We consider the stochastic solution to a Cauchy problem corresponding to a nonnegative diffusion with zero drift, which represents a price process under some risk-neutral measure . When the diffusion coefficient is locally Holder continuous with some exponent in (0, 1], the stochastic solution  is shown to be a classical solution . A comparison theorem for the Cauchy problem is also proved, without the\ge  linear growth condition on the diffusion coefficient. Moreover, we establish the equivalence: the stochastic solution is the unique classical solution to the Cauchy problem if, and only if, a comparison theorem holds. For the case where the stochastic solution may not be smooth, we characterize it as a limit of smooth  stochastic solutions associated with some approximating Cauchy problems .
We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing . When the diffusion coefficient of the underlying price process is locally H\"older continuous with exponent \delta\in (0, 1], the stochastic solution , which represents the price of a European option, is shown to be a classical solution to the Cauchy problem . This improves the standard requirement \delta\ge 1/2. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth , it is characterized as the limit of an approximating smooth stochastic solutions . In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale .
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consider
study
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corresponding to a nonnegative diffusion with zero drift, which represents a price process under some risk-neutral measure
for a degenerate parabolic equation arising from option pricing
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is locally Holder continuous with some exponent in
of the underlying price process is locally H\"older continuous with exponent \delta\in
A
, which represents the price of a European option,
R
. A comparison theorem for
to
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is also proved, without the
. This improves the standard requirement \delta
A
1/2. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual
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Moreover, we establish the equivalence: the
When the
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the unique classical solution to the Cauchy problem if, and only if, a comparison theorem holds. For the case where the stochastic solution may not be smooth, we characterize it as a limit of
not
A
, it is characterized as the limit of an approximating smooth
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associated with some approximating Cauchy problems
. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale