Volume 14, Issue 2

We present an algorithm for American option pricing based on stochastic approximation techniques. Besides working on a finite subset of the exercise dates (e.g. considering the associated Bermudean option), option pricing algorithms generally involve another step of discretization, either on the state space or on the underlying functional space. Our work, which is an application of a more general perturbed gradient algorithm introduced recently by the authors, consists in approximating the value functions of the classical dynamic programming equation at each time step by a linear combination of kernels. The so-called kernel-based stochastic gradient algorithm avoids any a priori discretization, besides the discretization of time. Thus, it converges toward the optimum of the non-discretized Bermudan option pricing problem. We present a comprehensive methodology to implement efficiently this algorithm, including discussions on the numerical tools used, like the Fast Gauss Transform, or Brownian bridge. We also compare our results to some existing methods, and provide empirical statistical results.

We investigate the problem of constructing small point sets with low star discrepancy in the s -dimensional unit cube. The size of the point set shall always be polynomial in the dimension s . Our particular focus is on extending the dimension of a given low-discrepancy point set. This results in a deterministic algorithm that constructs N -point sets with small discrepancy in a component-by-component fashion. The algorithm also provides the exact star discrepancy of the output set. Its run-time considerably improves on the run-times of the currently known deterministic algorithms that generate low-discrepancy point sets of comparable quality. We also study infinite sequences of points with infinitely many components such that all initial subsegments projected down to all finite dimensions have low discrepancy. To this end, we introduce the inverse of the star discrepancy of such a sequence, and derive upper bounds for it as well as for the star discrepancy of the projections of finite subsequences with explicitly given constants. In particular, we establish the existence of sequences whose inverse of the star discrepancy depends linearly on the dimension.

We introduce an infinite system of equations modeling the time evolution of the growth process of a network. The nodes are characterized by their degree k ∈ ℕ and a fitness parameter f ∈ [0, h ]. Every new node which emerges becomes a fitness f′ according to a given distribution P and attaches to an existing node with fitness f and degree k at rate f A k , where A k are positive coefficients, growing sublinearly in k . If the parameter f takes only one value, the dynamics of this process can be described by a variant of the Becker–Döring equations, where the growth of the size of clusters of size k occurs only with increment 1. In contrast to the established Becker–Döring equations, the system considered here is nonconservative, since mass (i.e. links) is continuously added. Nevertheless, it has the property of linearity, which is a natural consequence of the process which is being modeled. The purpose of this paper is to construct a solution of the system based on a stochastic approximation algorithm, which allows also a numerical simulation in order to get insight into its qualitative behaviour. In particular we show analytically and numerically the property of Bose–Einstein condensation, which was observed in the literature on random graphs.

In reviewing the foundations of probability, to establish the underpinnings of the Monte Carlo method, I found some basic results for which I could not find a simple, clear, and concise derivation in the literature. This paper is an attempt to present a sufficient collection of such results. The presentation is pitched at the level of Monte Carlo practitioners, rather than mathematical probabilists.