Discrete-time Setting Research Articles

Non-Parametric and Adaptive Modelling of Dynamic Periodicity and Trend with Heteroscedastic and Dependent Errors

SummaryPeriodicity and trend are features describing an observed sequence, and extracting these features is an important issue in many scientific fields. However, it is not an easy task for existing methods to analyse simultaneously the trend and dynamics of the periodicity such as time varying frequency and amplitude, and the adaptivity of the analysis to such dynamics and robustness to heteroscedastic dependent errors are not guaranteed. These tasks become even more challenging when there are multiple periodic components. We propose a non-parametric model to describe the dynamics of multicomponent periodicity and investigate the recently developed synchro-squeezing transform in extracting these features in the presence of a trend and heteroscedastic dependent errors. The identifiability problem of the non-parametric periodicity model is studied, and the adaptivity and robustness properties of the synchro-squeezing transform are theoretically justified in both discrete and continuous time settings. Consequently we have a new technique for decoupling the trend, periodicity and heteroscedastic, dependent error process in a general non-parametric set-up. Results of a series of simulations are provided, and the incidence time series of varicella and herpes zoster in Taiwan and respiratory signals observed from a sleep study are analysed.

Robust estimation with dynamic integral quadratic constraints: the discrete‐time case

The authors consider the problem of robust output estimation in the discrete-time setting. The uncertainty in the system is characterised by integral quadratic constraints with dynamic multipliers. The main technical contribution of this study is to derive two different synthesis conditions in terms of linear matrix inequalities (LMI) by two different approaches. The LMI condition derived by the first approach allows one to find the robust estimator and the performance certificate simultaneously, whereas the condition by the second approach has a smaller size but requires an additional step for recovering the estimator. The synthesis conditions are validated by numerical examples. The results verify the equivalence of the two approaches and show that the two-step approach is in fact computationally favourable when the system has large number of state variables.

Stochastic modelling of disability insurance in a multi-period framework

We propose a stochastic semi-Markovian framework for disability modelling in a multi-period discrete-time setting. The logistic transforms of disability inception and recovery probabilities are modelled by means of stochastic risk factors and basis functions, using counting processes and generalized linear models. The model for disability inception also takes IBNR claims into consideration. We fit various versions of the models into Swedish disability claims data.

Discrete-Time Approximation of Functionals in Models of Ornstein–Uhlenbeck Type, with Applications to Finance

The paper provides benchmark approximation techniques for handling in models in continuous-time functional relationships in variables that are sampled discretely over time. The methods are demonstrated for value-functional-type of expectations in models of Ornstein–Uhlenbeck type. Based on Laguerre reduction series, a three-step program for these functionals is shown to result in both a continuous and a discrete time setting. The program is illustrated in the options case and in models based on GIG-distributions, yielding novel series representations for calibration when variance is discretely-sampled in particular. By numerical examples it is shown how the series enable computation accuracies of some 3 decimal places, for example, with just a single digit number of terms; for this the paper considers discretely-sampled situations with dimensions of up to 4 digits, and even in these dimensions significant discrepancies with the continuously-sampled values are found to persist.

Alias-Free Digital Click Modulator

An alias-free, discrete-time click modulator is developed in this paper. Previous approaches rely on translating the continuous-time click modulator to a discrete-time setting, although a key component in the click modulator - the analytic exponential modulator-cannot be exactly transformed to the discrete domain. Every discrete-time version of the click modulator reported in the literature is prone to aliasing effects that are ameliorated, but not eliminated, using interpolation techniques. The precision of the switching times of the output square wave is critical to click modulation because uncertainties in their determination adversely affect the performance and SNR of the modulator. A novel method that uses frequency domain information to compute these switching times without error is also presented. This two techniques are used to develop an off-line discrete-time click modulator that achieves a SNR larger than 180 dB for both multitonal and bandpass signals.

Towards [formula omitted]-stability of discrete-time reset control systems via dissipativity theory

This paper proposes conditions on input–output stability of discrete-time reset systems by using some key dissipativity properties. In the continuous-time setting, dissipativity of the base linear system is preserved under reset actions if the storage function is decreasing at reset times. Indeed, when the reset system is a full reset system, the dissipativity of the base linear system ensures the dissipativity of the reset system. However, in the discrete-time setting, this condition on the storage function is not enough to ensure the dissipativity of the base linear system. We define some dissipativity properties of discrete-time reset systems and give an appropriate definition of the reset system in order to preserve the (Q,S,R)−dissipativity of the base linear system under reset actions. As a result, ℓ2-stability of feedback interconnected dissipative control reset systems is obtained.

Pricing and Hedging Contingent Claims with Liquidity Costs and Market Impact

We study the influence of taking liquidity costs and market impact into account when hedging a contingent claim, first in the discrete time setting, then in continuous time. In the latter case and in a complete market, we derive a fully non-linear pricing partial differential equation, and characterizes its parabolic nature according to the value of a numerical parameter naturally interpreted as a relaxation coefficient for market impact. We then investigate the more challenging case of stochastic volatility models, and prove the parabolicity of the pricing equation in a particular case.

