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Alexander Sokol

Publications and source records attributed to Alexander Sokol.

17 recordsLinked to original sources
arXiv2015

Revisiting the forward equations for inhomogeneous semi-Markov processes

In this paper, we consider a class of inhomogeneous semi-Markov processes directly based on intensity processes for marked point processes. We show that this class satisfies the semi-Markov properties defined elsewhere in the literature. We use the marked point process setting to derive strong upper bounds on various probabilities for semi-Markov processes. Using these bounds, we rigorously prove for the case of countably infinite state space that the transition intensities are right-derivatives of the transition probabilities, and we prove for the case of finite state space that the transition probabilities satisfy the forward equations, requiring only right-continuity of the transition intensities in the time and duration arguments and a boundedness condition. We also show relationships between several classes of semi-Markov processes considered in the literature, and we prove an integral representation for the left derivatives of the transition probabilities in the duration parameter.

math.PR↗
arXiv2015

A generic model for spouse's pensions with a view towards the calculation of liabilities

We introduce a generic model for spouse's pensions. The generic model allows for the modeling of various types of spouse's pensions with payments commencing at the death of the insured. We derive abstract formulas for cashflows and liabilities corresponding to common types of spouse's pensions. We show how the standard formulas from the Danish G82 concession can be obtained as a special case of our generic model. We also derive expressions for liabilities for spouse's pensions in models more advanced than found in the G82 concession. The generic nature of our model and results furthermore enable the calculation of cashflows and liabilities using simple estimates of marital behaviour among a population.

q-fin.RM↗
arXiv2014

Exponential martingales and changes of measure for counting processes

We give sufficient criteria for the Doléans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are adapted particularly to the case of counting processes and are sufficiently weak to be useful and verifiable, as we illustrate by several examples. In particular, the criteria allow for the construction of for example nonexplosive Hawkes processes as well as counting processes with stochastic intensities depending on diffusion processes.

math.PR↗
arXiv2014

Degrees of freedom for nonlinear least squares estimation

We give a general result on the effective degrees of freedom for nonlinear least squares estimation, which relates the degrees of freedom to the divergence of the estimator. We show that in a general framework, the divergence of the least squares estimator is a well defined but potentially negatively biased estimate of the degrees of freedom, and we give an exact representation of the bias. This implies that if we use the divergence as a plug-in estimate of the degrees of freedom in Stein's unbiased risk estimate (SURE), we generally underestimate the true risk. Our result applies, for instance, to model searching problems, yielding a finite sample characterization of how much the search contributes to the degrees of freedom. Motivated by the problem of fitting ODE models in systems biology, the general results are illustrated by the estimation of systems of linear ODEs. In this example the divergence turns out to be a useful estimate of degrees of freedom for $\ell_1$-constrained models.

math.ST↗
arXiv2014

Causal interpretation of stochastic differential equations

We give a causal interpretation of stochastic differential equations (SDEs) by defining the postintervention SDE resulting from an intervention in an SDE. We show that under Lipschitz conditions, the solution to the postintervention SDE is equal to a uniform limit in probability of postintervention structural equation models based on the Euler scheme of the original SDE, thus relating our definition to mainstream causal concepts. We prove that when the driving noise in the SDE is a Lévy process, the postintervention distribution is identifiable from the generator of the SDE.

math.PR↗
arXiv2014

Quantifying identifiability in independent component analysis

We are interested in consistent estimation of the mixing matrix in the ICA model, when the error distribution is close to (but different from) Gaussian. In particular, we consider $n$ independent samples from the ICA model $X = Aε$, where we assume that the coordinates of $ε$ are independent and identically distributed according to a contaminated Gaussian distribution, and the amount of contamination is allowed to depend on $n$. We then investigate how the ability to consistently estimate the mixing matrix depends on the amount of contamination. Our results suggest that in an asymptotic sense, if the amount of contamination decreases at rate $1/\sqrt{n}$ or faster, then the mixing matrix is only identifiable up to transpose products. These results also have implications for causal inference from linear structural equation models with near-Gaussian additive noise.

math.ST↗
arXiv2013

Intervention in Ornstein-Uhlenbeck SDEs

We introduce a notion of intervention for stochastic differential equations and a corresponding causal interpretation. For the case of the Ornstein-Uhlenbeck SDE, we show that the SDE resulting from a simple type of intervention again is an Ornstein-Uhlenbeck SDE. We discuss criteria for the existence of a stationary distribution for the solution to the intervened SDE. We illustrate the effect of interventions by calculating the mean and variance in the stationary distribution of an intervened process in a particularly simple case.

math.PR↗
arXiv2012

An extended Novikov-type criterion for local martingales with jumps

For local martingales with nonnegative jumps, we prove a sufficient criterion for the corresponding exponential martingale to be a true martingale. The criterion is in terms of exponential moments of a convex combination of the optional and predictable quadratic variation. The result extends earlier known criteria.

math.PR↗
arXiv2012

Optimal Novikov-type criteria for local martingales with jumps

We consider local martingales $M$ with jumps larger than $a$ for some $a$ larger than or equal to -1, and prove Novikov-type criteria for the corresponding exponential local martingale to be a uniformly integrable martingale. We obtain criteria using both the quadratic variation and the predictable quadratic variation. We prove optimality of the coefficients in the criteria. As a corollary, we obtain a verbatim extension of the classical Novikov criterion for continuous local martingales to the case of local martingales with nonnegative jumps.

