In this paper we study the situation faced by a videotape rental store which must
decide how many copies of a new videotape release it should purchase for rental to customers. A
model representing this process is developed and contrasted with the classical newsboy problem.

Barry A. Pasternack

Zvi  Drezner

The videotape rental model

The global properties of the classical three-dimensional Lotka-Volterra two prey-one
predator and one prey-two predator systems, under the assumption that competition can
be neglected, are analysed with the direct Lyapunov method. It is shown that, except for a
pathological case, one species is always driven to extinction, and the system behaves asymptotically
as a two-dimensional predator-prey Lotka-Volterra system. The same approach can be easily
extended to systems with many prey species and one predator, or many predator species and one
prey, and the same conclusion holds. The situation considered is common for New Zealand wild life,
where indigenous and introduced species interact with devastating consequences for the indigenous
species. According to our results the New Zealand indigenous species are definitely driven to
extinction, not only in consequence of unsuccessful competition, but even when competition is
absent. This result leads to a better understanding of the mechanism of natural selection, and
gives a new insight into pest control practice.

A.  Korobeinikov

G. C. Wake

Global properties of the three-dimensional
predator-prey Lotka-Volterra systems

For randomized block designs, nonparametric treatment comparisons
are usually made using the Friedman test for complete designs, and the Durbin test
for incomplete designs; see, for example, Conover (1998). This permits assessment
of only the mean rankings. Such comparisons are here extended to permit assessments
of bivariate effects such as the linear by linear effect and the quadratic by
linear, or umbrella effect.

J. C. W. Rayner

D. J. Best

Nonparametric tests for data in randomised blocks with Ordered
alternatives

We consider a risky economic project that may yield either profits or losses, depending
on random events. We study an insurance mechanism under which the plan of project implementation
maximizing the expected value of profits becomes optimal almost surely. The mechanism
is linear in the decision variables, &#8220;actuarially fair&#8221; and robust to changes in the utility function.The premium and the compensation in the insurance scheme are expressed through dual variables
associated with information constraints in the problem of maximization of expected profits. These
dual variables are interpreted as the shadow prices of information. Along with the general model,
several specialized models are considered in which the insurance mechanism and the shadow prices
are examined in detail.

I. V. Evstigneev

W. K. Klein Haneveld

L. J. Mirman

Robust insurance mechanisms and the shadow prices of information
constraints

On a queue with correlated arrivals

In the last few years new types of path-dependent options called corridor options
or range options have become well-known derivative instruments in European options markets.
Since the payout profiles of those options are based on occupation times of the underlying security
the purpose of this paper is to provide closed form pricing formulae of Black &#x26; Scholes type for
some significant representatives. Alternatively we demonstrate in this paper a relatively simple
derivation of the Black &#x26; Scholes price for a <i>single corridor option</i>  &#x2013; based on a static portfolio representation &#x2013; which does not make use of the distribution of occupation times (of Brownian
motion). However, knowledge of occupation times&#x27; distributions is a more powerful tool.

Andreas Pechtl

Some applications of occupation times of Brownian motion with
drift in mathematical finance

The purpose of this paper is to present a survey of recent developments concerning
the distributions of occupation times of Brownian motion and their applications in mathematical
finance. The main result is a closed form version for Akahori&#x27;s generalized arc-sine law which can
be exploited for pricing some innovative types of options in the Black &#x26; Scholes model. Moreover
a straightforward proof for Dassios&#x27; representation of the <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B1;</mml:mi></mml:math>
-quantile of Brownian motion with drift
shall be provided.

