




Contributed Papers Session / Communications libres (Kee Lam, Organizer)
 GHADA ALOBAIDI, Department of Applied Mathematics, University of Western
Ontario, London, Ontario N6A 5B7
Using Monte Carlo methods to evaluate suboptimal exercise
policies for American options

(joint work with R. Mallier)
Options are derivative financial instruments which give the holder the
right but not the obligation to buy (or sell) the underlying asset.
American options are options which can be exercised either on or before
apredetermined expiry date. For such options there is, therefore, the
possibility of early exercise, and the issue of whether and when to
exercise an American option is one of the bestknown problems in
mathematical finance, leading to an optimal exercise boundary and an
optimal exercise policy, the following of which will maximize the
expected return from the option.
In this study, we use a Monte Carlo scheme to look at several such
strategies that a somewhat illadvised investor might follow, and
calculate the expected return from the option using these strategies.
In addition to evaluating several naive strategies, we will also look
at how the expected return is affected by the ``frequency of
checking'', meaning how often the investor checks the value of the
option to see if his exercise criteria have been met.
 RALUCA BALAN, Department of Mathematics and Statistics, University of Ottawa,
Ottawa, Ontario K1N 6N5
A Markov property for setindexed processes

A special place in the modern theory of stochastic processes indexed by
partially ordered sets is taken by the theory of setindexed processes.
We will consider a certain type of Markov property (the ``setMarkov''
property) for setindexed processes, which has the merit of attaining
the following three goals: (i) all processes with independent
increments are setMarkov; (ii) there exists a systematic procedure
which allows us to construct a general setMarkov process; and
(iii) we can define a generator which completely characterizes the
finite dimensional distributions of a setMarkov process. We will show
that the setMarkov property implies a type of sharp Markov property. A
setMarkov process becomes Markov in the classical sense when it is
transported by a ``flow''. An example of a setMarkov process which
does not have independent increments is the empirical process.
 HOWARD E. BELL, Brock University, St. Catharines, Ontario L2S 3A1
Almostcommutativity in rings

A subset S of the ring R is called almost commutative if
each element of S centralizes all except finitely many elements of
S. We investigate commutativity in infinite rings in in infinite
rings in which certain infinite subsets of zero divisors are almost
commutative. We prove that if the set D of all zero divisors is
infinite and almost commutative, then D is commutative; and under
certain additional hypotheses, R is commutative.
 ALINA CARMEN COJOCARU, Department of Mathematics and Statistics, Queen's University,
Kingston, Ontario K7L 3N6
On the cyclicity of the reduction modulo p of an elliptic curve
over Q without complex multiplication

Inspired by a conjecture of Lang and Trotter, Serre considered the
problem of determining how often the reduction modulo a prime p of an
elliptic curve E defined over Q gives a cyclic group.
Following Hooley's work on Artin's primitive root conjecture, he
showed in 1976 that the number N(x) of such primes p £ x is
~ c [(x)/(logx)] for some constant c, assuming the
Generalized Riemann Hypothesis for Dedekind zeta functions (GRH). In
1980 Ram Murty removed this hypothesis for elliptic curves with complex
multiplication. In 1990 Rajiv Gupta and Ram Murty proved
unconditionally that for any elliptic curve over Q the
number N(x) is >> [(x)/(log^{2} x)]. In this paper we weaken GRH
for elliptic curves without complex multiplication, obtaining the same
density of primes as the one obtained by Serre.
This is joint work with Ram Murty (Queen's University).
 FRANCOIS DUBEAU, Departement de mathematiques et d'informatique,
Universite de Sherbrooke, Sherbrooke Quebec J1K 2R1
On impulsive ordinary and delay differential equations

Existence and uniqueness of the solution to
ordinary and delay differential equations with infinitely many
statedependent impulses are considered. A simple transformation
allows us to show that these problems are equivalent to problems
without impulse. A fixed point approach is then applied for an
appropriate norm.
 ZHAOSHENG FENG, Department of Mathematics, Texas A&M University, College
Station, Texas 77843, USA
The existence of the algebraic curve solution for second
order polynomial autonomous systems in the complex domain

