2015/09/13 by Paul M. N. Feehan, Feehan, Paul M. N., Ruoting Gong +3
Economics, Econometrics and Finance · Mathematics · Social Sciences · #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Insurance, Mortality, Demography, Risk Management
paper · pdf · doi:10.48550/arxiv.1509.03864
We prove Feynman-Kac formulas for solutions to elliptic and parabolic\nboundary value and obstacle problems associated with a general Markov diffusion\nprocess. Our diffusion model covers several popular stochastic volatility\nmodels, such as the Heston model, the CEV model and the SABR model, which are\nwidely used as asset pricing models in mathematical finance. The generator of\nthis Markov process with killing is a second-order, degenerate, elliptic\npartial differential operator, where the degeneracy in the operator symbol is\nproportional to the 2\α-power of the distance to the boundary of the\nhalf-plane, with \α\∈(0,1]. Our stochastic representation formulas\nprovide the unique solutions to the elliptic boundary value and obstacle\nproblems, when we seek solutions which are suitably smooth up to the boundary\nportion \Γ0 contained in the boundary of the upper half-plane. In the\ncase when the full Dirichlet condition is given, our stochastic representation\nformulas provide the unique solutions which are not guaranteed to be any more\nthan continuous up to the boundary portion \Γ0.\n