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.

Pigato, P. (2022). Density estimates and short-time asymptotics for a hypoelliptic diffusion process. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 145, 117-142 [10.1016/j.spa.2021.11.012].

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

Pigato, P
2022-01-01

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.
2022
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/06 - PROBABILITA' E STATISTICA MATEMATICA
Settore SECS-S/06 - METODI MATEMATICI DELL'ECONOMIA E DELLE SCIENZE ATTUARIALI E FINANZIARIE
Settore STAT-04/A - Metodi matematici dell'economia e delle scienze attuariali e finanziarie
Settore MATH-03/B - Probabilità e statistica matematica
English
Pigato, P. (2022). Density estimates and short-time asymptotics for a hypoelliptic diffusion process. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 145, 117-142 [10.1016/j.spa.2021.11.012].
Pigato, P
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/284833
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