2015/10/20 by Tahir Choulli, Choulli, Tahir, Catherine Daveloose +3
Economics, Econometrics and Finance · Social Sciences · #FOS: Economics and business #FOS: Mathematics #Financial Risk and Volatility Modeling #Insurance, Mortality, Demography, Risk Management #Mathematical Finance (q-fin.MF) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1510.05858
openalex publication_date 2015/10/20 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We consider a market model where there are two levels of information. The\npublic information generated by the financial assets, and a larger flow of\ninformation that contains additional knowledge about a random time. This random\ntime can represent many economic and financial settings, such as the default\ntime of a firm for credit risk, and the death time of an insured for life\ninsurance. By using the expansion of filtration, the random time uncertainty\nand its entailed risk are fully considered without any mathematical\nrestriction. In this context with no model's specification for the random time,\nthe main challenge lies in finding the dynamics and the structures for the\nvalue processes of defaultable or mortality and/or longevity securities which\nare vital for the insurance securitization. To overcome this obstacle, we\nelaborate our optional martingale representation results, which state that any\nmartingale in the large filtration stopped at the random time can be decomposed\ninto precise and unique orthogonal local martingales (i.e. local martingales\nwhose product remains a local martingale). This constitutes our first and\nprobably the principal contribution. Even though the driving motivation for\nthis representation resides in credit risk theory, our results are applicable\nto several other financial and economics contexts, such as life insurance and\nfinancial markets with random horizon. Thanks to this optional representation,\nwe decompose any defaultable or mortality and/or longevity liability into the\nsum of "non-correlated" risks using a risk basis. This constitutes our second\ncontribution.\n