Time consistent dynamic risk processes

By Jocelyne Bion-Nadal. 2009

112 citations recorded by the source corpus.

A crucial property for dynamic risk measures is the time consistency. In this paper, a characterization of time consistency in terms of a "cocycle condition" for the minimal penalty function is proved for general dynamic risk measures continuous from above. Then the question of the regularity of paths is addressed. It is shown that, for a time consistent dynamic risk measure normalized and non-degenerate, the process associated with any bounded random variable has a cadlag modification, under a mild condition always satisfied in the case of continuity from below. When normalization is not assumed, a right continuity condition on the penalty has to be added. Applying these results and using right continuous BMO martingales, families of not necessarily normalized dynamic risk measures leading to cadlag paths, and allowing for jumps, are exhibited.

Subfields: Demography, Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/f2f4c620a3ce7b7c6e54b1cf6c6db439d6538f66

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Citations in the corpus

Optimal stopping for non-linear expectations—Part I (2011, unknown): “The stability is also referred to as “fork convexity” in stochastic control theory, “m-stability” in stochastic analysis or “rectangularity” in decision theory (see the introduction of [5,2] for details).”
Capacities, Measurable Selection and Dynamic Programming Part I: Abstract Framework (2013, methodology): “Then under appropriate conditions and by similar arguments as in Theorem 4.6, we can easily obtain the time consistency of ( E s,t ) s ≤ t of the form This formulation is closed to the dynamic risk measure proposed by Bion-Nadal [3], where α is called the minimal penalty function.”
Optimal Stopping for Dynamic Convex Risk Measures (2009, background): “1) was shown for Q P rather than Q ∼ P in [4].”
Risk measuring under model uncertainty (2010, background): “Dynamic risk measures have then been studied in many papers, among them [13], [11] [24] [7] [8] [30].”
Dynamic risk measures (2010, background): “3 summarizes robust representation results from [8, 18, 22].”
Bid-Ask Dynamic Pricing in Financial Markets with Transaction Costs and Liquidity Risk (2007, methodology): “For papers on dynamic risk measures we also refer to Delbaen [25] for the study of coherent (or homogeneous) dynamic risk measures, to Detlefsen and Scandolo [29] and Föllmer and Penner [38] for dynamic risk measures in a discrete time setting and to Bion-Nadal [12] and [13] for dynamic risk”
Lectures on Stochastic Programming: Modeling and Theory (2009, background): “In the finance literature time consistency is usually defined via the composition property of dynamic risk measures; see, Artzner, Delbaen, Eber, Heath and Ku [9], BionNadal [20, 21], Cheridito, Delbaen and Kupper [38], Cheridito and Kupper [39], Föllmer and Penner [79], Frittelli and Scandolo [81],”