Robust Optimal Excess-of-Loss Reinsurance and Investment Problem with Delay and Dependent Risks

Zhang, Yan; Zhao, Peibiao

doi:https://doi.org/10.1155/2019/6805351

Discrete Dynamics in Nature and Society

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Abstract Introduction Model Formulation Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 6805351 | https://doi.org/10.1155/2019/6805351

Robust Optimal Excess-of-Loss Reinsurance and Investment Problem with Delay and Dependent Risks

Yan Zhang^1,2and Peibiao Zhao¹

Academic Editor: Youssef N. Raffoul

Received11 Dec 2018

Revised03 Jun 2019

Accepted26 Jun 2019

Published01 Aug 2019

Abstract

This paper investigates a robust optimal excess-of-loss reinsurance and investment problem with delay and dependent risks for an ambiguity-averse insurer (AAI). The AAI’s wealth process is assumed to be two dependent classes of insurance business. He/she can purchase excess-of-loss reinsurance from the reinsurer and invest in a risk-free asset and a risky asset whose price follows Heston model. We obtain the explicit expressions of the optimal excess-of-loss reinsurance and investment strategy by maximizing the expected exponential utility of AAI’s terminal wealth. Finally, we give the proof of the verification theorem. Our models and results posed here can be regarded as a generalization of the existing results in the literature.

1. Introduction

In the past decade, the topic about optimal investment and reinsurance problems has attracted a lot of attention. These optimal problems have been studied in terms of various objectives, for example, [1–4] considered the objective function of minimizing the ruin probability; [5–9] studied the optimal problems aiming to maximize the survival probability or the expected utility of terminal wealth; [10–13] investigated optimal reinsurance and investment problems under mean variance criterion.

Although so many notable scholars have considered the optimal reinsurance and investment problems, two important aspects are still being worthy of further exploration. One is lack of considering ambiguity, and the other one is the optimal control problems under delayed systems. On one hand, the model uncertainties do exist widely in finance, especially in insurance, the field of asset pricing, consumption, and portfolio selection. As a result, the ambiguity-averse insurer (AAI) has to look for a methodology to handle this uncertainty. One possible way is to use the robust approach, where some alternative models closed to the estimated model are introduced and the robust optimal strategy is obtained. Recently, some scholars paid more attention to optimal investment-reinsurance problems with ambiguity. Reference [14] assumed that the insurer’s wealth process follows a diffusion model, and they optimized a proportional reinsurance and investment problem with model uncertainty. Reference [15] obtained the robust optimal proportional reinsurance and investment strategies for an AAI; in their article, the surplus process is assumed be a Cramér–Lundberg risk model and the risky asset’s price follows a constant elasticity of variance (CEV) model. Reference [16] studied the robust optimal proportional reinsurance and investment strategies for both an insurer and a reinsurer. Reference [17] took default risk into account and derived the robust optimal control strategy under variance premium. Different from the above-mentioned literature, [18] analyzed a robust optimal problem of excess-of-loss reinsurance and investment in a model with jumps for an AAI.

On the other hand, the investors or the insurers make important future decisions only according to the present states of a system, but they do not consider the past states. However, the future states of a system usually may depend on its past states, which do exist in our real-world systems. For example, in the stock market, investors not only are concerned with the present stock price but also pay more attention to the trend of the stock price in the past periods. Thus, it is more realistic to take some past information of the system into account. Due to the structure of infinite-dimensional state space, generally speaking, it is difficult to solve these stochastic control problems with delay analytically. As a result, there is an explicit solution for this problem. Only those problems, in which some special forms of delay information are considered in the state process, are found to be finite-dimensional and then can be solved (see, for example, [19–21]). Moreover, the delay is first introduced into the optimal proportional reinsurance and investment problems by [22] under mean-variance criterion. Reference [23] optimized the delayed problem of excess-of-loss reinsurance and investment under maximizing the expected exponential utility of the insurer’s terminal wealth. Reference [24] took multiple dependent classes of insurance business into consideration, investigated the time-consistent reinsurance-investment problem with delay and derived the optimal strategy under the mean-variance criterion. As mentioned in [24], in fact, some insurance businesses are usually correlated by some way in practice. For example, a traffic accident (or fire accidents or car accidents or aviation accidents and so on) may cause property loss or medical claims or death claims; these insurance businesses will be correlated. Therefore, it is necessary to take dependent risks into account in the actuarial literature. References [25, 26] assumed that the insurer’s surplus process consists of two or more dependent classes of insurance business and the claim number processes are correlated through a common shock component, and they discussed optimal proportional reinsurance problems under the criterion of maximizing the expected utility of terminal wealth. For other research about dependent risks, we refer the readers to [27–31] and the references therein.

This paper takes excess-of-loss reinsurance into account, which is better than proportional reinsurance in most situations; see [32]. Suppose that the insurer’s wealth process consists of two dependent classes of insurance business. The insurer is allowed to purchase excess-of-loss reinsurance and invest in a financial market which consists of a risk-free asset and a risky asset. The risky asset’s price is described by Heston model. Moreover, it is assumed that there exists capital inflow into or outflow from the insurer’s current wealth. Given that the insurer’s claim process and the risky asset price (true model) may deviate from a relative good estimated model (reference model) in real-word, the model uncertainty should be taken into consideration. On the basis of the above setup, we first formulate a robust optimal control problem with delay and dependent risks and then investigate the optimal strategy for an AAI by maximizing the expected exponential utility of terminal wealth. This paper has the following main contributions: (i) an optimal excess-of-loss reinsurance and investment problem with dependent risks is studied; (ii) both ambiguity and the capital inflow/outflow are introduced into this problem; (iii) some special cases are provided, such as the case of investment-only, ambiguity-neutral insurer, and no delay, which demonstrates that our model and results can be considered as a generalization of the existing results in some literature, e.g., [23, 25].

The rest of this paper is structured as follows. We present the formulation of our model in Section 2. Section 3 discusses the robust optimal strategy and derives the optimal results. Section 4 is devoted to proving the verification theorem. Some special cases of our model are provided in Section 5. Section 6 concludes the paper. In Appendix, technical proofs are presented.

2. Model Formulation

We consider a filtered complete probability space , where represents the terminal time and is a positive finite constant and stands for the information of the market available up to time . Assume that all processes introduced below are well-defined and adapted processes in this space. In addition, suppose that trading takes place continuously and involves no taxes or transaction costs and that all securities are infinitely divisible.

2.1. Surplus Process

This section presents a risk model consisting of two dependent classes of insurance business. The insurer’s wealth process is modeled ofwhere the positive constant is the premium rate; is the ith claim size from the first class; are assumed to be i.i.d. positive random variables with common distribution function , finite first moment , and second moment is the ith claim size from the second class and are assumed to be i.i.d. positive random variables with common distribution function , finite first moment , and second moment , and are three independent Poisson processes with positive intensity parameters , , and , respectively.

The compound Poisson processes and represent the cumulative amount of claims for the first class and the second class in time interval , respectively. , , , and are mutually independent. Further, the insurer’s premium rate is calculated according to the expected value principle; i.e.where is the insurer’s safety loading from the ith claim.

In addition, we assume that for and for , where represents the maximum claim size from the first class; and for , and for , where represents the maximum claim size from the second class.

2.2. Excess-of-Loss Reinsurance

Suppose that the insurer can purchase excess-of-loss reinsurance by reducing the underlying claims risk. Denote by and the excess-of-loss retention levels, and letbe the parts of the first claims and the second claims held by the insurer, respectively. Then by (1), the wealth process becomeswith the premium ratewhere is the reinsurer’s safety loading from the ith claim. Assume that , which implies that the reinsurance is not cheap. According to [33], the wealth process (4) can be approximated by the following diffusion model: where and are two standard Brownian motions whose correlation coefficient isand is another standard Brownian motion dependent of and .

For convenience, let where ,

2.3. Financial Market

The insurer is assumed to invest in a risk-free asset whose price process is governed byand a risky asset whose price process follows Heston model, where positive constant is the risk-free interest rate, , and are all positive constants, and and are two standard Brownian motions with By standard Gaussian linear regression, can be rewritten as where is another standard Brownian motion. We assume that , , and are mutually independent. Moreover, we require to ensure that is almost surely nonnegative.

