Sergio Polidoro

SOME RESULTS ON PARTIAL DIFFERENTIAL EQUATIONS AND ASIAN OPTIONS

We analyze partial differential equations arising in the evaluation of Asian options. The equations are strongly degenerate partial differential equations in three dimensions. We show that the solution of the no-arbitrage partial differential equation is sufficiently regular and standard numerical methods can be employed to approximate it.

The obstacle problem for a class of hypoelliptic ultraparabolic equations

We study the obstacle problem for a class of degenerate parabolic operators with continuous coefficients. This problem arises in the Black–Scholes framework when considering path-dependent American options. We prove the existence of a unique strong solution u to the Cauchy and Cauchy–Dirichlet problems, under rather general assumptions on the obstacle function. We also show that u is a solution in the viscosity sense.

THE MOSER'S ITERATIVE METHOD FOR A CLASS OF ULTRAPARABOLIC EQUATIONS

We adapt the iterative scheme by Moser, to prove that the weak solutions to an ultraparabolic equation, with measurable coefficients, are locally bounded functions. Due to the strong degeneracy of the equation, our method differs from the classical one in that it is based on some ad hoc Sobolev type inequalities for solutions.

Gaussian estimates for hypoelliptic operators via optimal control

— We obtain Gaussian lower bounds for the fundamental solution of a class of hypoelliptic equations, by using repeatedly an invariant Harnack inequality. Our main result is given in terms of the value function of a suitable optimal control problem.

Hölder Regularity for Solutions of Ultraparabolic Equations in Divergence Form

AbstractWe consider the second order differential equation

On the Harnack inequality for a class of hypoelliptic evolution equations

\sum\limits_{i,j = 1}^{m_0 } {\partial _{x_i } (a_{i,j} (x,t)\partial _{x_j } u) + \sum\limits_{i,j = 1}^N {b_{i,j} x_i \partial _{x_j } u - \partial _t u = \sum\limits_{j = 1}^{m_0 } {\partial _{x_j } F_j (x,t)} } }

Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term

, where (x,t)∈ℝN+1, 0<m0⩽N, the coefficients ai,j belong to a suitable space of vanishing mean oscillation functions VMOL and B=(bi,j) is a constant real matrix. The aim of this paper is to study interior regularity for weak solutions to the above equation assuming that Fj belong to a function space of Morrey type.

A finite difference method for a boundary value problem related to the Kolmogorov equation

We prove a global Harnack inequality for a class of degenerate evolution operators by using repeatedly an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators.