Dynamic Risk Measures for Anticipated Backward Doubly Stochastic Volterra Integral Equations

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.


Introduction
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]. In addition to these static risk measures, a class of dynamic risk measures based on backward stochastic differential equations (BSDEs) has been widely studied.
The general theory of BSDEs, first established by [4], can be used to construct dynamic risk measures for risk evaluation in the field of finance and insurance. For instance, in a static and dynamic framework, Ref. [5] discussed a class of risk measures based on the theory of g-expectation of BSDEs, which was established by [6]. With the help of g-expectation of BSDEs, some equivalent characterizations of dynamic risk measures were provided in [7], and further developed by [8,9] to dynamic risk measures for processes. Ref. [10] extended dynamic risk measures induced by the g-expectation of BSDEs to the multidimensional case. Recently, Ref. [11] studied dynamic risk measures related to BSDEs with jumps under an enlargement of filtration, and presented a numerical approach for them. Moreover, under the framework of set-valued 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. 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. [14] stated that dynamic risk measures induced by BSVIEs are time-inconsistent. The time-inconsistent dynamic risk measures for BSVIEs with jumps were considered by [15,16]. Ref. [17] discussed equilibrium dynamic risk measures induced by quadratic BSVIEs and explored equilibrium recursive utility processes.
Dynamic risk measures based on the classical BSDEs and BSVIEs have their own advantages. Compared with risk measures by the classical BSDEs, risk measures by BSVIEs allow the terminal objective of a wealth process (which is commonly described by a random variable in the classical BSDEs seting) to be described by a stochastic process (which only needs to be measurable to the information at terminal time). Risk measures by BSVIEs also consider the time value of a wealth process. Unfortunately, risk measures by BSVIEs are time-inconsistent, not time-consistent, while risk measures based on the classical BSDEs are time-consistent in a sense.
In recent years, BSVIEs have been widely studied by many other researchers. For example, Ref. [18] proved the existence and uniqueness of BSVIEs with jumps in Hilbert spaces. The unique solvability of BSVIEs under more general stochastic non-Lipschitz conditions was shown in [19]. Furthermore, Ref. [20] provided some theoretical research on backward doubly stochastic Volterra integral equations (BDSVIEs), including the well-posedness of solutions, a comparison theorem, and a related optimization problem. The similar work for anticipated BSVIEs, compared with [20], was presented in [21]. Refs. [22,23] considered extended BSVIEs and some related control problems.
In this work, we introduce the following Volterra integral equation to simulate a wealth process, (1) Here, ξ is a given R n -valued stochastic process, satisfy some given conditions, and ζ(·) and δ(·) are some given R + -valued continuous functions satisfying s + ζ(s) ≤ T + K, s + δ(s) ≤ T + K, s ∈ [0, T]. The above equation is called anticipated backward doubly stochastic Volterra integral equation (ABDSVIE). ξ is called the terminal condition (or sometimes the free term), and f (·) and g(·) are called the generator or the driver of (1). ABDSVIEs can be seen as a generalized form of BSVIEs and BDSVIEs. ABDSVIEs can also be seen as a generalized form of the class of Volterra integral equations (VIEs) of the first kind, which belong to the class of so-called weakly regular integral equations. In [24], this monograph systematically provided the detailed related theories and methods, including VIEs of the first/second kind, the control and optimization problem, and the various applications of Volterra models. In [25], they applied some special nonlinear VIEs to an optimal control of a multifactor vintage capital model. To the best of our knowledge, no study about ABDSVIEs is available up to now. Now, we interpret the motivation for introducing ABDSVIEs. Some inside and future market information is considered in the financial market. First, there may be some inside information, which can be only observed by some specific investors in the derivatives market. This inside information decreases as the trading day approaches. In other words, this inside information will be observed by all investors on the trading day. Therefore, this inside information, which may not be observed in practice, can be simulated by a Brownian motion B with a backward Itô integral. Second, because these specific investors have more market information than ordinary investors, the wealth process Y can be predicted by these specific investors for a short period of time in the future. This shows that the generator contains not only the current market information, but also the future information. Based on the above idea, we introduce ABDSVIEs (1) to derive dynamic risk measures for the risk quantification of the wealth process of some investors.
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. The theory, including the existence, uniqueness and a comparison theorem for ABDSVIEs, is provided. Finally, dynamic risk measures by ABDSVIEs are presented. The theory of ABDSVIEs extends the model and results of [20,21]. This paper is organized as follows. Section 2 provides some preliminaries including the definition of M-solution, and some results on BDSVIEs. Section 3 contains our main results; that is, the theory for ABDSVIEs including the existence, uniqueness, a comparison theorem, and their applications in risk measures are presented. All proofs of the main results of this paper are addressed in Section 4. Finally, conclusions are summarized.
Let the class of P-null sets of F be denoted by N .
where F ϑ . Let H represent Euclidean space. |x| denotes its Euclidean norm for any x ∈ R n , and let |A| be defined by |A| = √ TrAA * for A ∈ R n×d . We introduce the following spaces. respectively. Similarly, we can define L 2 Consider the following equtions: where (ξ(·), f (·), g(·)) satisfies the following assumptions.
There exists a L > 0 and 0 < α < 1 T+2 such that for any y 1 , (2) is not a filtration. In [20], based on the idea of M-solution to BSVIEs, the authors defined a filtration G to introduce the M-solution to BDSVIEs.
(ii) For any t ∈ [0, T] and any nonnegative integrable functions f 1 (·) and f 2 (·), there has a constant Γ ≥ 0 with There has a L > 0 and 0 < α < 1 T+2 satisfying for any y 1 , Now, we define a class of dynamic risk measures for ABDSVIEs. By slightly modifying the definition of dynamic risk measures in [13], the terminal time of dynamic risk measures can be extended to the time T + K. The detail definition of dynamic risk measures is as follows.

Main Results
Now, we present some main results concerning ABDSVIEs. Namely, we first present the well-posedness of ABDSVIEs, and then a comparison theorem for ABDSVIEs will be given. Finally, dynamic convex risk measures via ABDSVIEs will be discussed.

Remark 3.
For t ∈ [0, T + K], ξ(t) is not F T∨t -measurable, but F t -adapted in Theorem 1 and Corollary 1. In fact, we suppose that ξ(t) is F t -adapted just for simplicity of presentation, and the results of Theorem 1 and Corollary 1 remain true when ξ(t) is F T∨t -measurable.
Furthermore, the second main result concerning a comparison theorem for ABDSVIEs will be presented in a one-dimensional setting.

s, Y(s), Z(t, s), Y(s + ζ(s)) ds
Let f (t, s, y, z, ψ) : where r(·) is R + -valued deterministic function. Now, we state dynamic risk measures via ABDSVIEs and give the following main theorem.
Thus, the result has been completed.

Conclusions
In this paper, based on the consideration of the existence 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. The theory, including the existence, uniqueness and a comparison theorem for ABDSVIEs, is provided. Finally, dynamic risk measures by ABDSVIEs are presented. Now, we are not sure whether ABDSVIEs can be used to optimize the multifactor vintage capital model of [25][26][27]. It is a very interesting and meaningful research topic for our future work.