Showing papers by "Pavel Chigansky published in 2018"

Statistical Analysis of the Mixed Fractional Ornstein--Uhlenbeck Process

Abstract: This paper addresses the problem of estimating the drift parameter of the Ornstein--Uhlenbeck-type process driven by the sum of independent standard and fractional Brownian motions. With the help of some recent results on the canonical representation and spectral structure of mixed processes, the maximum likelihood estimator is shown to be consistent and asymptotically normal in the large-sample limit.

Exact spectral asymptotics of fractional processes

Abstract: Eigenproblems frequently arise in theory and applications of stochastic processes, but only a few have explicit solutions. Those which do, are usually solved by reduction to the generalized Sturm--Liouville theory for differential operators. This includes the Brownian motion and a whole class of processes, which derive from it by means of linear transformations. The more general eigenproblem for the {\em fractional} Brownian motion (f.B.m.) is not solvable in closed form, but the exact asymptotics of its eigenvalues and eigenfunctions can be obtained, using a method based on analytic properties of the Laplace transform. In this paper we consider two processes closely related to the f.B.m.: the fractional Ornstein--Uhlenbeck process and the integrated fractional Brownian motion. While both derive from the f.B.m. by simple linear transformations, the corresponding eigenproblems turn out to be much more complex and their asymptotic structure exhibits new effects.

What can be observed in real time PCR and when does it show

Abstract: Real time, or quantitative, PCR typically starts from a very low concentration of initial DNA strands. During iterations the numbers increase, first essentially by doubling, later predominantly in a linear way. Observation of the number of DNA molecules in the experiment becomes possible only when it is substantially larger than initial numbers, and then possibly affected by the randomness in individual replication. Can the initial copy number still be determined? This is a classical problem and, indeed, a concrete special case of the general problem of determining the number of ancestors, mutants or invaders, of a population observed only later. We approach it through a generalised version of the branching process model introduced in Jagers and Klebaner (J Theor Biol 224(3):299-304, 2003. doi: 10.1016/S0022-5193(03)00166-8 ), and based on Michaelis-Menten type enzyme kinetical considerations from Schnell and Mendoza (J Theor Biol 184(4):433-440, 1997). A crucial role is played by the Michaelis-Menten constant being large, as compared to initial copy numbers. In a strange way, determination of the initial number turns out to be completely possible if the initial rate v is one, i.e all DNA strands replicate, but only partly so when [Formula: see text], and thus the initial rate or probability of succesful replication is lower than one. Then, the starting molecule number becomes hidden behind a "veil of uncertainty". This is a special case, of a hitherto unobserved general phenomenon in population growth processes, which will be adressed elsewhere.

Persistence of small noise and random initial conditions

Abstract: The effect of small noise in a smooth dynamical system is negligible on any finite time interval; in this paper we study situations where the effect persists on intervals increasing to ∞. Such an asymptotic regime occurs when the system starts from an initial condition that is sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to a solution of the unperturbed system started from a certain random initial condition. In this paper we consider the case of one-dimensional diffusions on the positive half-line; this case often arises as a scaling limit in population dynamics.

Persistence of Small Noise and Random initial conditions.

Abstract: The effect of small noise in a smooth dynamical system is negligible on any finite time interval. Here we study situations when it persists on intervals increasing to infinity. Such asymptotic regime occurs when the system starts from initial condition, sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to solution of the unperturbed system, started from a certain {\em random} initial condition. In this paper we consider the case of one dimensional diffusions on the positive half line, which often arise as scaling limits in population dynamics.

Persistence of Small Noise and Random initial conditions in the Wright-Fisher model

Abstract: The effect of small noise in a smooth dynamical system is negligible on any finite time interval. Here we study situations when it persists on intervals increasing to infinity. Such phenomenon occurs when the system starts from initial condition, sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to solution of the unperturbed system, started from a certain {\em random} initial condition. In this paper we consider the Wright-Fisher diffusion from population dynamics. We show that when the selection parameter is positive and the diffusion coefficient is small, the solution, which starts near zero, follows the logistic equation with a random initial condition that comes from an approximating Feller branching diffusion. We also consider the case of discrete time dynamical systems for which random initial condition involves the limit of iterates of the deterministic system and weighted random perturbations.

Populations with interaction and environmental dependence: from few, (almost) independent, members into deterministic evolution of high densities

Abstract: Many populations, e.g. of cells, bacteria, viruses, or replicating DNA molecules, start small, from a few individuals, and grow large into a noticeable fraction of the environmental carrying capacity $K$. Typically, the elements of the initiating, sparse set will not be hampering each other and their number will grow from $Z_0=z_0$ in a branching process or Malthusian like, roughly exponential fashion, $Z_t \sim a^tW$, where $Z_t$ is the size at discrete time $t\to\infty$, $a>1$ is the offspring mean per individual (at the low starting density of elements, and large $K$), and $W$ a sum of $z_0$ i.i.d. random variables. It will, thus, become detectable (i.e. of the same order as $K$) only after around $\log K$ generations, when its density $X_t:=Z_t/K$ will tend to be strictly positive. Typically, this entity will be random, even if the very beginning was not at all stochastic, as indicated by lower case $z_0$, due to variations during the early development. However, from that time onwards, law of large numbers effects will render the process deterministic, though initiated by the random density at time log $K$, expressed through the variable $W$. Thus, $W$ acts both as a random veil concealing the start and a stochastic initial value for later, deterministic population density development. We make such arguments precise, studying general density and also system-size dependent, processes, as $K\to\infty$. As an intrinsic size parameter, $K$ may also be chosen to be the time unit. The fundamental ideas are to couple the initial system to a branching process and to show that late densities develop very much like iterates of a conditional expectation operator.

Thresholds of mixed fractional Brownian motion

Abstract: This paper provides yet another look at the mixed fractional Brownian motion (fBm), this time, from the spectral perspective. We derive exact asymptotic formula for the eigenvalues of its covariance operator, which separates contributions of the standard and fractional parts. This, in turn, allows to obtain exact asymptotics of the small $L^2$-ball probabilities for the mixed fBm, which was previously known only on the logarithmic level. The obtained expressions show an interesting stratification of scales, which occurs at certain values, {\em thresholds}, of the Hurst parameter of the fractional component. Some of these values have been previously encountered in other problems involving such mixtures.