Nagai, Hideo Robust estimates of certain large deviation probabilities for controlled semi-martingales. (English) Zbl 1325.60027 Chojnowska-Michalik, Anna (ed.) et al., Stochastic analysis. Special volume in honour of Jerzy Zabczyk. Selected papers based on the presentations at the Banach Center conference on stochastic analysis and control, Bȩdlewo, Poland, May 6–10, 2013. Warsaw: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-28-7/pbk). Banach Center Publications 105, 159-192 (2015). Summary: Motivated by downside risk minimization on the wealth process in an incomplete market model, we have studied in the recent work the asymptotic behavior as time horizon \(T\to \infty\) of the minimizing probability that the empirical mean of a controlled semi-martingale falls below a certain level on the time horizon \(T\). This asymptotic behavior relates to a risk-sensitive stochastic control problem in the risk-averse case. Indeed, we obtained an expression of the decay rate of the probability by the Legendre transform of the limit value of the value function of the stochastic control problem, which is characterized as the solution to the H-J-B equation of ergodic type. In the current work, we present the results on its robust version, admitting model uncertainty.For the entire collection see [Zbl 1323.60004]. MSC: 60F10 Large deviations 60G48 Generalizations of martingales 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H20 Stochastic integral equations 93E20 Optimal stochastic control 91G80 Financial applications of other theories 91G10 Portfolio theory 35K55 Nonlinear parabolic equations 35K10 Second-order parabolic equations Keywords:large deviation probabilities; controlled semi-martingales; Legendre transform; HJB equation; model uncertainty; risk minimization; wealth process; incomplete market model PDFBibTeX XMLCite \textit{H. Nagai}, Banach Cent. Publ. 105, 159--192 (2015; Zbl 1325.60027) Full Text: DOI