In the present paper, we are concerned with the following polynomial
autonomous systems
where f(x) and g(x) are the polynomials in x in the complex
domain \mathbb C.
It is wellknown that the polynomial autonomous system (1) plays an
important role in the qualitative theory of ordinary differential
equations, because many practical problems can be converted to (1), and
it also can be widely applied in many scientific fields such as
Engineering, Control Theory, Fluid Mechcanics, and so on. For example,
when f(x) = e(x^{2}1) and g(x) = x, (1) is equivalent to the
famous Van der pol equation

d ^{2} x dt^{2}

+e(x^{2}1) 
dx dt

+x = 0 
 (2) 
Unfortunately, in general, (1) is not solvable^{[1]}, so numerical
analysis is a common method by engineers and physicists. Recently, a
good result had been presented in paper [3] that a polynomial
autonomous system is not integrable if it does not have any algebraic
curve solution in \mathbb C. Therefore, the problem that under
what special conditions (1) has the algebraic curve solution in
C has become a very interesting research topic during the
past years^{[310]}.
In this paper, we are trying to use a new approch which we currently
call AA method (Algebraic Analytics method) to investigate the
existence of the algebraic curve solution of the second order
polynomial autonomous systems in the complex domain C. We
obtain a few theorems for the existence of the algebraic curve solution
of (1). These results are not only very important in the qualitative
theory of polynomial autonomous systems, but also very useful in
investigating the integrability of nonlinear ordinary differential
equations such as Lienard equations.
 GAIL IVANOFF, Department of Mathematics and Statistics, University of
Ottawa, Ottawa, Ontario K1N 6N5
Random censoring in setindexed survival analysis

Using the theory of setindexed martingales, we develop a general model
for survival analysis with censored data which is parameterized by sets
instead of time points. We define a setindexed NelsonAalen estimator
for the integrated hazard function with the presence of a censoring by
a random set which is a stopping set. We prove that this estimator is
asymptotically unbiased and consistent. A central limit theorem is
given. This model can be applied to cases when censoring occurs in
geometrical objects or patterns, and is a generalization of models with
multidimensional failure times.
 TOMASZ KACZYNSKI, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1
Recursive coboundary formula for cycles in acyclic chain
complexes

Given an (m1)dimensional cycle z in a finitely generated acyclic
chain complex (for example a triangulation of a polyhedron, cubical
grid or a finite cellular complex) we want to construct an
mdimensional chain COB(z) whose algebraic boundary is z. The
acyclicity of the chain complex implies that a solution exists (it is
not unique!) but the traditional linear algebra methods of finding it
lead to a high complexity of computation. We are searching for more
efficient algorithms based on geometric considerations.
The main motivation for studying this problem comes from the
topological and computational dynamics, namely, from designing general
algorithms computing the homomorphism induced in homology by a
continuous map. This, for turn, is an essential step in computing such
invariants of dynamical properties of a map as Conley index or
Lefschetz number. Another potential motivation is in the relationship
of our problem to the problem of finding minimal surfaces of closed
curves.
 ROLAND MALLIER, Department of Applied Mathematics,
University of Western Ontario, London, Ontario N6A 5B7
Laplace transforms and American call options

(joint work with Ghada Alobaidi)
Options are derivative securities which are used in financial markets.
They give the holder the right (but not the obligation) to buy (a call)
or sell (a put) some other underlying security. European options can
only be exercised at expiry, but American options can be exercised at
or prior to expiry, and this leads to a free boundary problem for the
optimal exercise boundary as the holder of an American option must
constantly decide whether to exercise the option or retain it. Starting
from the BlackScholes partial differential equation (which describes
the value of a derivative security), we use Laplace transform
techniques to derive an (Urysohn) integral equation giving the location
of this optimal exercise boundary for an American call option with a
constant dividend yield. It is necessary to modify the definition of
the transform slightly because of the presence of the free boundary. We
also give expressions for the transform of the value of the option in
terms of the optimal exercise boundary.
 DOUG PITNEY, University of Western Australia, Nedlands 6907,
Western Australia
Webbased mathematics instruction: two Australian models