math.PR↗
arXiv2012

Proving existence results in martingale theory using a subsequence principle

New proofs are given of the existence of the compensator (or dual predictable projection) of a locally integrable cádlág adapted process of finite variation and of the existence of the quadratic variation process for a cádlág local martingale. Both proofs apply a functional analytic subsequence principle. After presenting the proofs, we discuss their application in giving a simplified account of the construction of the stochastic integral of a locally bounded predictable process with respect to a semimartingale.

math.PR↗
arXiv1996

Isotropic Spin Wave Theory of Short-Range Magnetic Order

We present an isotropic spin wave (ISW) theory of short-range order in Heisenberg magnets, and apply it to square lattice S=1/2 and S=1 antiferromagnets. Our theory has three identical (isotropic) spin wave modes, whereas the conventional spin wave theory has two transverse and one longitudinal mode. We calculate temperature dependences of various thermodynamic observables analytically and find good (several per cent) agreement with independently obtained numerical results in a broad temperature range.

cond-mat↗
arXiv1995

Crossover Phase Diagram of 2D Heisenberg Ferro- and Antiferromagnets

We propose a single crossover phase diagram applicable to 2D collinear Heisenberg antiferromagnets (AFMs) and ferromagnets (FMs), and show that the scaling regimes of AFMs and FMs are in one-to-one correspondence. The phase diagram is split into classical and quantum regions. Our two key results are: (i) in the classical region, the AFMs and FMs exhibit nearly identical behavior near their respective ordering wavevectors, which we observe for S=1 and higher using series expansions; and (ii) in the quantum region, quantum critical (QC) regime is present not only for the AFMs, but for FMs as well.

cond-mat↗
arXiv1995

Magnetic Behavior of the Cuprate Superconductors

I review recent work on magnetic dynamics of the high temperature superconductors using a model that combines two weakly interacting species of low-energy excitations: the antiferromagnetic spin waves which carry spin-1 and no charge, and Fermi-liquid-like quasiparticles which carry spin-1/2 and charge e. The model allows conversion of spin waves into electron-hole pairs; however, the low-energy spin waves are not collective modes of the quasiparticles near the Fermi surface, but rather are a separate branch of the low-energy spectrum. With certain experimentally justified assumptions, this theory is remarkably universal: the dependence on the detailed microscopic Hamiltonian and on doping can be absorbed into several experimentally measurable parameters. The z=1 theory of the insulators and z=2 theory of the overdoped materials, are both reproduced as limiting cases of the theory described here, which predicts that the underdoped materials remain in z=1 universality class at sufficiently high temperature. This theory provides a framework for understanding both the experimental results and microscopic calculations, and in particular yields a possible explanation of the spin gap phenomenon. I also discuss some of the important unresolved issues.

cond-mat↗
arXiv1994

Crossover and scaling in a nearly antiferromagnetic Fermi liquid in two dimensions

We consider two-dimensional Fermi liquids in the vicinity of a quantum transition to a phase with commensurate, antiferromagnetic long-range order. Depending upon the Fermi surface topology, mean-field spin-density-wave theory predicts two different types of such transitions, with mean-field dynamic critical exponents $z=1$ (when the Fermi surface does not cross the magnetic zone boundary, type $A$) and $z=2$ (when the Fermi surface crosses the magnetic zone boundary, type $B$). The type $A$ system only displays $z=1$ behavior at all energies and its scaling properties are similar (though not identical) to those of an insulating Heisenberg antiferromagnet. Under suitable conditions precisely stated in this paper, the type $B$ system displays a crossover from relaxational behavior at low energies to type $A$ behavior at high energies. A scaling hypothesis is proposed to describe this crossover: we postulate a universal scaling function which determines the entire, temperature-, wavevector-, and frequency-dependent, dynamic, staggered spin susceptibility in terms of 4 measurable, $T=0$, parameters (determining the distance, energy, and order parameter scales, plus one crossover parameter). The scaling function contains the full scaling behavior in all regimes for both type $A$ and $B$ systems. The crossover behavior of the uniform susceptibility and the specific heat is somewhat more complicated and is also discussed. Explicit computation of the crossover functions is carried out in a large $N$ expansion on a mean-field model. Some new results for the critical properties on the ordered side of the transition are also obtained in a spin-density wave formalism. The possible relevance of our results to the doped cuprate compounds is briefly discussed.

cond-mat↗
arXiv1993

Quantum Critical Scaling in a Moderately Doped Antiferromagnet

Using high temperature expansions for the equal time correlator $S(q)$ and static susceptibility $χ(q)$ for the t-J model, we present evidence for quantum critical (QC), $z\!=\!1$, behavior at intermediate temperatures in a broad range of $t/J$ ratio, doping, and temperatures. We find that the dynamical susceptibility is very close to the universal scaling function computable for the asymptotic QC regime, and that the dominant energy scale is temperature. Our results are in excellent agreement with measurements of the spin-echo decay rate, $1/T_{\rm 2G}$, in La$_2$CuO$_4$, and provide qualitative understanding of both $1/T_1$ and $1/T_{\rm 2G}$ nuclear relaxation rates in doped cuprates.

cond-mat↗