Abstract
In this paper, we propose the notion of continuous-time dynamic spectral risk measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk measures, which are obtained by iterating a given spectral risk measure (such as expected shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk measures driven by lattice random walks, under suitable scaling and vanishing temporal and spatial mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.
Highlights
Financial analysis and decision making rely on quantification and modelling of future risk exposures
We consider in this article a new class of such continuous-time dynamic coherent risk measures, which we propose to call dynamic spectral risk measures (DSRs)
Due to its continuous-time domain of definition, a DSR is, in contrast, independent of a grid structure. While the latter holds for any continuous-time risk measure, we show that DSRs emerge as the limits of such iterated spectral risk measures when the time-step vanishes and under appropriate scaling of the parameters, by establishing a functional limit theorem
Summary
Financial analysis and decision making rely on quantification and modelling of future risk exposures. A subsequent breakthrough was the development and application of the notion of backward stochastic differential equations (BSDEs) in the context of risk analysis, which gave rise to the (strongly) time-consistent extension of coherent risk measures to continuous-time dynamic settings [39, 42]. The notion of strong time-consistency in economics goes back at least as far as [46] and has been standard in the economics literature ever since; see for instance [9, 20, 23, 24, 27, 32, 33] Due to their recursive structure, financial optimisation problems, such as utility optimisation under the entropic risk measure and related robust portfolio optimisation problems, satisfy the dynamic programming principle and admit time-consistent dynamically optimal strategies (see for instance [5, 36] and references therein).
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