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Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical finance

  • Panagiota Daskalopoulos
  • Paul M. N. Feehan
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    The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is 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. With the aid of weighted Sobolev spaces, we prove existence, uniqueness, and global regularity of solutions to stationary variational inequalities and obstacle problems for the elliptic Heston operator on unbounded subdomains of the half-plane. In mathematical finance, solutions to obstacle problems for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.

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    File URL: http://arxiv.org/pdf/1109.1075
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    Paper provided by arXiv.org in its series Papers with number 1109.1075.

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    Date of creation: Sep 2011
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    Handle: RePEc:arx:papers:1109.1075
    Contact details of provider: Web page: http://arxiv.org/

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    1. Xinfu Chen & John Chadam & Lishang Jiang & Weian Zheng, 2008. "Convexity Of The Exercise Boundary Of The American Put Option On A Zero Dividend Asset," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 185-197.
    2. JosE Da Fonseca & Martino Grasselli & Claudio Tebaldi, 2008. "A multifactor volatility Heston model," Quantitative Finance, Taylor & Francis Journals, vol. 8(6), pages 591-604.
    3. Ekström, Erik, 2004. "Properties of American option prices," Stochastic Processes and their Applications, Elsevier, vol. 114(2), pages 265-278, December.
    4. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
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