MULTIDIMENSIONAL DYNAMIC RISK MEASURE VIA CONDITIONAL g‐EXPECTATION

Abstract

This paper deals with multidimensional dynamic risk measures induced by conditional g-expectations. A notion of multidimensional g-expectation is proposed to provide a multidimensional version of nonlinear expectations. By a technical result on explicit expressions for the comparison theorem, uniqueness theorem, and viability on a rectangle of solutions to multidimensional backward stochastic differential equations, some necessary and sufficient conditions are given for the constancy, monotonicity, positivity, and translatability properties of multidimensional conditional gexpectations and multidimensional dynamic risk measures; we prove that a multidimensional dynamic g-risk measure is nonincreasingly convex if and only if the generator g satisfies a quasi-monotone increasingly convex condition. A general dual representation is given for the multidimensional dynamic convex g-risk measure in which the penalty term is expressed more precisely. It is shown that model uncertainty leads to the convexity of risk measures. As to applications, we show how this multidimensional approach can be applied to measure the insolvency risk of a firm with interacting subsidiaries; optimal risk sharing for γ -tolerant g-risk measures, and risk contribution for coherent g-risk measures are investigated. Insurance g-risk measure and other ways to induce g-risk measures are also studied at the end of the paper.

Optimal Trade Execution with Instantaneous Price Impact and Stochastic Resilience

“…A necessary and sufficient condition under which the comparison theorem holds for multidimensional BSDEs has been first given by Hu and Peng [14]. The equivalent quasi-monotonicity condition (iv) below can be found in [24,Theorem 3.1]. The comparison results in [14,24] are stated under an additional continuity condition on the drivers that is not satisfied in our model.…”

“…The equivalent quasi-monotonicity condition (iv) below can be found in [24,Theorem 3.1]. The comparison results in [14,24] are stated under an additional continuity condition on the drivers that is not satisfied in our model. However, the continuity condition is only needed to prove that if a comparison principle holds, then the system is necessarily quasi-monotone.…”

On the problem of stochastic decisions with backward stochastic viability property

“…Moreover, such risk measures are widely used for evaluating the risk of uncertain future outcomes or costs, and also assisting with stipulating minimum interventions for risk management (e.g., see [6], [3], [7] or [4] for related discussions). On the other hand, for decision problems involving multi-dimensional BSDEs coupled with forward-SDEs, there are some interesting studies, based on the stochastic backward viability property, that establish condition for the solutions of the associated PDEs (e.g., see [8] and [9] for additional discussions; see also [10] and [11] on the notion of viability properties for SDEs and inclusions). Note that the rationale behind our framework, which follows in some sense the settings of these papers, is to show how a backward stochastic viability property can be systematically used to obtain consistently optimal decision solutions.…”

Dynamic Risk Measures for Processes via Backward Stochastic Differential Equations Associated with Lévy Processes

“…Penner and Réveillac [34] established a link between risk measures for processes and BSDEs and studied the corresponding time-consistent dynamic risk measures for processes induced by BSDEs. Xu [35] studied multidimensional dynamic convex risk measures induced by conditional g-expectations. Ji et al [36] provided some time-consistent dynamic risk measures for processes via BSDEs, and established the oneto-one relationship between the generators BSDEs and the corresponding dynamic risk measures for processes.…”