659 research outputs found
Closed-form expression for finite predictor coefficients of multivariate ARMA processes
We derive a closed-form expression for the finite predictor coefficients of
multivariate ARMA (autoregressive moving-average) processes. The expression is
given in terms of several explicit matrices that are of fixed sizes independent
of the number of observations. The significance of the expression is that it
provides us with a linear-time algorithm to compute the finite predictor
coefficients. In the proof of the expression, a correspondence result between
two relevant matrix-valued outer functions plays a key role. We apply the
expression to determine the asymptotic behavior of a sum that appears in the
autoregressive model fitting and the autoregressive sieve bootstrap. The
results are new even for univariate ARMA processes.Comment: Journal of Multivariate Analysis, to appea
Optimal long term investment model with memory
We consider a financial market model driven by an R^n-valued Gaussian process
with stationary increments which is different from Brownian motion. This
driving noise process consists of independent components, and each
component has memory described by two parameters. For this market model, we
explicitly solve optimal investment problems. These include (i) Merton's
portfolio optimization problem; (ii) the maximization of growth rate of
expected utility of wealth over the infinite horizon; (iii) the maximization of
the large deviation probability that the wealth grows at a higher rate than a
given benchmark. The estimation of paremeters is also considered.Comment: 25 pages, 3 figures. To appear in Applied Mathematics and
Optimizatio
Baxter's inequality for fractional Brownian motion-type processes with Hurst index less than 1/2
The aim of this paper is to prove an analogue of Baxter's inequality for
fractional Brownian motion-type processes with Hurst index less than 1/2. This
inequality is concerned with the norm estimate of the difference between
finite- and infinite-past predictor coefficients.Comment: 7 page
Binary market models with memory
We construct a binary market model with memory that approximates a
continuous-time market model driven by a Gaussian process equivalent to
Brownian motion. We give a sufficient conditions for the binary market to be
arbitrage-free. In a case when arbitrage opportunities exist, we present the
rate at which the arbitrage probability tends to zero as the number of periods
goes to infinity.Comment: 13 page
Applications of a finite-dimensional duality principle to some prediction problems
Some of the most important results in prediction theory and time series
analysis when finitely many values are removed from or added to its infinite
past have been obtained using difficult and diverse techniques ranging from
duality in Hilbert spaces of analytic functions (Nakazi, 1984) to linear
regression in statistics (Box and Tiao, 1975). We unify these results via a
finite-dimensional duality lemma and elementary ideas from the linear algebra.
The approach reveals the inherent finite-dimensional character of many
difficult prediction problems, the role of duality and biorthogonality for a
finite set of random variables. The lemma is particularly useful when the
number of missing values is small, like one or two, as in the case of
Kolmogorov and Nakazi prediction problems. The stationarity of the underlying
process is not a requirement. It opens up the possibility of extending such
results to nonstationary processes.Comment: 15 page
The intersection of past and future for multivariate stationary processes
We consider an intersection of past and future property of multivariate
stationary processes which is the key to deriving various representation
theorems for their linear predictor coefficient matrices. We extend useful
spectral characterizations for this property from univariate processes to
multivariate processes.Comment: 8 page
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