### Abstract

We prove existence and uniqueness of stochastic representations for solutions to elliptic and parabolic boundary value and obstacle problems associated with a degenerate Markov diffusion process. In particular, our article focuses on the Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate, elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to terminal/boundary value or obstacle problems for the parabolic Heston operator correspond to value functions for American-style options on the underlying asset.

Original language | English (US) |
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Pages (from-to) | 981-1031 |

Number of pages | 51 |

Journal | Transactions of the American Mathematical Society |

Volume | 367 |

Issue number | 2 |

DOIs | |

State | Published - 2015 |

### Keywords

- Degenerate diffusion process
- Degenerate elliptic and parabolic differential operators
- Degenerate stochastic differential equation
- Feller square root process
- Feynman-Kac formula
- Heston stochastic volatility process
- Mathematical finance
- Stochastic representation formula

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## Cite this

*Transactions of the American Mathematical Society*,

*367*(2), 981-1031. https://doi.org/10.1090/s0002-9947-2014-06043-1