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We develop exact Markov chain Monte Carlo methods for discretely-sampled, directly and indirectly observed diffusions. The qualification "exact" refers to the fact that the invariant and limiting distribution of the Markov chains is the…
In this paper we consider parameter estimation for discretely observed diffusion processes. In particular, we focus on data that are observed at low frequency and methodology that can estimate parameters with uncertainty quantification.…
A general model of catalytic branching process (CBP) with any finite number of catalysis centers in a discrete space is studied. More exactly, it is assumed that particles move in this space according to a specified Markov chain and they…
Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems involving multiple time-scales, by a…
Accurately assessing financial risk requires capturing both individual asset volatility and the complex, asymmetric dependence structures that emerge during extreme market events. While modern diffusion-based models have advanced…
Diffusion theory is a central tool of modern population genetics, yielding simple expressions for fixation probabilities and other quantities that are not easily derived from the underlying Wright-Fisher model. Unfortunately, the textbook…
Consider a single server queue with renewal arrivals and i.i.d. service times in which the server operates under a processor sharing service discipline. To describe the evolution of this system, we use a measure valued process that keeps…
In this paper, a modification of the conventional approximations to the quasi-maximum likelihood method is introduced for the parameter estimation of diffusion processes from discrete observations. This is based on a convergent…
This paper considers the relative frequencies of distinct types of individuals in multitype branching processes. We prove that the frequencies are asymptotically multivariate normal when the initial number of ancestors is large and the time…
We study the convergence of the new family of mimetic finite difference schemes for linear diffusion problems recently proposed in [38]. In contrast to the conventional approach, the diffusion coefficient enters both the primary mimetic…
The paper studies a class of critical Markov branching processes with infinite variance of the offspring distribution. The processes admit also an immigration component at the jump-points of a non-homogeneous Poisson process, assuming that…
The paper is concerned with the approximation of the deterministic the mean field type control system by a mean field Markov chain. It turns out that the dynamics of the distribution in the approximating system is described by a system of…
We develop diffusion approximations for parallel-queueing systems with the randomized longest-queue-first scheduling algorithm by establishing new mean-field limit theorems as the number of buffers $n\to\infty$. We achieve this by allowing…
This work provides a brief introduction to continuous-state branching processes (CB-processes) and continuous-state branching processes with immigration (CBI-processes) accessible to graduate students with reasonable background in…
The long time behavior of an absorbed Markov process is well described by the limiting distribution of the process conditioned to not be killed when it is observed. Our aim is to give an approximation's method of this limit, when the…
Density dependent families of Markov chains, such as the stochastic models of mass-action chemical kinetics, converge for large values of the indexing parameter $N$ to deterministic systems of differential equations (Kurtz, 1970). Moreover…
Single-cell lineage tracking strategies enabled by recent experimental technologies have produced significant insights into cell fate decisions, but lack the quantitative framework necessary for rigorous statistical analysis of mechanistic…
In this note we prove the strong Feller property of a strong Markov quasi diffusion process corresponding to an elliptic operator with merely bounded measurable coefficients. We also prove H\"older continuity of harmonic functions…
We study a catalytic branching process (CBP) with any finite set of catalysts. This model describes a system of particles where the movement is governed by a Markov chain with arbitrary finite or countable state space and the branching may…
A general continuous-state branching processes in random environment (CBRE-process) is defined as the strong solution of a stochastic integral equation. The environment is determined by a L\'evy process with no jump less than $-1$. We give…