A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance

We present a simple and easy to implement method for the numerical solution of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many cases, the considered problems have only a viscosity solution, to which, fortunately, many intuitive (e.g. finite difference based) discretisations can be shown to converge. However, especially when using fully implicit time stepping schemes with their desirable stability properties, one is still faced with the considerable task of solving the resulting nonlinear discrete system. In this paper, we introduce a penalty method which approximates the nonlinear discrete system to first order in the penalty parameter, and we show that an iterative scheme can be used to solve the penalised discrete problem in finitely many steps. We include a number of examples from mathematical finance for which the described approach yields a rigorous numerical scheme and present numerical results.Comment: 18 Pages, 4 Figures. This updated version has a slightly more detailed introduction. In the current form, the paper will appear in SIAM Journal on Numerical Analysi

Isomonodromic deformation theory and the next-to-diagonal correlations of the anisotropic square lattice Ising model

In 1980 Jimbo and Miwa evaluated the diagonal two-point correlation function of the square lattice Ising model as a $\tau$-function of the sixth Painlev\'e system by constructing an associated isomonodromic system within their theory of holonomic quantum fields. More recently an alternative isomonodromy theory was constructed based on bi-orthogonal polynomials on the unit circle with regular semi-classical weights, for which the diagonal Ising correlations arise as the leading coefficient of the polynomials specialised appropriately. Here we demonstrate that the next-to-diagonal correlations of the anisotropic Ising model are evaluated as one of the elements of this isomonodromic system or essentially as the Cauchy-Hilbert transform of one of the bi-orthogonal polynomials.Comment: 11 pages, 1 figur

Phase transitions during membrane formation of polylactides. I. A morphological study of membranes obtained from the system polylactide-chloroform-methanol

The influence of solid-liquid demixing, liquid-liquid demixing and vitrification on the morphology of polylactide membranes has been investigated. To study the effects of crystallization of polylactides on the membrane and morphology, polylactides of varying stereoregularity were used. The polymers applied were poly--lactide (PLLA) and copolymers with different molar ratios of -lactide and -lactide [poly-L95/D5-lactide (PLA95), poly-L80/D20-lactide (PLA80) and poly-L50/D50-lactide (PDLLA)]. Solutions of polylactides in chloroform cast on a glass plate were immersed in methanol. From solutions containing the slowly crystallizing PLA80 or uncrystallizable PDLLA porous membranes were obtained if the phase separated system was removed from the nonsolvent bath within a few hours after immersion. After longer equilibration times in methanol the structure collapsed. The swelling in the nonsolvent methanol was too high to allow stabilization of the liquid-liquid demixed structure by vitrification. Stable membranes were easily obtained with more rapidly crystallizing polymers like PLLA. Casting solutions with low PLLA concentrations gave membranes with a cellular morphology due to liquid-liquid demixing by nucleation and growth of a polymer poor phase. Crystallization only played a role in the fixation of the liquid-liquid demixed structure. At increasing PLLA concentrations the demixing sequence gradually reversed to crystallization followed by liquid-liquid demixing. In these cases membranes with porous spherulites or spherulites surrounded with a cellular layer were obtained

A morphological study of membranes obtained from the systems polylactide-dioxane-methanol, polylactide-dioxane-water and polylactide-N-methyl pyrrolidone-water

The influence of liquid-liquid demixing, solid-liquid demixing, and vitrification on the membrane morphologies obtained from several polylactide-solvent-nonsolvent systems has been investigated. The polymers investigated were the semicrystalline poly-L-lactide (PLLA) and the amorphous poly-DL-lactide (PDLLA). The solvent-nonsolvent systems used were dioxane-water, N-methyl pyrrolidone-water and dioxane-methanol. For each of these systems it was attempted to relate the membrane morphology to the ternary phase diagram at 25°C. It was demonstrated that for the amorphous poly-DL-lactide the intersection of a glass transition and a liquid-liquid miscibility gap in the phase diagram was a prerequisite for the formation of stable membrane structures. For the semicrystalline PLLA a wide variety of morphologies could be obtained ranging from cellular to spherulitical structures. For membrane-forming combinations that show delayed demixing, trends expected on the basis of phase diagrams were in reasonable agreement with the observed membrane morphologies. Only for the rapidly precipitating system PLLA-N-methyl pyrrolidone-water were structures due to liquid-liquid demixing obtained when structures due to solid-liquid demixing were expected. Probably, rapid precipitation conditions promote solid-liquid demixing over liquid-liquid demixing, because the activation energy necessary for liquid-liquid demixing is lower than that for crystallization