## Differential and Integral Equations

- Differential Integral Equations
- Volume 14, Number 6 (2001), 701-738.

### On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance

Giovanna Citti, Andrea Pascucci, and Sergio Polidoro

#### Abstract

We consider the following nonlinear degenerate parabolic equation which arises in some recent problems of mathematical finance: $$ \partial_{xx} u + u \partial_{y} u - \partial_{t} u =f. $$ Using a harmonic analysis technique on Lie groups, we prove that, if the solution $u$ satisfies condition $\partial_x u \neq 0$ in an open set $\Omega \subset \mathbb R^3$ and $f \in C^{\infty}(\Omega)$, then $u \in C^{\infty}(\Omega)$.

#### Article information

**Source**

Differential Integral Equations, Volume 14, Number 6 (2001), 701-738.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356123243

**Mathematical Reviews number (MathSciNet)**

MR1826957

**Zentralblatt MATH identifier**

1013.35046

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35A30: Geometric theory, characteristics, transformations [See also 58J70, 58J72] 35K65: Degenerate parabolic equations 35Q80: PDEs in connection with classical thermodynamics and heat transfer 91B16: Utility theory

#### Citation

Citti, Giovanna; Pascucci, Andrea; Polidoro, Sergio. On the regularity of solutions to a nonlinear ultraparabolic equation arising in mathematical finance. Differential Integral Equations 14 (2001), no. 6, 701--738. https://projecteuclid.org/euclid.die/1356123243