Linear and nonlinear ultraparabolic equations of Kolmogorov type arising in diffusion theory and in finance

By Lanconelli Ermanno, Pascucci Andrea, Sergio Polidoro. 2002

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This paper contains a survey on a series of papers by the authors, dealing with linear and non linear Kolmogorov-type operators, arising in diffusion theory, probability and finance. Some new results, about existence for Cauchy problems, regularity properties and pointwise estimates of solutions, are also announced. 1 The Kolmogorov equation and the rank condition In its simplest form, Kolmogorov’s equation can be written as ∂ x1u + x1∂x2u− ∂tu = 0, (x1, x2, t) = (x, t) ∈ R × R. (1.1) The second order part in (1.1) is strongly degenerate due to the presence in it of the only term ∂2 x1 . However Kolmogorov constructed already in 1934 an explicit fundamental solution of (1.1) which is a C∞ function outside the diagonal [24]. This implies that (1.1) is hypoelliptic, i.e. every distributional solution to (1.1) in an open subset Ω of R3 actually is a C∞(Ω) function. As Hormander pointed out in the introduction of his celebrated paper on hypoelliptic second order differential equations [22], the Komogorov method can also be applied to the more general operator L = div(AD) + 〈x,BD〉 − ∂t, x ∈ R , t ∈ R, (1.2) where D = (∂x1 , . . . , ∂xN ) and 〈·, ·〉 denote, respectively, the gradient and the inner product in RN ; A = (aij) and B = (bij) are N ×N constant real matrices, and A is symmetric and non-negative defined. We remark that equation (1.1) can be written in the form (1.2) with A = ( 1 0 0 0 ) and B = ( 0 1 0 0 ) . ∗Piazza di Porta S. Donato 5, 40126 Bologna (Italy). E-mail: lanconel@dm.unibo.it, pascucci@dm.unibo.it, polidoro@dm.unibo.it †Investigation supported by the University of Bologna. Funds for selected research topics.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e350c668a810e971225c7665e83f9e5e0192ebca

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Citations in the corpus

Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term (2008, background): “7) where ∆Rm is the Laplace operator on R and B is a constant N ×N real matrix (see [15] and its bibliography for a survey on known results on Kolmogorov type operators.”
On the complete model with stochastic volatility by Hobson and Rogers (2004, background): “As in the classical theory, constant-coefficients KEs serve as an essential class of prototypes and many results can be extended to the general situation of variable coefficients by perturbation arguments (see, for instance, the survey [17]).”