models. In particular, we provide explicit details for continuous processes whose marginals equate with the popular VG, CGMY and NIG price models. We also show how to perfectly sample the marginal distributions of more general classes of convoluted subordinators when L is in a sub-class of generalized gamma convolutions, which is relevant for pricing of European style options.

We study the inverse boundary crossing problem for diffusions. Given a diffusion process X_t, and a survival distribution p on [0, ∞), we demonstrate that there exists a boundary b(t) such that p(t) = ℙ[τ > t], where τ is the first hitting time of X_t to the boundary b(t). The approach taken is analytic, based on solving a parabolic variational inequality to find b. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary b does indeed have p as its boundary crossing distribution. Since little is known regarding the regularity of b arising from the variational inequality, this requires a detailed study of the problem of computing the boundary crossing distribution of X_t to a rough boundary. Results regarding the formulation of this problem in terms of weak solutions to the corresponding Kolmogorov forward equation are presented.

A voter sits on each vertex of an infinite tree of degree k, and has to decide between two alternative opinions. At each time step, each voter switches to the opinion of the majority of her neighbors. We analyze this majority process when opinions are initialized to independent and identically distributed random variables.

In particular, we bound the threshold value of the initial bias such that the process converges to consensus. In order to prove an upper bound, we characterize the process of a single node in the large k-limit. This approach is inspired by the theory of mean field spin-glass and can potentially be generalized to a wider class of models. We also derive a lower bound that is nontrivial for small, odd values of k.

We formulate the notion of the classical Fisher–Kolmogorov–Petrovskii–Piscounov (FKPP) reaction diffusion equation associated with a homogeneous conservative fragmentation process and study its traveling waves. Specifically, we establish existence, uniqueness and asymptotics. In the spirit of classical works such as McKean [Comm. Pure Appl. Math. 28 (1975) 323–331] and [Comm. Pure Appl. Math. 29 (1976) 553–554], Neveu [In Seminar on Stochastic Processes (1988) 223–242 Birkhäuser] and Chauvin [Ann. Probab. 19 (1991) 1195–1205], our analysis exposes the relation between traveling waves and certain additive and multiplicative martingales via laws of large numbers which have been previously studied in the context of Crump–Mode–Jagers (CMJ) processes by Nerman [Z. Wahrsch. Verw. Gebiete 57 (1981) 365–395] and in the context of fragmentation processes by Bertoin and Martinez [Adv. in Appl. Probab. 37 (2005) 553–570] and Harris, Knobloch and Kyprianou [Ann. Inst. H. Poincaré Probab. Statist. 46 (2010) 119–134]. The conclusions and methodology presented here appeal to a number of concepts coming from the theory of branching random walks and branching Brownian motion (cf. Harris [Proc. Roy. Soc. Edinburgh Sect. A 129 (1999) 503–517] and Biggins and Kyprianou [Electr. J. Probab. 10 (2005) 609–631]) showing their mathematical robustness even within the context of fragmentation theory.

Suppose we have three independent copies of a regular diffusion on [0, 1] with absorbing boundaries. Of these diffusions, either at least two are absorbed at the upper boundary or at least two at the lower boundary. In this way, they determine a majority decision between 0 and 1. We show that the strategy that always runs the diffusion whose value is currently between the other two reveals the majority decision whilst minimizing the total time spent running the processes.

We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here the extra stress tensor of the fluid is given by a polynomial of degree p − 1 of the rate of strain tensor, while the colored noise is considered as a random force. We investigate the existence and the uniqueness of weak solutions to this SPDE.

In this paper we prove the central limit theorem for Hotelling’s T² statistic when the dimension of the random vectors is proportional to the sample size.

We consider finite and infinite systems of particles on the real line and half-line evolving in continuous time. Hereby, the particles are driven by i.i.d. Lévy processes endowed with rank-dependent drift and diffusion coefficients. In the finite systems we show that the processes of gaps in the respective particle configurations possess unique invariant distributions and prove the convergence of the gap processes to the latter in the total variation distance, assuming a bound on the jumps of the Lévy processes. In the infinite case we show that the gap process of the particle system on the half-line is tight for appropriate initial conditions and same drift and diffusion coefficients for all particles. Applications of such processes include the modeling of capital distributions among the ranked participants in a financial market, the stability of certain stochastic queueing and storage networks and the study of the Sherrington–Kirkpatrick model of spin glasses.

This article deals with the numerical resolution of Markovian backward stochastic differential equations (BSDEs) with drivers of quadratic growth with respect to z and bounded terminal conditions. We first show some bound estimates on the process Z and we specify the Zhang’s path regularity theorem. Then we give a new time discretization scheme with a nonuniform time net for such BSDEs and we obtain an explicit convergence rate for this scheme.

In this paper, we consider multistopping problems for finite discrete time sequences X₁, …, X_n. m-stops are allowed and the aim is to maximize the expected value of the best of these m stops. The random variables are neither assumed to be independent not to be identically distributed. The basic assumption is convergence of a related imbedded point process to a continuous time Poisson process in the plane, which serves as a limiting model for the stopping problem. The optimal m-stopping curves for this limiting model are determined by differential equations of first order. A general approximation result is established which ensures convergence of the finite discrete time m-stopping problem to that in the limit model. This allows the construction of approximative solutions of the discrete time m-stopping problem. In detail, the case of i.i.d. sequences with discount and observation costs is discussed and explicit results are obtained.

Let S_n = 1/n X_nX_n* where X_n = {X_ij} is a p × n matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_{n}(\mathbf{t}_{1},\mathbf{t}_{2},\sigma)=\sqrt{p}({\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}(S_{n}+\sigma I)^{-1}{\mathbf{x}}_{n}(\mathbf{t}_{2})-{\mathbf{x}}_{n}(\mathbf{t}_{1})^{*}{\mathbf{x}}_{n}(\mathbf{t}_{2})m_{n}(\sigma))$ in which σ > 0 and m_n(σ) = ∫d F_yn(x)/(x + σ) where F_yn(x) is the Marčenko–Pastur law with parameter y_n = p/n; which converges to a positive constant as n → ∞, and x_n(t₁) and x_n(t₂) are unit vectors in ${\mathbb{C}}^{p}$, having indices t₁ and t₂, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence Y_n(t₁, t₂, σ) converges weakly to a (2m + 1)-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenmatrix of S_n is asymptotically close to that of a Haar-distributed unitary matrix.

This paper is devoted to studying constrained continuous-time Markov decision processes (MDPs) in the class of randomized policies depending on state histories. The transition rates may be unbounded, the reward and costs are admitted to be unbounded from above and from below, and the state and action spaces are Polish spaces. The optimality criterion to be maximized is the expected discounted rewards, and the constraints can be imposed on the expected discounted costs. First, we give conditions for the nonexplosion of underlying processes and the finiteness of the expected discounted rewards/costs. Second, using a technique of occupation measures, we prove that the constrained optimality of continuous-time MDPs can be transformed to an equivalent (optimality) problem over a class of probability measures. Based on the equivalent problem and a so-called w̄-weak convergence of probability measures developed in this paper, we show the existence of a constrained optimal policy. Third, by providing a linear programming formulation of the equivalent problem, we show the solvability of constrained optimal policies. Finally, we use two computable examples to illustrate our main results.