A supermartingale relation for multivariate risk measures

Feinstein, Zachary; Rudloff, Birgit

doi:10.1080/14697688.2018.1459810

Cited by 13 publications

(21 citation statements)

References 56 publications

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“…Additional properties for multiportfolio time-consistency were studied in e.g. Feinstein and Rudloff [17,[20][21][22] and have been utilised directly for computing such risk measures in discrete time in Feinstein and Rudloff [19]. Though not the focus of this work, multiportfolio time-consistency has been extended to risk measures for processes in Chen and Hu [10].…”

Section: Literature Reviewmentioning

confidence: 99%

Set-valued risk measures as backward stochastic difference inclusions and equations

Ararat

Feinstein

2020

Finance Stoch

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Scalar dynamic risk measures for univariate positions in continuous time are commonly represented via backward stochastic differential equations. In the multivariate setting, dynamic risk measures have been defined and studied as families of set-valued functionals in the recent literature. There are two possible extensions of scalar backward stochastic differential equations for the set-valued framework:(1) backward stochastic differential inclusions, which evaluate the risk dynamics on the selectors of acceptable capital allocations; or (2) set-valued backward stochastic differential equations, which evaluate the risk dynamics on the full set of acceptable capital allocations as a singular object. In this work, the discrete-time setting is investigated with difference inclusions and difference equations in order to provide insights for such differential representations for set-valued dynamic risk measures in continuous time.

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Section: Literature Reviewmentioning

confidence: 99%

Set-valued risk measures as backward stochastic difference inclusions and equations

Ararat

Feinstein

2020

Finance Stoch

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“…From Theorems 3.1 and 4.2 and using results in [13,14], one can steadily deduce some equivalent characterizations of multiportfolio time consistency for set-valued dynamic risk measures for processes, such as the cocycle condition on the sum of minimal penalty function and supermartingale relation. By monotonicity, (ρ, R) is multiportfolio time consistent.…”

Section: Equivalence Of Multiportfolio Time Consistencymentioning

confidence: 99%

“…However, as demonstrated in those works, the equivalence of risk measures for processes and risk measures for random vectors no longer holds except under certain strong assumptions. This limits the application of known results for, e.g., multiportfolio time consistency as proven in [14,15] to set-valued risk measures for processes without rigorous direct proof.…”

Section: Introductionmentioning

confidence: 99%

Set-Valued Dynamic Risk Measures for Bounded Discrete-Time Processes

Chen

2020

Int. J. Theor. Appl. Finan.

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In this paper, we study how to evaluate the risk of a financial portfolio, whose components may be dependent and come from different markets or involve more than one kind of currencies, while we also take into consideration the uncertainty about the time value of money. Namely, we introduce a new class of risk measures, named set-valued dynamic risk measures for bounded discrete-time processes that are adapted to a given filtration. The time horizon can be finite or infinite. We investigate the representation results for them by making full use of Legendre–Fenchel conjugation theory for set-valued functions. Finally, some examples such as the set-valued dynamic average value at risk and the entropic risk measure for bounded discrete-time processes are also given.

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“…For the role of the optimal filtration and its use in defining risk measures of processes the interested reader should consult (Acciaio et al 2012). For recent results on dynamic aspects of systemic risk measures we refer to Biagini et al (2019), Feinstein et al (2017), Feinstein and Rudloff (2018) and Armenti et al (2018).…”

Section: Introductionmentioning

confidence: 99%

Dynamic systemic risk measures for bounded discrete time processes

2019

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The question of measuring and managing systemic risk-especially in view of the recent financial crisis-became more and more important. We study systemic risk by taking the perspective of a financial regulator and considering the axiomatic approach originally introduced in Chen et al. (Manag Sci 59(6):1373-1388, 2013 and extended in Kromer et al. (Math Methods Oper Res 84:323-357, 2016). The aim of this paper is to generalize the static approach in Kromer et al. ( 2016) and analyze systemic risk measures in a dynamic setting. We work in the framework of Cheridito et al. (Electron J Probab 11:57-106, 2006) who consider risk measures for bounded discrete-time processes. Apart from the possibility to consider the "evolution of financial values", another important advantage of the dynamic approach is the possibility to incorporate information in the risk measurement and management process. In context of this dynamic setting we also discuss the arising question of time-consistency for our dynamic systemic risk measures.

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A supermartingale relation for multivariate risk measures

Cited by 13 publications

References 56 publications

Set-valued risk measures as backward stochastic difference inclusions and equations

Set-valued risk measures as backward stochastic difference inclusions and equations

Set-Valued Dynamic Risk Measures for Bounded Discrete-Time Processes

Dynamic systemic risk measures for bounded discrete time processes

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