2.4. Wealth Process with Delay

Let be the reinsurance-investment strategy, where is the excess-of-loss retention level for ith claim at time t, where means “no reinsurance” and means “full reinsurance,” and is the money amount invested in the risky asset at time t, the amount of money invested in the risk-free asset at time t is , and here is the insurer’s wealth after adopting strategy Thus, the evolution of is governed by

It is noted that the wealth process is traditionally formulated as (14), which is a stochastic differential equation (SDE) without delay. In the sections below, we will formulate a wealth process with delay, which is caused by the instantaneous capital inflow into or outflow from the insurer’s current wealth. The delayed wealth process is still denoted by Let , , and be the delayed wealth and average and pointwise performance of the wealth in the past horizon , respectively, i.e.for , where is an average parameter and h > 0 is the delay parameter. Denote by the function the capital inflow/outflow amount, where represents the absolute performance of wealth between t and , and stands for the average performance of the wealth in Such capital inflow or outflow, which is related to the past performance of the wealth, may come out in various situations. For example, a good past performance of the wealth may bring the insurer more gain. On the contrary, a poor past performance of the wealth may force the insurer to seek further capital injection to cover the loss so as to achieve the final performance objective. Following [22, 23], when we consider such a capital inflow/outflow function, the wealth process can be given as follows:

Such capital inflow/outflow, which is related to the past performance of the wealth, may come out in various situations. For example, a good past performance of the wealth may bring the insurer more gain and further the insurer can pay a part of the gain as dividend to his/her stakeholders, in this case On the contrary, a poor past performance of the wealth may force the insurer to seek further capital injection to cover the loss so as to achieve the final performance objective and in this case To make this problem solvable, we assume that the amount of the capital inflow/outflow is proportional to the past performance of the insurer’s wealth, i.e. where and are two nonnegative constants. Inserting (6), (11), (12), and (19) into (18) leads to the following stochastic differential delay equation (SDDE):whereFurthermore, assume that , which means that the insurer is endowed with the initial wealth at time and does not start the investment and (re)insurance business until time 0. Therefore, the initial value of the average performance wealth is

Given that the investment performance has an effect on the insurer’s wealth, this paper assumes that the insurer is concerned with and in the time interval Moreover, suppose that the insurer has the following exponential utility function defined by where and are constants. Here represents a constant absolute risk aversion coefficient which plays a vital role in insurance practice and actuarial mathematics. Note that is the weight of , so will impact the average performance of the terminal wealth. We consider the integrated delayed wealth rather than the average one directly. Define as the transformed weight, which can be considered as the weight between and . Thus (23) is equivalent to considering For simplicity, we call the combination the terminal wealth. In fact, our modeling framework for the term is consistent with the classical literature (e.g., [34, 35]) on stochastic control problem with delay. Reference [23] considered a utility function, which is similar to (23) in our paper and studied an optimal problem with delay for an insurer; in their article, the insurer’s surplus process is assumed to follow the classical Cramér-Lundberg model. Compared with [23], we not only incorporate model uncertainty into our study which will be introduced later but also assume that the wealth process consists of two classes of insurance business, in which the two claim processes are dependent.

In traditional, it is assumed that the insurer is ambiguity-neutral with the following objective function:where is the expectation under the probability measure P and is the set of admissible strategies which will be defined in Definition 1. However, many insurers are ambiguity-averse and always try to guard themselves against worse-case scenarios. Thus, it is reasonable to suppose an insurer is ambiguity-averse in the field of insurance. In what follows, we present a robust portfolio choice with uncertainty for an AAI. Suppose that the AAI has a relative good estimated model (also called reference model) to describe the risky assets prices and his/her claim process, but he/she is always skeptical about this reference model and hopes to take alternative models into account. According to [36], the alternative models are defined by a set of probability measures which are equivalent to the as follows:

Definition 1. For any fixed , the strategy is said to be admissible if it is -progressively measurable and satisfies
(i) ;
(ii) , where ;
(iii) , the SDDE (20) has a pathwise unique solution with , where is the chosen model to describe the worst case, is the condition expectation given , , Let be the set of all admissible strategies.

Next, define a process satisfying that

(i) is –measurable, ;

(ii) , where We denote for the space of all such processes

For , we define a real-valued process on bywhere By Ito’s differentiation rule,

Thus is a P-martingale. Hence, For , a new real-word probability measure absolutely continuous to on is defined by

So far, we have constructed a family of real-world probability measures parameterized by Applying Girsanov’s theorem, we can see thatunder the alternative measure , is a standard three-dimension Brownian motion, where Note that the alternative models in class only differ in the drift terms. Thus, the risky asset’s price (12) under isAnd the wealth process (20) under is rewritten as

Inspired by [37, 38], we formulate the following robust control problem to modify problem (25), i.e., where and is calculated under . In (34), the deviations from the reference model are penalized by the second term in the expectation. In fact, this penalty term depends on the relative entropy arising from diffusion risk. In addition, the parameter in (35) represents the strength of the preference for robustness. For analytical tractability, suppose that the parameter in (35) is state-dependent. In particular, following [37, 39], we setwhere stands for the ambiguity-aversion coefficient and describes the AAI’s attitude to the diffusion risk.

