Stable mixed moving averages

Summary The class of (non-Gaussian) stable moving average processes is extended by introducing an appropriate joint randomization of the filter function and of the stable noise, leading to stable mixed moving averages. Their distribution determines a certain combination of the filter function and the mixing measure, leading to a generalization of a theorem of Kanter (1973) for usual moving averages. Stable mixed moving averages contain sums of independent stable moving averages, are ergodic and are not harmonizable. Also a class of stable mixed moving averages is constructed with the reflection positivity property.Research supported by AFSOR Contract 91-0030Research also supported by ARO DAAL-91-G-0176Research also supported by AFOSR 90-0168Research also supported by ONR N00014-91-J-0277     

Properties of moving averages of asset prices

We derive the stochastic differential equation (sde) governinga quantity M which is the ratio of an exponentially weightedmoving average of a share price to the share price itself. Thesdes for other path-dependent random variables are also given.We then present the (ordinary and partial) differential equationsfor the expected first exit time and the cumulative distributionfunction for the first exit time. The boundary conditions forthese equations are discussed in some depth. Some numericalsolutions are presented. We consider the crossing of two movingaverages with different weights, and the application to thepricing of pathdependent options.     

Using composite moving averages to forecast sales

Combining moving averages has been suggested as a simple and practical means to improve sales forecasting. Here we present a natural extension whereby combinations of all possible moving averages up to a given number of periods are employed. We evaluate the method's performance relative to other methods, such as simple moving averages and exponentially-weighted moving averages, on two industrial data sets. Particular attention is placed on methods for selecting the number of periods employed, and on handling noisy data.     

Optimality and robustness of combinations of moving averages

A combination of moving averages has been shown previously to be more accurate than simple moving averages, under certain conditions, and to be more robust to non-optimal parameter specification. However, the use of the method depends on specification of three parameters: length of greater moving average, length of shorter moving average, and the weighting given to the former. In this paper, expressions are derived for the optimal values of the three parameters, under the conditions of a steady state model. These expressions reduce a three-parameter search to a single-parameter search. An expression is given for the variance of the sampling error of the optimal combination of moving averages and this is shown to be marginally greater than that for exponentially weighted moving averages (EWMA). Similar expressions for optimal parameters and the resultant variance are derived for equally weighted combinations. The sampling variance of the mean of such combinations is shown to be almost identical to the optimal general combination, thus simplifying the use of combinations further. It is demonstrated that equal weight combinations are more robust than EWMA to noise to signal ratios lower than expected, but less robust to noise to signal ratios higher than expected.     

Weak and universal consistency of moving weighted averages

The properties of weighted averages as linear estimators of a regression function and its derivatives are investigated for the fixed design case. Results on weak consistency and on universal consistency are derived, using a modification of the definition of Stone [10]. As examples we consider kernel estimates and weighted local regression estimators and show that the general results apply.     

Gaussian moving averages,semimartingales and option pricing

We provide a characterization of the Gaussian processes with stationary increments that can be represented as a moving average with respect to a two-sided Brownian motion. For such a process we give a necessary and sufficient condition to be a semimartingale with respect to the filtration generated by the two-sided Brownian motion. Furthermore, we show that this condition implies that the process is either of finite variation or a multiple of a Brownian motion with respect to an equivalent probability measure. As an application we discuss the problem of option pricing in financial models driven by Gaussian moving averages with stationary increments. In particular, we derive option prices in a regularized fractional version of the Black–Scholes model.     

Stable processes: moving averages versus fourier transforms

Summary We study the relationship between Fourier transforms and moving averages of stable processes. The two classes have a large intersection, however as soon as the noise is required to be bounded they become disjoint.Research supported by the Air Force Office of Scientific Research Contract No. F49620 85C 0144 and by the Office of Naval Research Grant No. N00014 86C 0227This work was done in part when the author was visiting the Center for Computational Statistics, George Mason University, Fairfax, VA 22030-4444     

Lévy driven moving averages and semimartingales

The aim of the present paper is to study the semimartingale property of continuous time moving averages driven by Lévy processes. We provide necessary and sufficient conditions on the kernel for the moving average to be a semimartingale in the natural filtration of the Lévy process, and when this is the case we also provide a useful representation. Assuming that the driving Lévy process is of unbounded variation, we show that the moving average is a semimartingale if and only if the kernel is absolutely continuous with a density satisfying an integrability condition.     

New classes of self-similar symmetric stable random fields

We construct two new classes of symmetric stable self-similar random fields with stationary increments, one of the moving average type, the other of the harmonizable type. The fields are defined through an integral representation whose kernel involves a norm on ⁿ . We examine how the choice of the norm affects the finite-dimensional distributions. We also study the processes which are obtained by projecting the random fields on a one-dimensional subspace. We compare these projection processes with each other and with other well-known self-similar processes and we characterize their asymptotic dependence structure.The research was done at Boston University while the first author was on leave from the Hugo Steinhaus Center, Poland. The second author was partially supported by the ONR Grant N00014-90-J-1287 at Boston University and by a grant of the United States-Israel Binational Science Foundation.     

Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

More limit theory for the sample correlation function of moving averages

Let X_t = Σ^∞_j=-∞ c_jZ_{t - j} be a moving average process where {Z_t} is iid with common distribution in the domain of attraction of a stable law with index , 0 < < 2. If 0 < < 2, E|Z₁| < ∞ and the distribution of |Z₁|and |Z₁Z₂| are tail equivalent then the sample correlation function of {X₁} suitably normalized converges in distribution to the ratio of two dependent stable random variables with indices and /2. This is in sharp contrast to the case E|Z₁| ^{= ∞} where the limit distribution is that of the ratio of two independent stable variables. Proofs rely heavily on point process techniques. We also consider the case when the sample correlations are asymptotically normal and extend slightly the classical result.     

The Laplacian on p.c.f. self-similar sets via the method of averages

We show how the symmetric Laplacian on p.c.f. self-similar sets, together with its associated Dirichlet form and harmonic functions, can be defined entirely in terms of average values of a function over basic sets. The approach combined the constructive limit-of-difference-quotients method of Kigami and the method of averages introduced by Kusuoka and Zhou for the Sierpinski carpet. We consider well-known examples, such as the unit interval, the Vicsek set,the hexagasket, and SG⁴. This paper has generalized the results in [11], [13] and [14], but a different proof is needed.     

A class of asymptotically self-similar stable processes with stationary increments

We generalize the BM-local time fractional symmetric α$\alpha$-stable motion introduced in Cohen and Samorodnitsky (2006) by replacing the local time with a general continuous additive functional (CAF). We show that the resulting process is again symmetric α$\alpha$-stable with stationary increments. Depending on the CAF, the process is either self-similar or lies in the domain of attraction of the BM-local time fractional symmetric α$\alpha$-stable motion. We also show that the process arises as a weak limit of a discrete “random rewards scheme” similar to the one described by Cohen and Samorodnitsky.     

A robust forecasting system,based on the combination of two simple moving averages

For series with negligible growth and seasonality, simple moving averages are frequently used to estimate the current level of a process, and the resultant value projected as a forecast for future observations. This paper shows that a linear combination of two simple moving averages (SMA) can provide an improved estimate of the underlying level of the process. The proposition is demonstrated by simulation, and good combinations are listed. The theory underlying the improvement is developed. The general rules are then illustrated through an application in an inventory situation.