On the integro-differential equations of purely discontinuous Markoff processes

Purely Discontinuous Asset Price Processes

This paper presents the case for modeling asset price processes as purely discontinuous processes of ̄nite variation with an in ̄nite arrival rate of jumps that have arrival rates completely monotone in the jump size. The arguments address both the empirical realities of asset returns and the implications of the economic principle of no arbitrage. Two classes of economic models meeting these con...

On Discontinuous Differential Equations

Theorem A. Let g : [0, T ]× IR 7→ IR be a bounded function. (i) If the map t 7→ g(t, x) is measurable for each x and the map x 7→ g(t, x) is continuous for each t, then the Cauchy problem (1.1) has at least one solution. (ii) If the map t 7→ g(t, x) is measurable for each x and the map x 7→ g(t, x) is Lipschitz continuous for each t, with a uniform Lipschitz constant, then the Cauchy problem (1...

Nonlinear Integro - Differential Equations

Backward Stochastic Differential Equations Associated with Lévy Processes and Partial Integro-differential Equations

In this paper, we deal with a class of backward stochastic differential equations driven by Teugels martingales associated with a Lévy process (BSDELs). The comparison theorem is obtained. It is also shown that the solution of BSDE provides a viscosity solution of the associated system with partial integro-differential equations.

Application of the block backward differential formula for numerical solution of Volterra integro-differential equations

In this paper, we consider an implicit block backward differentiation formula (BBDF) for solving Volterra Integro-Differential Equations (VIDEs). The approach given in this paper leads to numerical methods for solving VIDEs which avoid the need for special starting procedures. Convergence order and linear stability properties of the methods are analyzed. Also, methods with extensive stability r...