For convenience, we first provide some notations. Let be an open set and Let be the space of such that and its partial derivatives , , , , , , are continuous on To make the problem (34) solvable, by dynamic programming principle, the robust Hamilton-Jacobi-Bellman (HJB) equation for (34) can be derived as (see [39, 40]):for with boundary condition where is the generator of (33) under and is defined asHere, is a short notation for

3. Robust Optimal Results with Delay

The aim of this section is to find the robust optimal control strategy for problem (34) under the exponential utility. As mentioned above, in general, the delayed control problem is infinite-dimensional. In order to make this problem be finite-dimensional and solvable, according to [23], we assume the parameters satisfies the following conditions:It is noted that the above two conditions are the sufficient conditions for the optimal control problem with delay, which guarantee that the HJB equation (37) has a closed-form solution. Furthermore, they help us explore the implication of the problems with delay and without delay.

In what follows, we try to solve the HJB equation (37) with boundary condition (38). We conjecture that the value function has the following form: with , , Let , , be the partial derivatives of From (42), we get

Step 1. Putting (43) into (37) and rearranging terms, since , we get

Step 2. According to (44), fixing and maximizing over yields the first-order condition for the minimum point as follows:Replacing (45) back into (44) leads to where By the first-order condition for , we getPlugging (48) into (46) derives Let To find the minimizer and of , we need to take the first and the second derivatives of with respect to and It is assumed that and are continuous and differentiable, and For any , differentiating with respect to , yields We first consider that , then and Moreover, suppose that there exists at least one point satisfying the following equation:Taking into the second derivatives of , we will get the following Hessian matrix:which means this Hessian matrix is positive definite at the point Therefore, if we can find such that (52) holds, then the point is the minimizer of

Step 3. Inserting into (49) yields With the assumptions (40) and (41), we have and , and then (54) can be transformed intoAccording to the arbitrariness of and , (55) is equivalent to the two following equations: For (56), taking the boundary condition into account yields

In order to determine the point clearly, (52) is transformed intoor equivalently, Next, we need to define three following auxiliary functions:For convenience, we assume that It is easy to verify that both and are strictly increasing functions for , so their inverse functions and exist. From (60), we get , and thenPlugging (62) and (58) into the second equation of (52), we obtainTherefore, letIf the equation has a solution on , the solution is indeed we try to derive, and as a result, will be easily determined. Then, the robust optimal reinsurance strategy can be derived and summarized in the following theorem.

Theorem 2. Assume that , and letFor the robust optimal control problem (34), the optimal excess-of-loss reinsurance strategy is given as follows.
(i) If or , we have For , the optimal excess-of-loss reinsurance strategy is , whereand is the unique solution to if holds.
For , if the solution to exists, the optimal excess-of-loss reinsurance strategy is , where(ii) If , we have For , the expression of optimal excess-of-loss reinsurance strategy is

Proof. (i) If or , then we have
For , we have Since and are increasing functions on , it is easy to verify that increases on What is more, we can get by the fact that
If , the equation admits a unique solution , and thus the optimal excess-of-loss strategy isDue to , we obtain If , it implies that the equation has a unique solution on Due to , we choose At this time, substituting back into in (50) and taking the first derivative of with respect to derives the minimizer of isConsequently, from (68), (69), and (70), and can be expressed as and in (67) and (68), respectively.
For , holds; then we have Meanwhile, strictly increases on and As a result, the equation has no solution on In other words, there does not exist the solution satisfying (52) when and However, it does not mean that (52) has no solution on and It is not difficult to prove that is a convex function on So for , if the solution of the equation exists, it will only be obtained on which is indeed we try to derive.
Because , and is a convex function on , we can derive that on What is more, is strictly increasing and , so we obtain from (62) that Due to , we get Plugging into in (50) and taking the first derivative of with respect to derives the minimizer of isTo sum up, for the case of , we derive (68).
(ii) If , we have Thus for , the inequality holds. The optimal problem is similar to that of in case (i). At this time, we obtain the optimal reinsurance strategy for is (68). This ends the proof of Theorem 2.

