no. 3
Variational analysis for the Black and Scholes equation with stochastic volatility
ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 3, pp. 373-395.

We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation 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 Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods.

DOI : 10.1051/m2an:2002018
Classification : 91B28, 91B24, 35K65, 65M06, 65M60
Mots clés : degenerate parabolic equations, european options, weighted Sobolev spaces, finite element and finite difference method 
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     title = {Variational analysis for the {Black} and {Scholes} equation with stochastic volatility},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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     volume = {36},
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Achdou, Yves; Tchou, Nicoletta. Variational analysis for the Black and Scholes equation with stochastic volatility. ESAIM: Modélisation mathématique et analyse numérique, Tome 36 (2002) no. 3, pp. 373-395. doi : 10.1051/m2an:2002018. http://archive.numdam.org/articles/10.1051/m2an:2002018/

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