2017/12/18 by D. J. Daley, Daley, Daryl J., Masakiyo Miyazawa +1 · 1 citation
Economics, Econometrics and Finance · Social Sciences · #60J27 #60K25 #60K37 #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1712.06278
openalex publication_date 2017/12/18 · openalex created_date 2018/01/05 · openalex updated_date 2026/07/28
Martingales constitute a basic tool in stochastic analysis; this paper considers their application to counting processes. We use this tool to revisit a renewal theorem and its extensions for various counting processes. We first consider a renewal process as a pilot example, deriving a new semimartingale representation that differs from the standard decomposition via the stochastic intensity function. We then revisit Blackwell's renewal theorem, its refinements and extensions. Based on these observations, we extend the semimartingale representation to a general counting process, and give conditions under which asymptotic behaviour similar to Blackwell's renewal theorem holds.