G5 Doctoral dissertation (article)
Weighted BMO, Riemann-Liouville type operators, and approximation of stochastic integrals in models with jumps (2020)

Nguyen, T. T. (2020). Weighted BMO, Riemann-Liouville type operators, and approximation of stochastic integrals in models with jumps [Doctoral dissertation]. Jyväskylän yliopisto. JYU dissertations, 328. http://urn.fi/URN:ISBN:978-951-39-8442-7

JYU authors or editors

Publication details

All authors or editors: Nguyen, Tran Thuan

eISBN: 978-951-39-8442-7

Journal or series: JYU dissertations

eISSN: 2489-9003

Publication year: 2020

Number in series: 328

Number of pages in the book: 1 verkkoaineisto (vi, 67 sivua, 121 sivua useina numerointijaksoina)

Publisher: Jyväskylän yliopisto

Place of Publication: Jyväskylä

Publication country: Finland

Publication language: English

Persistent website address: http://urn.fi/URN:ISBN:978-951-39-8442-7

Publication open access: Openly available

Publication channel open access: Open Access channel

Abstract

This thesis investigates the interplay between weighted bounded mean oscillation (BMO), Riemann–Liouville type operators applied to càdlàg processes, real interpolation, gradient type estimates for functionals on the Lévy–Itô space, and approximation for stochastic integrals with jumps. There are two main parts included in this thesis. The first part discusses the connections between the approximation problem in L2 or in weighted BMO, Riemann–Liouville type operators, and the real interpolation theory in a general framework (Chapter 3). The second part provides various applications of results in the first part to several models: diffusions in the Brownian setting (Section 3.5) and certain jump models (Chapter 4) for which the (exponential) Lévy settings are typical examples (Chapter 6 and Chapter 7). Especially, for the models with jumps we propose a new approximation scheme based on an adjustment of the Riemann approximation of stochastic integrals so that one can effectively exploit the features of weighted BMO. In our context, making a bridge from the first to the second part requires gradient type estimates for a semigroup acting on Hölder functions in both the Brownian setting (Section 3.5) and the (exponential) Lévy setting (Chapter 5). In the latter case, we consider a kind of gradient processes appearing naturally from the Malliavin derivative of functionals of the Lévy process, and we show how the gradient behaves in time depending on the “direction” one tests.

Keywords: stochastic processes; integral calculus; interpolation; approximation

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2020