Distributions of occupation times of Brownian motion with drift

This paper is concerned with the application of an asymptotic quasi-likelihood
practical procedure to estimate the unknown parameters in linear stochastic models of the form
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>y</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#x03B8;</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#x03B8;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>..</mml:mn><mml:mo>,</mml:mo><mml:mi>T</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>
, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow></mml:math>
 is a linear predictable process of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B8;</mml:mi></mml:math>
 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>M</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math>
 is an
error term such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>|</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>
 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mi>M</mml:mi><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>|</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x003C;</mml:mo><mml:mi>&#x221E;</mml:mi></mml:math>
 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>F</mml:mi></mml:math>
is a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C3;</mml:mi></mml:math>-field generated
from<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>s</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>&#x2264;</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>
. For this model, to estimate the parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B8;</mml:mi><mml:mo>&#x2208;</mml:mo><mml:mi>&#x0398;</mml:mi></mml:math>, the ordinary least squares
method is usually inappropriate (if there is only one observable path of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:math> and if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msubsup><mml:mi>M</mml:mi><mml:mi>t</mml:mi><mml:mn>2</mml:mn></mml:msubsup></mml:mrow><mml:mo>|</mml:mo><mml:mrow /></mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>t</mml:mi></mml:msub><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>
is not a constant) and the maximum likelihood method either does not exist or is mathematically
intractable. If the finite dimensional distribution of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>M</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math>
 is unknown, to obtain a good estimate of
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03B8;</mml:mi></mml:math>
 an appropriate predictable process <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>g</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math> should be determined. In this paper, criteria for determining
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>g</mml:mi><mml:mi>t</mml:mi></mml:msub></mml:math>
 are introduced which, if satisfied, provide more accurate estimates of the parameters via the
asymptotic quasi-likelihood method.

Riccardo  Biondini

Yan-Xia  Lin

Sifa  Mvoi

A practical procedure for estimation of linear models via
asymptotic quasi-likelihood

Historically, utilities have been granted a natural monopoly status through the
regulatory process. Under such conditions, utilities need to prove to their regulators that their
expenditures were necessary to comply with imposed &#x201C;obligation to serve.&#x201D; When these
prudency arguments are successful, the utilities may recover their costs plus a rate of return.Some have argued that this structure has not created an environment that fosters productive
efficiency. With deregulation on the horizon, the utility business is changing. To survive the
21st century, utilities need to find ways to improve their efficiency. One such avenue is
strategic staffing.

M. Sami Khawaja

Strategic staffing model for electric utilities

Nagumo&#39;s equation is a third order non-linear ordinary differential equation
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>&#x2212;</mml:mo><mml:mi>c</mml:mi><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:msup><mml:mi>f</mml:mi><mml:mo>&#x2032;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mfrac><mml:mrow><mml:mi>d</mml:mi><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mfrac><mml:mo>&#x2212;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mi>b</mml:mi><mml:mo>/</mml:mo><mml:mi>c</mml:mi></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:math>
 where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>u</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>&#x2212;</mml:mo><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>u</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mi>&#x03B1;</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>
, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#x003C;</mml:mo><mml:mi>&#x03B1;</mml:mi><mml:mo>&#x003C;</mml:mo><mml:mn>1</mml:mn></mml:math>
. In this paper we
have developed a technique to determine those values of the parameters <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>a</mml:mi><mml:mo>,</mml:mo><mml:mi>b</mml:mi></mml:math>
 and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi></mml:math>
 which permit
non-constant bounded solutions.

M.  Iqbal

Classroom Note: Numerical solutions of Nagumo's equation

Speed of sap flow in plants and trees is of interest to botanists and environmentalists
because of its connection with the rate of utilisation of nutrients in the soil. An established method
uses the transport of heat where an impulsive heat source is introduced along a radial line by a
probe in the trunk sapwood. The temperature is monitored, upstream and downstream, and, by
solving the heat flow equation in the moving fluid, the sap velocity may be deduced indirectly
under some simplifying assumptions which chiefly render the method most useful when applied to
trees of relatively large diameter. Transform methods are used to obtain the appropriate threedimensional
time-dependent solution in explicit form and values for the resulting sap velocity are
compared with the existing two-dimensional theory.

Ron  Gribben

On the heat impulse method for deducing sap flow

The sample correlation coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> is almost universally used to estimate the
population correlation coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C1;</mml:mi></mml:math>. If the pair <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>Y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>
has a bivariate normal distribution, this
would not cause any trouble. However, if the marginals are nonnormal, particularly if they
have high skewness and kurtosis, the estimated value from a sample may be quite different from
the population correlation coefficient <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C1;</mml:mi></mml:math>.The bivariate lognormal is chosen as our case study for this robustness study. Two approaches
are used: (i) by simulation and (ii) numerical computations.Our simulation analysis indicates that for the bivariate lognormal, the bias in estimating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#x03C1;</mml:mi></mml:math> can
be very large if <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#x03C1;</mml:mi><mml:mo>&#x2260;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>, and it can be substantially reduced only after a large number (three to four
million) of observations. This phenomenon, though unexpected at first, was found to be
consistent to our findings by our numerical analysis.

C. D. Lai

T. P. Hutchinson

Robustness of the sample correlation - the bivariate lognormal case

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