Convex Synthesis of Multivariable Static Discrete-time Anti-windup via the Jury-Lee Criterion

Due to its ease of application, the circle criterion has been widely used to guarantee the stability of many anti-windup schemes. While the Popov criterion gives less conservative results, it has been conjectured in the literature that it cannot be used for convex anti-windup synthesis. This paper shows that the conjecture does not necessarily apply in the discrete-time setting. We show how the search for optimal parameters corresponding to the Jury-Lee criterion (a discrete counterpart of the Popov criterion) can be formulated as a convex search via a linear matrix inequality (LMI). The result is then extended to two existing multivariable static anti-windup schemes with stable open-loop plants. Two numerical examples of multivariable anti-windup controller synthesis are provided, and it is shown that in both cases the synthesis using the Jury-Lee criterion can allow better performance than existing methods which use the circle criterion alone.

Variability and singularity arising from poor compliance in a pharmacokinetic model I: the multi-IV case

We consider a simple multi-IV model for drug concentration in the case of poor patient compliance. The model is a stochastic one, and is thus able to take into account an irregular drug intake schedule. Under some assumptions, we study features of the drug concentration relevant for practical purposes such as its variability or the regularity of its cumulative probability distribution. We consider five variants: random instants for drug intake with either deterministic or random doses, both in continuous and discrete-time settings, plus a model with stochastically varying elimination rate. Our computations make it possible to assess in a precise way the effect of various significant parameters such as the mean rate of intake, the elimination rate, and the mean dose. They also quantify how much poor compliance will affect the regimen: in that view, we provide precise comparisons with the variability of concentration in the cases of (a) a fully compliant patient and (b) a population of fully compliant patients with lognormally distributed elimination rates. The time discretized version of our models reveal unexpected links with measures known as infinite Bernoulli convolutions. Our findings help in understanding the consequences of poor compliance, and may have practical outcomes in terms of drug dosing and scheduling.

Super-Replication of Financial Derivatives Via Convex Programming

We give a method based on convex programming to calculate the optimal super-replicating and sub-replicating prices and corresponding hedging portfolios of a financial derivative in terms of other financial derivatives in a discrete-time setting. Our method produces a model that matches the super-replicating (or sub-replicating) price within an arbitrary precision and is consistent with the other financial derivatives prices. Applications include robust replication in terms of call prices with various strikes and maturities of forward start options, volatility and variance swaps and derivatives, cliquets calls, barrier options, lookback and Asian options. Numerical examples show that, in some cases, the best super-replicating and/or sub-replicating prices are within 10% of the price obtained by a standard model, but considerably differ from it in other cases. Our method can incorporate bid-ask spreads, interest rates and dividends and various limitations to the diffusion model.

DISCRETE-TIME WILSON FRAMES WITH GENERAL LATTICES

In this paper, we focus on the construction of Wilson frames and their dual frames for general lattices of volume [Formula: see text] (K even) in the discrete-time setting. We obtain a necessary and sufficient condition for two Bessel sequences having Wilson structure to be dual frames for l2(ℤ). When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for l2(ℤ), show that a Wilson frame for l2(ℤ) can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying Gabor frame.

A Unified Approach for the Identification of SISO/MIMO Wiener and Hammerstein Systems

Hammerstein and Wiener models are nonlinear representations of systems composed by the coupling of a static nonlinearity N and a linear system L in the form N-L and L-N respectively. These models can represent real processes which made them popular in the last decades. The problem of identifying the static nonlinearity and linear system is not a trivial task, and has attracted a lot of research interest. It has been studied in the available literature either for Hammerstein or Wiener systems, and either in a discrete-time or continuous-time setting. The objective of this paper is to present a unified framework for the identification of these systems that is valid for SISO and MIMO systems, discrete and continuous-time setting, and with the only a priori knowledge that the system is either Wiener or Hammerstein.

Steady-State Analysis and Regulation of Discrete-Time Nonlinear Systems

This paper presents results on steady-state analysis and regulation for nonlinear discrete-time systems subject to time-varying excitations. In the analysis part of the paper, for convergent nonlinear systems (which have uniquely defined steady-state responses to excitations) we provide a complete characterization of the steady-state responses to excitations generated by an exosystem. As a corollary, we obtain a nonlinear frequency response function which extends the well-known FRF defined for linear systems to the class of nonlinear convergent systems. In the control part of the paper, we present a characterization of all controllers solving the global output regulation problem. All these results are obtained using the machinery of convergent systems, extended to the discrete-time setting. For piecewise affine systems, general results are supplied with a constructive design procedure.

Time-changed geometric fractional Brownian motion and option pricing with transaction costs

This paper deals with the problem of discrete time option pricing by a fractional subdiffusive Black–Scholes model. The price of the underlying stock follows a time-changed geometric fractional Brownian motion. By a mean self-financing delta-hedging argument, the pricing formula for the European call option in discrete time setting is obtained.

