Preservation of convexity of solutions to parabolic equations

Abstract

In the present paper we find necessary and sufficient conditions on the coefficients of a parabolic equation for convexity to be preserved. A parabolic equation is said to preserve convexity if given a convex initial condition, any solution of moderate growth remains a convex function of the spatial variables for each fixed time.

General properties of solutions to inhomogeneous Black–Scholes equations with discontinuous maturity payoffs

“…The theorem 3 generalizes the well-known convexity preservation results of standard Black-Scholes equations to inhomogenous cases. Janson et al [12] studied preservation of convexity of solutions to general inhomogeneous parabolic equations but their result didn't cover ours because of regularity assumption of coefficients.…”

Superreplication of Options on Several Underlying Assets

“…Remark 2.1. The assumption that β is C 3 is unnecessarily strong -see Section 3 of Janson and Tysk (2004). To clarify the presentation, however, we will continue to assume this.…”

“…This equation is a fully nonlinear parabolic equation, which, for convex onedimensional claims, reduces to the usual Black-Scholes equation, once again illustrating the fact that superhedging is a much simpler problem in one dimension than in several. Janson and Tysk (2004) considered second-order parabolic differential equations of the form G(x, t), where the differential operator…”

“…If L satisfies the condition it is said to be locally convexity preserving (LCP). In the present paper we use some of the results of Janson and Tysk (2004) to study the problem of convexity-preserving models in finance. The results of Janson and Tysk cannot be used directly in our setting, however.…”

General Properties of Solutions to Inhomogeneous Black-Scholes Equations with Discontinuous Maturity Payoffs and Application

“…Ekström et al [7] provided a counter example of the solution not preserving convexity when the stock price follows a jump-diffusion process and studied the condition to preserve convexity. All the above mentioned works dealt with homogenous Black-Scholes (integro-) differential equations while Janson et al [12] studied preservation of convexity of solutions to general (inhomogeneous) parabolic equations under a regularity assumption of coefficients.…”