Risk measures for processes and BSDEs

Authors: Irina Penner Anthony Réveillac

Publish Date: 2014/09/17

Volume: 19, Issue: 1, Pages: 23-66

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Abstract

The paper analyzes risk assessment for cash flow processes in continuous time We combine the framework of convex risk measures for processes with a decomposition result for optional and predictable measures to provide a systematic approach to the issues of model ambiguity and uncertainty about the time value of money We also establish a link between risk measures for processes and BSDEsWe thank Kostas Kardaras and Michael Kupper for the very helpful comments and discussions during our work on this paper We also thank the referee the Associate Editor and the CoEditor for their comments and suggestions which improved the paper significantly Financial support from the DFG Research Center Matheon is gratefully acknowledged by both authorsWe first prove an auxiliary result on the extension of local martingales We apply here terminology and results from 27 Chap V For a given nondecreasing sequence of stopping times tau n ninmathbbN such that τ=lim n τ n is a predictable stopping time we consider a stochastic interval of the form Open image in new window The interval can be either open or closed at the right boundary τ Defining B=⋂ n τ n τ we have that Open image in new window on B and Open image in new window on B c We call a process L a local martingale resp a semimartingale a supermartingale on Open image in new window if for any stopping time σ such that Open image in new window the stopped process L σ is a local martingale resp a semimartingale a supermartingale The following lemma shows how a local martingale on the stochastic interval Open image in new window can be extended beyond it a related result for stochastic intervals of the simpler form 〚0τ〚 can be found in 11 Proposition A4 cf also 7 Lemma 610Let tau n ninmathbbN be a nondecreasing sequence of stopping times such that τ=lim n τ n is a predictable stopping time Assume further that L is a nonnegative local martingale on the stochastic interval Open image in new window Then there exists a càdlàg local martingale tildeL=tilde L t tin 0T such that tildeL=tildeLtau and L=tildeL on Open image in new window If we assume in addition that tildeL tildeD satisfies property 5 of the theorem we obtain also tildeL=tildeLtau=Ltau=L on Δa τ 0∪D τ−=0 and tildeD=tildeDtau=Dtau=D on Δa τ =0∩D τ−0 which proves equality in the sense of indistinguishability on 0TThe proof of the “if” part follows exactly as in Step 5 of the proof of Theorem 41 Obviously the process a defined by property 4′ is predictable and since D is predictable the equality A15 holds in the same way for mathbbE int 0sigma nL twedgesigma ndD tWe provide here estimates for the BSDEs 73 and 74 that are used in the proof of Lemma 81 The results for the quadratic BSDE 73 follow basically from 5 20 the results for the reflected BSDE 74 might be known but since we did not find them explicitly written in the literature we give the proofs here Throughout this section we consider the BSDE 73 under assumptions H1–H4 and the RBSDE 74 under assumptions H1’–H4

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