Representation for Martingales Living after a Random Time with Applications

Tahir Choulli and Ferdoos Alharbi

Chapter 7 in Peter Carr Gedenkschrift:Research Advances in Mathematical Finance, 2023, pp 211-264 from World Scientific Publishing Co. Pte. Ltd.

Abstract: Our financial setting consists of a market model with two flows of information. The smallest flow 𝔽 is the “public” flow of information which is available to all agents, while the larger flow 𝔾 has additional information about the occurrence of a random time τ. This random time can model the default time in credit risk or death time in life insurance. Hence the filtration 𝔾 is the progressive enlargement of 𝔽 with τ. In this framework, when τ is a finite honest time, we describe explicitly how 𝔾-local martingales can be represented in terms of 𝔽-local martingales and parameters of τ. This representation complements the recent work of Choulli, Daveloose and Vanmaele to the case when martingales live “after τ.” Under some mild assumptions on the pair (𝔽, τ), we fully elaborate the application of these results to the explicit parametrization of all deflators under 𝔾. The results are illustrated in the case of a jump-diffusion model and a discrete-time market model.

Keywords: Mathematical Finance; Quantitative Finance; Option Pricing; Derivatives; No Arbitrage; Asset Price Bubbles; Asset Pricing; Equilibrium; Volatility; Diffusion Processes; Jump Processes; Stochastic Integration; Trading Strategies; Portfolio Theory; Optimization; Securities; Bonds; Commodities; Futures (search for similar items in EconPapers)
JEL-codes: C02 C6 (search for similar items in EconPapers)
Date: 2023
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