Density estimates and short-time asymptotics for a hypoelliptic diffusion process

2020/04/14 by Paolo Pigato, Pigato, Paolo
Economics, Econometrics and Finance · Mathematics · #60F05 #60H07 #60H10 #60H30 #60J60 #91G20 #Complex Systems and Time Series Analysis #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2004.06541

openalex publication_date 2020/04/14 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We study a system of n differential equations, each in dimension d. Only the first equation is forced by a Brownian motion and the dependence structure is such that, under a local weak Hörmander condition, the noise propagates to the whole system. We prove upper bounds for the transition density (heat kernel) and its derivatives of any order. Then we give precise short-time asymptotics of the density at a suitable central limit time scale. Both these results account for the different non-diffusive scales of propagation in the various components. Finally, we provide a valuation formula for short-maturity at-the-money Asian basket options under correlated local volatility dynamics.

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