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We introduce a simple but efficient PDE method that makes use of interpolation wavelets for their advantages in compression and interpolation in order to define a sparse computational domain. It uses finite difference filters for approximate differentiation, which provide us with a simple and sparse stiffness matrix for the discrete system. Since the method only uses a nodal basis, the application of non-constant terms, boundary conditions and free-boundary conditions is straightforward. We give empirical results for financial products from the equity and fixed income markets in 1, 2 and 3 dimensions and show a speed-up factor between 2 and 4 with no significant reduction of precision.

Recent times have seen the transfer of risks to individual households as the unintended consequences of pension reforms and new rules for regulation of financial advisory companies. We present new theory and technology for individual Asset Liability Management (iALM) which addresses these critical issues for individuals. The iALM tool uses a cutting edge dynamic stochastic programming formulation based on ideas from behavioural finance and decision theory. The resulting system will be Illustrated in terms of solutions for representative households. The technological challenges of practical mass implementation requiring current HPC solutions such as the cloud and the progress in this direction made to date will also be discussed.

Dynamic multistage stochastic linear programming has many practical applications forproblems whose current decisions have to be made under future uncertainty. There are avariety of methods for solving these problems including nested Benders decomposition. Inthis method, recently shown to be superior to the alternatives for large problems, the problemis decomposed into a set of smaller linear programming problems. These problems can bevisualised as being attached to the nodes of a tree which is formed from the realizations ofthe random data vectors determining the uncertainty in the problem. The tree is traversedforwards and backwards, with information from the solutions to each nodal linear programmingproblem being passed to its immediate descendants by the formation of their righthand sides and to its immediate ancestor in the form of cuts. Problems in the same timeperiod can be solved independently and it is this inherent parallelism that is exploited inour parallel nested Benders algorithm. A parallel version of the MSLiP nested Benders codehas been developed and tested on various types of MIMD machines. The differing structuresof the test problems cause differing levels of speed-up. Results show that problems withfew variables and constraints per node do not gain from this parallelization. Stage aggregationhas been successfully exploited for such problems to improve their parallel solutionefficiency by increasing the size of the nodes and therefore the time spent calculating relativeto the time spent communicating between processors.

This paper provides a survey of the now considerable academic theory relating to the practice of dynamic asset allocation. This work is scattered through the literature and many of the key ideas are not as accessible or well known as they deserve to be. The paper begins by providing a definition of what is meant by dynamic asset allocation and a description of its most significant features. Next it develops the concept of path independence and its relationship to efficient diversification through time. It is shown that this principle also applies to funds whose performance is appraised relative to an index benchmark. The final sections of the paper describe the implications of recent work on market equilibrium and on performance measurement.

The fluctuation of stock prices is modelled as a sequence of temporary equilibria on a financial market with different types of agents. I summarize joint work with M. Schweizer on the class of Ornstein-Uhlenbeck processes in a random environment which appears in the diffusion limit. Moreover, it is shown how the random environment may be generated by the interaction of a large set of agents modelled by Markov chains as they appear in the theory of probabilistic cellular automata.

This paper states the ‘general mean-variance portfolio analysis problem’ and its solution, and briefly discusses its use in practice.

My aim is to make some comments of a quite general nature about the relation between applied mathematics and finance, theoretical and practical. I shall begin with a brief description of a case in which ‘technology transfer’ from a quite different area of mathematics, the Stefan problem, was helpful with a financial problem, namely the Black-Scholes approach to an American option. I then discuss some more general issues about the role of this kind of mathematics in finance and suggest some possible avenues for future progress.

Capacity expansion is the process of providing new facilities over time to meet rising demand. A general mathematical model of this process is presented, incorporating uncertain future demand (including the possibility of ‘surprises’), non-zero lead times and random cost overruns. In this model the decision maker controls the rate of investment in the current expansion project. Optimization is studied by methods of stochastic control theory. Numerical algorithms are presented which determine the optimal policy in some simple cases.

This paper concerns an algorithm, proposed by C. C. Gotlieb (4) and modified by J. Csima (1;2), for a recent combinatorial problem whose application includes the construction of school time-tables. Theoretically, the problem is related to systems of distinct subset representatives, the construction of Latin arrays, the colouring of graphs, and flows in networks (1;2;3). I t was conjectured by Gotlieb and Csima that if solutions to a given time-table problem existed, i.e. if time-tables incorporating certain pre-assigned meetings existed, their algorithm would find one.

Bei der Budgetierung ergeben sich die wirkungsvollsten Koordinierungsmechanismen zweifellos aus der Verteilung der Rollen auf die masgeblichen Beteiligten. Rollen — als Erwartungen uber das an institutionelle Positionen geknupfte Verhalten — stellen einen Teil der Arbeitsteilung dar. Sie sind Berechnungsmechanismen. Im amerikanischen Regierungssystem handeln die Verwaltungsdienststellen auf Bundesebene als Befurworter hoherer Ausgaben. Das Buro fur Management und Budget (OMB) 1) versteht sich als Anwalt des Prasidenten mit einem Hang zu Kurzungen. Das Bewilligungskomitee des Hauses ubt die Funktion des Huters des Schatzamtes aus, und das Bewilligungskomitee des Senates diejenige einer Berufungsinstanz, welcher die Dienststellen ihre abweichenden Auffassungen zur Politik des Hauses vortragen. Die Rollen greifen ineinander und fuhren zu einem Gefuge gegenseitiger Erwartungen, welche die Berechnungen fur die Beteiligten erheblich erleichtern. Da man davon ausgehen kann, das die Dienststellen alle Vorschlage mit einer gewissen Aussicht auf Unterstutzung einbringen werden, konnen sich das Buro fur Management und Budget und die Bewilligungskomitees darauf konzentrieren, diese Vorschlage in das Programm des Prasidenten einzufugen, oder aber sie zu beschneiden.

The requirement for methods of optimization arises from the mathematical complexity necessary to describe the theory of systems, processes, equipment, and devices which occur in practice. Even quite simple systems must sometimes be represented by theory which may contain approximations, by parameters which change with time, or by parameters that vary in a random manner. For many reasons the theory is imperfect, yet it must be used to predict the optimum operating conditions of a system such that some performance criterion is satisfied. At best such theory can predict only that the system is near to the desired optimum. Optimization methods are then used to explore the local region of operation and predict the way that the system parameters should be adjusted to bring the system to optimum.

Some of the problems encountered in the minimization of functions of n variables were introduced in Chapter 1. One additional problem, best illustrated by an example, is the enormity of hyperspace. Consider a 10 dimensional unit hypercube in which a search has determined that the volume from the origin to the point ½ in each dimension does not contain the minimum. The volume of the ½ unit cube eliminated from the search is equal to (½)10 ≈ 1/1000. The portion of the unit hypercube which remains is therefore 99.9% of the whole. Any form of complete search is clearly out of the question and attention must be given to the local area of a first estimate to the solution. All methods can therefore guarantee only to find the local minimum.

The problem considered in this chapter is that of minimizing the value $$ {\text{minimize}}\,E = f(x). $$