Abstract
Inspired by the consideration of some inside and future market information in financial market, a class of anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs) are introduced to induce dynamic risk measures for risk quantification. The theory, including the existence, uniqueness and a comparison theorem for ABDSVIEs, is provided. Finally, dynamic convex risk measures by ABDSVIEs are discussed.
Highlights
It is well known that the concept of coherent risk measures with four axioms was first proposed to evaluate a risk position by [1], and further extended to convex risk measures by [2,3]
Under the framework of set-valued backward stochastic differential equations (BSDEs), Ref. [12] introduced and studied set-valued risk measures. Another kind of dynamic risk measures based on backward stochastic Volterra integral equations (BSVIEs) are worth exploring and studying
The theory, including the existence, uniqueness and a comparison theorem for anticipated backward doubly stochastic Volterra integral equation (ABDSVIE), is provided
Summary
It is well known that the concept of coherent risk measures with four axioms was first proposed to evaluate a risk position by [1], and further extended to convex risk measures by [2,3]. [12] introduced and studied set-valued risk measures Another kind of dynamic risk measures based on backward stochastic Volterra integral equations (BSVIEs) are worth exploring and studying. BSVIEs, as a generalized form of BSDEs, were initially considered to induce dynamic risk measures in [13], where the concept of M-solution was introduced to solve the problem of uniqueness to BSVIEs. Ref. In this work, inspired by the consideration of some inside and future market information in the financial market, a class of ABDSVIEs are introduced and used to induce dynamic risk measures for risk quantification.
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