arXiv subjects
Publications and source records attributed to F. Klebaner.
To model amplification Polymerase Chain Reaction (PCR) techniques targeting DNA sequences of several types, we introduce a multitype PCR branching process as a generalized version of the Michaelis-Menten-based branching process model introduced in Jagers-Klebaner, 2003. We establish two limit theorems extending the results of Chigansky-Jagers-Klebaner, 2018 to the multitype case.
We consider stochastic dynamics of a population which starts from a small colony on a habitat with large but limited carrying capacity. A common heuristics suggests that such population grows initially as a Galton-Watson branching process and then its size follows an almost deterministic path until reaching its maximum, sustainable by the habitat. In this paper we put forward an alternative and, in fact, more accurate approximation which suggests that the population size behaves as a special nonlinear transformation of the Galton Watson process from the very beginning.
How long does it take for an initially advantageous mutant to establish itself in a resident population, and what does the population composition look like then? We approach these questions in the framework of the so called Bare Bones evolution model Klebaner et al (2011) that provides a simplified approach to the adaptive population dynamics of binary splitting cells. As the mutant population grows, cell division becomes less probable, and it may in fact turn less likely than that of residents. Our analysis rests on the assumption of the process starting from resident population, with sizes proportional to a large carrying capacity $K$. Actually, we assume carrying capacities to be $a_1K$ and $a_2K$ for the resident and the mutant populations, respectively, and study the dynamics for $K\to\infty$. We find conditions for the mutant to be successful in establishing itself alongside the resident. The time it takes turns out to be proportional to $\log K$. We introduce the time of establishment through the asymptotic behavior of the stochastic nonlinear dynamics describing the evolution, and show that it is indeed $\log K/\log ρ$, where $ρ>1$ is twice the probability of successful division of the mutant at its appearance. Looking at the composition of the population, at times $\log K/\log ρ+n, n \in \mathbb{Z}_+$, we find that the densities (i.e. sizes relative to carrying capacities) of both populations follow closely the corresponding two dimensional nonlinear deterministic dynamics that starts at {\it a random point}. We characterise this random initial condition in terms of the scaling limit of the corresponding dynamics, and the limit of the properly scaled initial binary splitting process of the mutant. The deterministic approximation with random initial condition is in fact valid asymptotically at all times $\log K/\log ρ+n$ with $n\in \mathbb{Z}$.
Let $\mathfrak{z}$ be a stochastic exponential, i.e., $\mathfrak{z}_t=1+\int_0^t\mathfrak{z}_{s-}dM_s$, of a local martingale $M$ with jumps $\triangle M_t>-1$. Then $\mathfrak{z}$ is a nonnegative local martingale with $\E\mathfrak{z}_t\le 1$. If $\E\mathfrak{z}_T= 1$, then $\mathfrak{z}$ is a martingale on the time interval $[0,T]$. Martingale property plays an important role in many applications. It is therefore of interest to give natural and easy verifiable conditions for the martingale property. In this paper, the property $\E\mathfrak{z}_{_T}=1$ is verified with the so-called linear growth conditions involved in the definition of parameters of $M$, proposed by Girsanov \cite{Girs}. These conditions generalize the Beneŝ idea, \cite{Benes}, and avoid the technology of piece-wise approximation. These conditions are applicable even if Novikov, \cite{Novikov}, and Kazamaki, \cite{Kaz}, conditions fail. They are effective for Markov processes that explode, Markov processes with jumps and also non Markov processes. Our approach is different to recently published papers \cite{CFY} and \cite{MiUr}.
The CEV model is given by the stochastic differential equation $X_t=X_0+\int_0^tμX_sds+\int_0^tσ(X^+_s)^pdW_s$, $\frac{1}{2}\le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0\le t\le T$, in the Skorokhod metric. We give a new approximation by continuous processes which allows to relax some technical conditions in the proof of weak convergence in \cite{HZa} done in terms of discrete time martingale problem. We calculate ruin probabilities as an example of such approximation. We establish that the ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the point zero is a discontinuity point of the limiting distribution. To establish such convergence we use the Levy metric, and also confirm the convergence numerically. Although the result is given for the specific model, our method works in a more general case of non-Lipschitz diffusion with absorbtion.
We give asymptotic analysis for probability of absorbtion $\mathsf{P}(τ_0\le T)$ on the interval $[0,T]$, where $ τ_0=\inf\{t:X_t=0\}$ and $X_t$ is a nonnegative diffusion process relative to Brownian motion $B_t$, dX_t&=μX_tdt+σX^γ_tdB_t. X_0&=K>0 Diffusion parameter $σx^γ$, $γ\in [{1/2},1)$ is not Lipschitz continuous and assures $\mathsf{P}(τ_0>T)>0$. Our main result: $$ \lim\limits_{K\to\infty} \frac{1}{K^{2(1-γ)}}\log\mathsf{P}(τ_{0}\le T) =-\frac{1}{2\E M^2_T}, $$ where $ M_T=\int_0^Tσ(1-γ)e^{-(1-γ)μs}dB_s $. Moreover we describe the most likely path to absorbtion of the normed process $\frac{X_t}{K}$ for $K\to\infty$.
We consider a continuous time version of Cramer's theorem with nonnegative summands $ S_t=\frac{1}{t}\sum_{i:τ_i\le t}ξ_i, t \to\infty, $ where $(τ_i,ξ_i)_{i\ge 1}$ is a sequence of random variables such that $tS_t$ is a random process with independent increments.
The Large Deviation Principle is established for stochastic models defined by past-dependent non linear recursions with small noise. In the Markov case we use the result to obtain an explicit expression for the asymptotics of exit time.
We give an explicit formula for the most likely path to extinction for the Galton-Watson processes with large initial population. We establish this result with the help of the large deviation principle (LDP) which also recovers the asymptotics of extinction probability. Due to the nonnegativity of the Galton-Watson processes, the proof of LDP verification at the point of extinction uses a nonstandard argument of independent interest.
We propose an adaptive algorithm for tracking of historical volatility. The algorithm is built under the assumption that the historical volatility function belongs to the Stone-Ibragimov-Khasminskii class of $k$ times differentiable functions with bounded highest derivative and its subclass of functions satisfying a differential inequalities. We construct an estimator of the Kalman filter type and show optimality of the estimator's convergence rate to zero as sample size $n\to\infty$. This estimator is in the framework of GARCH design, but a tuning procedure of its parameters is faster than with traditional GARCH techniques.