In order to get the expressions for and the value function , we have to derive the expressions of and in (57). By (58), we can rewrite (57) as

Now we discuss this problem in two cases as follows.

Case 1. If , for , the optimal reinsurance strategy for the problem (34) is (67). Denoting by the function in (73), we haveFor equation (74), we conjecture a solution of the following formwith , Then (74) is transformed into That is where with the boundary condition

According to the arbitrariness of l, decomposing (77) intoandAccording to [41], taking the boundary condition , into account yieldsandwhere , that is,and

Similarly, for , the optimal excess-of-loss strategy is (68). Denote the function by in (73), and let Similar to (77)—(82), we obtainand Hence for , we haveSince is continuous at , we haveand we derive

Based on (80), (81), (82), (83), (84), (85), (86), (87), (89), and (90), it is not difficult to find that Moreover, since , we obtainAs a result, (87) can be rewritten as By (48), (75), and (84), we can see that the optimal investment strategy for is the same as that for , which is

Case 2. If , we have For , the optimal excess-of-loss strategy is (68). Denoting by the function in (73), we havewith boundary condition , Similar to the analysis for in Case 1, we conjecture a solution to (94) of the following formwith , Then (94) is transformed into where with the boundary condition

Then employing the same method to solve the optimal problem as Case 1, a direct calculation yields , andThus, for , we can obtain the expression of optimal investment strategy which is the same as (93) and the corresponding value function.

The following theorem summarizes the above analysis.

Theorem 3. Recall functions and defined in (81), (82), (86), (92), and (97), respectively. Then, for the problem (34), we have the robust optimal investment strategy and (i) if holds, the optimal value function is(ii) if holds, the optimal value function is

Remark 4. Form Theorems 2 and 3, we find that
(1) in the light of assumptions (40) and (41), the robust optimal strategy depends on the parameters , , , and
(2) the robust optimal value function depends on parameters , , , , and the average past performance .

4. Verification Theorem

In this section, we will verify the candidate optimal strategies and given by (45), Theorems 2 and 3 are indeed optimal, and the candidate value function (42) is just the value function defined in (34). The main theorem is summarized as follows.

Theorem 5. If there exists a function and a control strategy such that
(1) for ;
(2) for ;
(3) ;
(4) for ;
(5) for and are uniformly integrable, where denotes the set of stopping times
Then and is an optimal control strategy.

Proof. It is similar to the proof of Theorem 3.2 in [42]. So we omit it here.

Next, conditions (1)–(5) in Theorem 5 will be checked. We first give two lemmas, which are useful for verifying Theorem 8.

Lemma 6. If the parameters satisfy then we have

Proof. ee Appendix.

Lemma 7. For problem (34), is the solution to equation (37) with boundary condition , if the parameters satisfy condition (101) and then

Proof. Se Appendix.

Theorem 8. For problem (34), if the parameters satisfy the conditions in Lemma 7 and is the solution to (37) with boundary condition , then the optimal value function is , and the optimal strategy is , which is given in Theorems 2 and 3.

Proof. By the proofs of Theorems 2 and 3, conditions (1)–(4) in Theorem 5 hold for By Lemma 7, we know that condition (5) in Theorem 5 also holds for Then, according to Theorem 5, the results of Theorem 8 are obtained.

5. Special Cases

In this section, we consider the robust optimal problem (34) without insurance (investment-only case), with ambiguity-neutral insurer (ANI) and without delay, respectively. Since they are all special cases of (34), we only provide the results here without giving the proofs.

5.1. Investment-Only Case

If there is no reinsurance, i.e., , , then we have , and the wealth process (6) can be rewritten as According to (33), the above wealth process with delay for an AAI under the probability is and the HJB equation becomeswhere is a short notation for , which represents the optimal value function of the investment-only problem with the boundary condition , and

Theorem 9. For the investment-only problem with delay, i.e., , , under the assumptions (23), (36), (40), and (41), the robust optimal investment strategy is and the optimal value function is for , where and are given by (81) and (83), and

In addition, if , , it means that we only consider one class of insurance business; in other words, there are no dependent risks in our model.

Corollary 10. For the investment-only problem with delay, if , , the robust optimal investment strategy is (108), and the optimal value function is where

Remark 11. According to Theorems 3 and 9 and Corollary 10, we find that the robust optimal investment strategy of investment-only case is the same as that of reinsurance-investment case, which implies that the robust optimal reinsurance strategy and the robust optimal investment strategy can be separated.

