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Contributed Papers Session / Communications libres
(Kee Lam, Organizer)

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GHADA ALOBAIDI, Department of Applied Mathematics, University of Western Ontario, London, Ontario  N6A 5B7
Using Monte Carlo methods to evaluate sub-optimal 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 apre-determined 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 best-known 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 ill-advised 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 set-indexed processes

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

HOWARD E. BELL, Brock University, St. Catharines, Ontario  L2S 3A1
Almost-commutativity 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)/(log² 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 state-dependent 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

ì í î x¢ = y y¢ = f(x)y+g(x)

(1)

where f(x) and g(x) are the polynomials in x in the complex domain \mathbb C.

It is well-known 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(x²-1) and g(x) = x, (1) is equivalent to the famous Van der pol equation

d 2 x dt2 +e(x2-1) 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^[3-10].

In this paper, we are trying to use a new approch which we currently call A-A 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 set-indexed survival analysis

Using the theory of set-indexed martingales, we develop a general model for survival analysis with censored data which is parameterized by sets instead of time points. We define a set-indexed Nelson-Aalen 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 (m-1)-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 m-dimensional 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 Black-Scholes 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
Web-based mathematics instruction: two Australian models

This talk will describe two web-based 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 web-based 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 web-based 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 web-based programs will be presented.

VOLKER RUNDE, University of Alberta, Edmonton, Alberta  T6G 2G1
Quasi-central, 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 quasi-central if

m ea - ea m ® 0     æ è m Î M(A) ö ø .

Every Arens regular Banach algebra, i.e. in particular every C^*-algebra, has a quasi-central bounded approximate identity. A. M. Sinclair raised the question for which locally compact groups G the group algebra L¹(G) has a quasi-central bounded approximate identity. Clearly, whenever G is a [SIN]-group, L¹(G) has a quasi-central, bounded approximate identity. It has been an open question whether there are locally compact groups G such that L¹(G) has a quasi-central, 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 parallelograms-properties 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 Barsotti-Tate Galois representations

We employ the methods of Breuil-Conrad-Diamond-Taylor to prove the modularity of a collection of potentially Barsotti-Tate l-adic Galois representations, contingent on the modularity of their residual representations. These results are a special case of a conjecture of Conrad-Diamond-Taylor. The proof involves extensive calculations using C. Breuil's classification of killed-by-l finite flat group schemes over highly ramified base schemes.