Abstract. We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jump-diffusion processes, which are fully nonlinear integro-partial differential equations. Two main results are obtained: (i) error bounds for a class of monotone approximation schemes, which under some assumptions includes finite difference schemes, and (ii) bounds on the error induced when the original Lévy measure is replaced by a finite measure with compact support, an approximation process that is commonly used when designing numerial schemes for integro-partial differential equations. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen.

We study the numerical approximation of viscosity solutions for integro-differential, possibly degenerate, parabolic problems. Similar models arise in option pricing, to generalize the celebrated Black-Scholes equation, when the processes which generate the underlying stock returns may contain both a continuous part and jumps. Convergence is proven for monotone schemes and numerical tests are presented and discussed.Mathematics Subject Classification (1991): 65M12, 35K55, 49L25

Abstract. We prove a comparison principle for unbounded semicontinuous viscosity sub-and supersolutions of nonlinear degenerate parabolic integro-partial differential equations coming from applications in mathematical finance in which geometric Lévy processes act as the underlying stochastic processes for the assets dynamics. As a consequence of the "geometric form" of these processes, the comparison principle holds without assigning spatial boundary data. We present applications of our result to (i) backward stochastic differential equations and (ii) pricing of European and American derivatives via backward stochastic differential equations. Regarding (i), we extend previous results on backward stochastic differential equations in a Lévy setting and the connection to semilinear integro-partial differential equations.

We study a finite-dimensional approach to the Heath-Jarrow-Morton model for interest rate and introduce a notion of approximate consistency for a family of functions in a deterministic and stochastic framework. This amounts to asking the decrease of the minimum distance in least squares sense. We start from a general linearly parameterized set of functions and extend the theory to a nonlinear Nelson-Siegel family. Necessary and sufficient condition to have approximately consistency are given as well as a criterion of stability for the approximation.

We describe a model for the optimization of the issuances of Public Debt securities developed together with the Italian Ministry of Economy and Finance. The goal is to determine the composition of the portfolio issued every month which minimizes a specific "cost function". Mathematically speaking, this is a stochastic optimal control problem with strong constraints imposed by national regulations and the Maastricht treaty. The stochastic component of the problem is represented by the evolution of interest rates. At this time the optimizer employs classic Linear Programming techniques. However more sophisticated techniques based on Model Predictive Control strategies are under development.