 
Summary: MATHEMATICS OF COMPUTATION
Volume 00, Number 0, Pages 000{000
S 00255718(XX)00000
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A PARTIAL DIFFERENTIAL EQUATION CONNECTED TO
OPTION PRICING WITH STOCHASTIC VOLATILITY:
REGULARITY RESULTS AND DISCRETIZATION
YVES ACHDOU, BRUNO FRANCHI, AND NICOLETTA TCHOU
Abstract. This paper completes a previous work [1] on a Black and Scholes
equation with stochastic volatility. This is a degenerate parabolic equation,
which gives the price of a European option as a function of the time, of the
price of the underlying asset and of the volatility, when the volatility is a
function of a mean reverting OrsteinUhlenbeck process, possibly correlated
with the underlying asset. The analysis involves weighted Sobolev spaces. We
give a characterization of the domain of the operator, which permits to use
results from the theory of semigroups. We then study a related model elliptic
problem and we propose a nite element method with a regular mesh with
respect to the intrinsic metric associated with the degenerate operator. For
the error estimate, we need to prove an approximation result.
1. Introduction
