F. Delbaen

Arbitrage possibilities in Bessel processes and their relations to local martingales

SummaryWe show that, if we allow general admissible integrands as trading strategies, the three dimensional Bessel process, Bes3, admits arbitrage possibilities. This is in contrast with the fact that the inverse process is a local martingale and hence is arbitrage free. This leads to some economic interpretation for the analysis of the property of arbitrage in foreign exchange rates. This notion (relative to general admissible integrands) does depend on the fact, which of the two currencies under consideration is chosen as numéraire. The results rely on a general construction of strictly positive local martingales. The construction is related to the Föllmer measure of a positive super-martingale.

Classical risk theory in an economic environment

Abstract In the present paper the influence of macro-economic factors such as interest and inflation on the classical risk and surplus process is studied.

Convex games and extreme points

The theory of games in characteristic function form is well developed. However not so much work has been done on the continuity properties of the core of these games. The aim of this paper is to prove that for convex games the core behaves nicely. (We also present some counterexamples for non convex games.) In Section 1 we give some definitions and notations and prove the fundamental lemma of our paper. Section 2 lists different characterizations of convex games. In Section 3 we prove the so-called projection theorem. Roughly this theorem states that the set of core restrictions is the core of the subgame. In Section 4 we give a characterization of the extreme points of the core of a finite game; although the results of this section are already discovered in [14], we use a different method. Sections 5 and 6 contain weak and strong continuity properties of the core correspondence. Section 8 gives some information on u-continuous games. We also answer a question raised in [l 11. Section 9 is devoted to selection theorems. For an application of selection theorems to mathematical economics we refer to a forthcoming paper of Sondermann [15]. Sections 7 and 10 give some approximation theorems; the purpose of these theorems is to approximate a game with infinitely many players by a game with finitely many players. Section 11 is an attempt to describe the structure of the core of a-continuous games. We hope that our results will lead to a better understanding of the core of a game with infinitely many players. We know that for finite games our results are not general but the paper has to be seen as a paper on infinite games.

Term structure of interest rates: The martingale approach

Martingale methods are used to study interest rate risk in a market with two fundamental assets: savings accounts and zero coupon bonds. Discounted prices of bonds have to be a martingale for a risk-neutral probability. Specifications are given when the instantaneous rate of interest is adapted to a Brownian motion or follows a diffusion.

The laplace transform of annuities certain with exponential time distribution

Abstract By means of Wiener processes, randomness in interest rates for annuities can be modelled. This paper wants to give an expression for the Laplace transform of annuities certain, when time is exponentially distributed.

Long-term returns in stochastic interest rate models

In this paper, we observe the convergence of the long-term return, using an extension of the Cox-Ingersoll-Ross [1985] stochastic model of the short interest rate r. Using the theory of Bessel processes, we are able to prove the convergence almost everywhere of \(frac{1}{t}{\int {_{_0}} ^t}{X_\delta }ds\) with the X a generalized Besselsquare process with drift with stochastic reversion level. In order to make some approximations, we observe the convergence in law of the sequence \(left({\sqrt {{{\frac{{ - 2\beta }}{{\delta n}}}^3}} _0}^{nt}\left( {{X_u} + \frac{{{\delta _u}}}{{2\beta }}} \right)du)t \geqslant 0\). By Aldous’ criterion, we are able to prove that this sequence converges in law to a Brownian motion.

A remark on the moments of ruin time in classical risk theory

Abstract We prove that the p th moment of the ruin time in a classical risk process exists if and only if the ( p +1)th moment of the claim size exists. The proof uses a theorem of Erdos on the speed of convergence in the law of large numbers. The result can also be obtained using fractional derivatives in Laplace transforms. Our proof is computational and again it shows the link between the limit behaviour in the (continuous time) compound Poisson process and the usual (discrete time) limit theorems in probability theory.

Estimation of the yield curve and the forward rate curve starting from a finite number of observations

Abstract The yield curve and the forward rate curve are estimated from given values of the yield-to-maturity. Using data from 1986 through 1991 a few illustrations of yield curves and forward rate curves are presented.

Limit theorems for the present value of the surplus of an insurance portfolio

Abstract It is shown that under general assumptions the present value of the surplus, submitted to macro-economic factors such as interest and inflation, converges to a limit at infinity.

Inversed martingales in risk theory

Abstract The theory of inversed martingales is used in order to prove a generalization of a result of H. Cramer on the probability of non-ruin for a classical surplus process if the initial reserve is positive.