Reflected backward stochastic differential equations and a class of non-linear dynamic pricing rule

In this paper, we provide a characterization of solutions of specific reflected backward stochastic dfferential Equations, whose generator g has quadratic growth w.r.t its variable z: this is achieved by introducing an extended notion of g-Snell envelope. In the case when g is convex (w.r.t z), the solution is related to the notion of convex risk measures and, more specifically here, to the robust representation of one class of dynamic monetary concave functionals already introduced in a discrete-time setting. In the final section, further applications to finance are provided.

Elasticity approach to asset allocation in discrete time

In this paper, we apply the elasticity approach to optimal asset allocation problems in discrete-time setting. In particu- lar, firstly, for a portfolio optimization problem, which targets to maximize the expected utility of the terminal wealth of a portfo- lio of an option, the underlying stock, and the risk-free bond, the elasticity approach can decompose this problem into a reduced optimization problem, consisting of only the stock and bond, and a pure delta neutral hedging problem. This decomposition provides a discrete-time version of the optimal alternative to the delta hedge, which was initially proposed in continuous time. Moreover, the general principle given by the pure delta neutral strategy is analyzed in our setting. Secondly, the same approach is applied to an optimal investment problem with defaultable securities, and show that this problem is essentially the same as the above mentioned reduced optimization problem. This work can be regarded as an extension of the elasticity approach in Kraft (Mathematical Methods of Operations Research 58(1) (2003), 159-182) to discrete-time models, and it shows that this approach can largely deduce the asset allocation problems in complete market.

Quasi maximum likelihood estimation for strongly mixing state space models and multivariate Lévy-driven CARMA processes

We consider quasi maximum likelihood (QML) estimation for general non-Gaussian discrete-ime linear state space models and equidistantly observed multivariate L\'evy-driven continuoustime autoregressive moving average (MCARMA) processes. In the discrete-time setting, we prove strong consistency and asymptotic normality of the QML estimator under standard moment assumptions and a strong-mixing condition on the output process of the state space model. In the second part of the paper, we investigate probabilistic and analytical properties of equidistantly sampled continuous-time state space models and apply our results from the discrete-time setting to derive the asymptotic properties of the QML estimator of discretely recorded MCARMA processes. Under natural identifiability conditions, the estimators are again consistent and asymptotically normally distributed for any sampling frequency. We also demonstrate the practical applicability of our method through a simulation study and a data example from econometrics.

On geometric records: rate of appearance and magnitude

We study the long-term behavior of geometric records from a sequence{Xn}n ≥ 1 of independent, nonnegative, random observations, with common continuous distribution functionF. Given aparameter k > 1, thenth observationXn is a geometricrecord if Xn > kmax{X1,..., Xn − 1}, that is, if Xn is k times greater than all preceding observations. This concept was introduced by Eliazar in 2005 (Physica A 348 181), where the question of waiting times was addressed. We consider the numberNn of geometricrecords among X1,..., Xn,and show that Nn increases to a finite random limit , for very light-tailed F. Formedium and heavy-tailed F, we prove that Nn diverges to infinity, establish its growth rate and give conditions for asymptotic normality.We also analyze the magnitude of geometric records, pointing out an unexpectedrelationship with models of paralyzable counters in particle physics. Our results arepresented in a discrete-time setting but we show how they can be translated intocontinuous time. Examples of applications to common families of distributions, such asFréchet systems, are also provided.

Pricing European option with transaction costs under the fractional long memory stochastic volatility model

This paper deals with the problem of discrete time option pricing using the fractional long memory stochastic volatility model with transaction costs. Through the ‘anchoring and adjustment’ argument in a discrete time setting, a European call option pricing formula is obtained.

Isotonic estimation of survival under a misattribution of cause of death.

Several authors have indicated that incorrectly classified cause of death for prostate cancer survivors may have played a role in the observed recent peak and decline of prostate cancer mortality. Motivated by the suggestion we studied a competing risks model where other cause of death may be misattributed as a death of interest. We first consider a naïve approach using unconstrained nonparametric maximum likelihood estimation (NPMLE), and then present the constrained NPMLE where the survival function is forced to be monotonic. Surprising observations were made as we studied their small-sample and asymptotic properties in continuous and discrete situations. Contrary to the common belief that the non-monotonicity of a survival function NPMLE is a small-sample problem, the constrained NPMLE is asymptotically biased in the continuous setting. Other isotonic approaches, the supremum (SUP) method and the Pooled-Adjacent-Violators (PAV) algorithm, and the EM algorithm are also considered. We found that the EM algorithm is equivalent to the constrained NPMLE. Both SUP method and PAV algorithm deliver consistent and asymptotically unbiased estimator. All methods behave well asymptotically in the discrete time setting. Data from the Surveillance, Epidemiology and End Results (SEER) database are used to illustrate the proposed estimators.