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We prove the existence of the fundamental solution of the degenerate second order partial differential equation related to Geman–Yor stochastic processes, that arise in models for option pricing theory in finance. We then prove pointwise lower and upper bounds for such fundamental solution. Lower bounds are obtained by using repeatedly an invariant Harnack inequality and by solving an associated optimal control problem with quadratic cost. Upper bounds are obtained by the fact that the cost satisfies a specific Hamilton–Jacobi–Bellman equation.

We consider possibly degenerate parabolic operators in the form $$ \sum_{k=1}^{m}X_{k}^{2}+X_{0}-\partial_{t}, $$ that are naturally associated to a suitable family of stochastic differential equations, and satisfying the H\"ormander condition. Note that, under this assumption, the operators in the form $\L$ has a smooth fundamental solution that agrees with the density of the corresponding stochastic process. We describe a method based on Harnack inequalities and on the construction of Harnack chains to prove lower bounds for the fundamental solution. We also briefly discuss PDE and SDE methods to prove analogous upper bounds. We eventually give a list of meaningful examples of operators to which the method applies., Comment: 23 pages, 5 figures, proceedings of the conference "Modern methods of stochastic analysis and statistics" (Moscow, 30 May - 1 June 2016)