We study the value of European security derivatives in the Black–Scholes model when the underlying asset $\xi $ is approximated by random walks ${\xi }^{(n)} $. We obtain an explicit error formula, up to a term of order $ \mathcal{O} ({n}^{- 3/ 2} )$, which is valid for general approximating schemes and general payoff functions. We show how this error formula can be used to find random walks ${\xi }^{(n)} $ for which option values converge at a speed of $ \mathcal{O} ({n}^{- 3/ 2} )$.

A two component classical Coulomb system is considered, in which particles of charge +q and + 2q are constrained to lie on a circle and interact via the two-dimensional Coulomb potential. At a special value of the coupling constant the correlation functions are calculated exactly and the asymptotic form of the truncated charge-charge correlation is found to obey Jancovici's sum rule.

In this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

We propose and analyse a method based on the Riccati transformation for solving the evolutionary Hamilton–Jacobi–Bellman equation arising from the dynamic stochastic optimal allocation problem. We show how the fully nonlinear Hamilton–Jacobi–Bellman equation can be transformed into a quasilinear parabolic equation whose diffusion function is obtained as the value function of a certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence and uniqueness and derive useful bounds of classical Hölder smooth solutions. Furthermore, we construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit travelling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 index as an example of the application of the method.

In this work we shall study the existence of extremal solutions for an impulsive problem with functional-boundary conditions and weak regularity assumptions, not only on the right-hand side of the equation and on the functions that define the boundary conditions, but also on the impulse functions, which will be required to be nondecreasing, but not continuous as well, as is customary in the literature.

Moreover, in order to prove one of our results we shall study a general impulsive linearproblem, giving a complete characterisation of resonance for it.

The problem of finding critical initial data which separate conditions leading to blow-up from those which give solutions tending to the (stable) minimal solution is considered. New criteria for blow-up and global existence are found; these are equivalent to obtaining upper and lower bounds respectively for the set of critical initial data.

The Einstein field equations have been solved for Bianchi type VI0 spacetimes with viscous fluid source. Four cosmological models are derived. They have nonzero expansion and shear. One of them have nonzero constant shear viscosity coefficient.

The Mach-number series expansion of the potential function for the two-dimensional flow of an inviscid, compressible, perfect, diatomic gas past a circular cylinder is obtained to 29 terms. Analysis of this expansion allows the critical Mach number, at which flow first becomes locally sonic, to be estimated as M* = 0.39823780 ± 0.00000001. Analysis also permits the following estimate of the radius of convergence of the series for the maximum velocity to be made: Mc = 0.402667605 ± 0.00000005, though we have been unable to determine the nature of the singularity of M = Mc. Since Mc exceeds M* by some 1.1%, it follows that this particular “airfoil” can possess a continuous range of shock-free potential flows above the critical Mach number. This result hopefully resolves a 70-year old controversy.

In a recent paper Weber et al. [9] examined the propagation of combustion waves in a semi-infinite gaseous or solid medium. Whereas their main concern was the behaviour of waves once they had been initiated, we concentrate here on the initiation of such waves in a solid medium and have not examined in detail the steadiness or otherwise of the waves subsequent to their formation. The investigation includes calculations for finite systems. The results for a slab, cylinder and sphere are compared.

Critical conditions for initiation of ignition by a power source are established. For a slab the energy input is spread uniformly over one boundary surface. In the case of cylindrical or spherical symmetry it originates from a cylindrical core or from a small, central sphere, respectively. The size of source and reactant body is important in the last two cases. With the exception of the initial temperature distribution, the equations investigated are similar in form to those of Weber et al. [5,9] and, as a prelude to the present study, with very simple adaptation, it has been possible to reproduce the results of the earlier work. We then go on to report the result of calculations for the initiation of ignition under different geometries with various initial and boundary conditions.

This paper is concerned with a reinvestigation of the problem of water wave scattering by a wall with multiple gaps by using the solution of a singular integral equation with a combination of logarithmic and power (Cauchy-type) kernels in disjoint multiple intervals. Use of Havelock's expansion of water wave potential reduces the problem to such an integral equation in the horizontal velocity across the gaps. The solution of the integral equation is obtained by utilizing the solutions of Cauchy-type integral equations in (0,∞) and also in multiple disjoint intervals. An explicit expression for the reflection coefficient is obtained for a wall with n gaps and supplemented by numerical results for up to three gaps.

In this paper, we use an ordinary differential equation approach to study the existence of similarity solutions for the equation u1 = Δ(uα) + θu–β in Rn × (0, ∞) where β > 0, θ ∈ [0, 1}, and n ≥ 1. This includes the slow diffusion equation when α > = 1, and the standard heat equation when α = 1, and the fast diffusion equation when 0 < α < 1. We prove that there are forward self-similar solutions for this equation with initial data of the form c|x|p, where p = 2/(α + β) if θ = 1; p ≥ 0 and 2 + (1 – α)p > 0 if θ = 0, for some positive constant c.

Certain definite integrals involving spherical Bessel functions are treated by relating them to Fourier integrals of the point multipoles of potential theory. The main result (apparently new) concerns

where l1, l2 and N are non-negative integers, and r1 and r2 are real; it is interpreted as a generalized function derived by differential operations from the delta function δ(r1 − r2). An ancillary theorem is presented which expresses the gradient ∇2nYlm(∇) of a spherical harmonic function g(r)YLM(Ω) in a form that separates angular and radial variables. A simple means of translating such a function is also derived.

We consider a family of two-point quadrature formulae, using some Euler-type identities. A number of inequalities, for functions whose derivatives are either functions of bounded variation, Lipschitzian functions or R-integrable functions, are proved.

The storage of bagasse, which is principally cellulose, presents many problems for the sugar industry, one of which is bagasse loss due to spontaneous combustion. This is an expensive problem for the industry as bagasse is used as a fuel by sugar mills, and for cogeneration of electricity. Self-heating occurs in the pile through an oxidation mechanism as well as a moisture dependent reaction. The latter reaction is now known to exhibit a local maximum, similar to the heat release curves found in cool-flame problems. Bagasse typically contains 45–55% by weight of water when milling is completed and the question of how to reduce the moisture content is important for two reasons. Firstly, wet bagasse does not burn nearly as efficiently as dry bagasse, and secondly, self-heating is greatly enhanced in the presence of water, for temperatures less than 60–70°C.

An existing mathematical model is used, but modified to take into account the newly observed peak in the moisture dependent reaction. Most of the previously reported complex bifurcation behaviour possible in this model is not realized when physically realistic parameter values are used. The bifurcation diagram describing the long-time steady-state solution is the familiar S-shaped hysteresis curve. In the presence of the new form of the moisture dependent reaction, an intermediate state can be found which is not a true steady-state of the system as, in reality, the characteristics of the pile slowly change as water is lost. This state corresponds to observations of an elevated temperature (around 60–70°C) which persists for long periods of time. Approximate equations can then be defined which predict this intermediate state, and hence a different hysteresis curve is found. A simple explanation for the process by which water is lost from the pile is obtained from these equations and an analytical expression is given for the exponential decay of water levels in the pile.

The known linear model reference adaptive control (MRAC) technique is extended to cover nonlinear and nonlinearizable systems (several equilibria, etc) and used to stabilize the system about a model. The method proposed applies the same Liapunov Design Technique but avoids the classical error equation. Instead it operates in the product of the state spaces of plant and model, aiming at convergence to a diagonal set. Control program, Liapunov functions and adaptive law are specified. The case is illustrated on a two-degrees of freedom robotic manipulator.

This paper considers the improvement of approximate eigenvalues and eigenfunctions of integral equations using the method of deferred correction. A convergence theorem is proved and a numerical example illustrating the theory is given.

Our purpose in this paper is to display the stability analysis of Runge–Kutta methods applied to a Volterra integral equation of a simple form. As prerequisite we define, and then develop the structure of, the class of Runge–Kutta methods considered. The test equation is taken as the “;basic” equation ; the simple form of this equation permits ready insight into features which are more obscure when considering (as elsewhere [1], [2], [6]) equations of a more complicated form. Due to the structure of the methods and the nature of the test equation, the stability analysis reduces to the study of recurrence relations of the form Фk + 1 = MФ k + γk (k = 0, 1, 2, …) which are common in stability discussions in numerical analysis.

The convexity assumptions for a minimax fractional programming problem of variational type are relaxed to those of a generalised invexity situation. Sufficient optimality conditions are established under some specific assumptions. Employing the existence of a solution for the minimax variational fractional problem, three dual models, the Wolfe type dual, the Mond-Weir type dual and a one parameter dual type, are constructed. Several duality theorems concerning weak, strong and strict converse duality under the framework of invexity are proved.

The bow flow generated by a wide flat-bottomed ship moving in water of finite depth is examined. Solutions obtained using an integral equation technique are presented for a range of different depths and for a range of angles of the front of the bow. The solution for the limiting case of infinite Froude number is obtained as an integral, and numerical solutions are found for the nonlinear problem in which the Froude number is finite. Solutions with smooth separation are shown to exist for all values of Froude number greater than unity, for any bow slope.

The steady response of the free surface of a fluid in a porous medium is considered during extraction of the fluid through a line sink. A conformal-mapping approach is used to derive exact solutions to a family of problems in which the line sink is placed at the apex of a wedge-shaped impermeable base, including the limiting cases of an unbounded aquifer and a flat-bottomed aquifer of finite depth. Both critical cusp solutions and sub-critical solutions are computed exactly as Fourier sine series.