This talk will describe two webbased mathematics programs developed at
the University of Western Australia. The first program is an
interactive simulation of a large mathematics class that provides
lecturers with instructional alternatives based on student and expert
lecturer interviews. The second program involves first year calculus
and statistics courses with two webbased components: digitised video
lectures and computer generated assignments.
The interactive lecture program is a professional development package
for anyone who teaches in a large lecture situation. It consists of
live footage of a large mathematics lecture, commentary from the
lecturer on specific interactions, interviews with award winning
lecturers, interviews with students, and abstracts from pertinent
research.
The second program features calculus and statistics lectures presented
in a large lecture theatre at the Perth campus of the University of
Western Australia. Instead of a traditional overhead projector, a
visualiser equipped with a video camera is used to capture the
lecture material. They are digitally recorded, compressed and made
available to students studying at remote locations (e.g. home,
libraries, university computer labs, etc). The webbased
assignments are computer generated and marked by comparing student
solutions to Mathematica solutions.
In this talk, details of the instructional design and statistics for
both webbased programs will be presented.
 VOLKER RUNDE, University of Alberta, Edmonton, Alberta T6G 2G1
Quasicentral, bounded approximate identities for group algebras

Let A be a Banach algebra with multiplier algebra
M(A). A bounded approximate identity
(e_{a})_{a} for A is called quasicentral
if
m e_{a}  e_{a} m ® 0 
æ è

m Î M(A) 
ö ø

. 

Every Arens regular Banach algebra, i.e. in particular every
C^{*}algebra, has a quasicentral bounded approximate identity.
A. M. Sinclair raised the question for which locally compact groups G
the group algebra L^{1}(G) has a quasicentral bounded approximate
identity. Clearly, whenever G is a [SIN]group, L^{1}(G) has a
quasicentral, bounded approximate identity. It has been an open
question whether there are locally compact groups G such that
L^{1}(G) has a quasicentral, bounded approximate identity, but which
fail to be [SIN]groups. We exhibit an example of such a group.
 DIETER RUOFF, Department of Mathematics and Statistics, University of Regina,
Regina, Saskatchewan S4S 0A2
Hyperbolic parallelogramsproperties and applications

The hyperbolic parallelogram, the quadrilateral whose opposite sides
determine boundary parallel lines, is an undeservedly overlooked
figure, both because of its remarkable properties, and because of its
suitability for applications. One of the four vertices of a
parallelogram, the outer vertex, determines, together with the adjacent
sides, an angle whose interior contains the opposite vertex. If A is
the outer vertex of parallelogram ABCD then AB > DC, AD > BC, and AB+ BC = AD + DC. There are several, rather different ways of proving
this. One of them is based on the figure of a hyperbolic rhombus in
which the two sides sharing the outer vertex, as well as the other two
sides are congruent.
Not surprisingly the figures related to a number of hyperbolic theorems
contain parallelograms, or can be furnished with parallelograms. And
indeed the above mentioned formulas are an invaluable help in the
proofs of these theorems.
 DAVID SAVITT, Department of Mathematics, Harvard University, Cambridge,
Massachusetts 02138, USA
Modularity of some potentially BarsottiTate Galois
representations

We employ the methods of BreuilConradDiamondTaylor to prove the
modularity of a collection of potentially BarsottiTate ladic Galois
representations, contingent on the modularity of their residual
representations. These results are a special case of a conjecture of
ConradDiamondTaylor. The proof involves extensive calculations using
C. Breuil's classification of killedbyl finite flat group
schemes over highly ramified base schemes.