In what follows, some special cases of the investment-only problem without delay are provided.

Corollary 12. For the investment-only problem without delay, i.e., ,
(1) if , , the robust optimal investment strategy isand the optimal value function isfor , where(2) If , , for , the robust optimal investment strategy is (113), and the optimal value function is where

5.2. ANI Case

If ambiguity-aversion coefficient , our model will reduce to an optimal control problem for an ANI. Then, the wealth process under probability measurers is given by (20). Denote the optimal value function bywhere , and the corresponding HJB equation isHere, is the short notation for with boundary condition Let Note that if and , according to the proof of Theorem 2, by the similar analysis that the equation in (64) has a unique solution when , it is easy to prove that equation has a unique solution

Theorem 13. When , for the optimal control problem (118) of an ANI who ignores ambiguity with utility (23), we obtain (i) If or , holds. For , we derive that the optimal reinsurance strategy is expressed byFor , if the solution to exists, we obtain the optimal reinsurance strategy isand the optimal value function is (ii) If , in this case For , the optimal reinsurance strategy is expressed by (123), and optimal value function is given bywhereand is given by (47).

Furthermore, if in Theorem 13, the optimal strategy here along with the optimal value function will coincide with Theorem 3.1 in [23]; i.e., our model extends the results in [23] to the case of robust optimal formulation under dependent risks.

5.3. No Delay Case

If , our model will reduce to a robust optimal control problem without delay. Then, the wealth process under probability measurer with strategy isSuppose the AAI has an exponential utility as follows: Denote the value function bywhere Then, the corresponding HJB equation is given bywhere is a short notation for with boundary condition , and , which represents the generator of the process (127) under , is defined byLet

Theorem 14. When , for problem (129) with the utility (128), we obtain the robust optimal investment strategy is and (i) if or holds, for , the robust optimal excess-of-loss reinsurance strategy is where is the unique solution to the equation if holds.
For , if the solution of exists, the robust optimal excess-of-loss reinsurance strategy isand the corresponding optimal value function is(ii) if holds, in this case For , the robust optimal reinsurance strategy is (136), and the corresponding value function iswhere and are given by (81) and (83), respectively, and

Remark 15. Following Corollary 12 and Theorem 14, we find there is a similar conclusion to Remark 11 for the problem without delay.

6. Conclusion

For the optimal control problems in insurance, most papers only consider the control systems without delay, while this paper studies a robust optimal reinsurance-investment problem with delay and dependent risks when the risky asset’s price is described by Heston model. To make the optimal control problem closer to reality, we furthermore consider some possible extensions of this paper. For example, we can consider the robust equilibrium reinsurance-investment problem for a mean-variance insurer with other kinds of dependent risks, such as copulas, which is a very challenging problem.

Appendix

Lemma A.1. is a decreasing function with respect to and satisfies where

Proof. For , differentiating (81) with respect to yieldsAccording to (84), it is true that for In addition, for ,For , ,for , ,As a result, decreases with respect to . Then plugging 0 and into (81) yields (A.1).

Proof of Lemma 6. Due to (A.1), with an appropriate constant ,we get Since is deterministic and bounded on , the first estimate in (A.7) is valid. The second inequality holds according to condition (101), and by Theorem 5.1 in [43], the last estimate is easily derived.

Proof of Lemma 7. Substituting and into (33) derivesInserting (A.8) into (42), we obtain the following estimate with appropriate constants ,whereand Because and are deterministic and bounded on , we obtain that the first estimate in (A.9) is valid. Since , , , , and are deterministic and bounded on , the second inequality holds. The third inequality follows the fact that is bounded.
Now we firstly consider the integrals and Note that and are bounded on , and then we get It is not difficult to verify that is a martingale under , and thus
Secondly, for , note that is deterministic and bounded; then by the Lemma 4.3 in [43], it is easily seen that is a martingale under , and, consequently,Further, following Theorem 5.1 in [43], a sufficient condition for is obtained, which is According to (A.1) and condition (102), we getConsequently, (A.14) holds.
Applying and , and by (A.9), we arrive at Hence, we have proved the first part of this lemma.
Plugging and into (35) leads toAccording to Cauchy-Schwarz inequality, we obtain the first estimate and the last one can be derived by (A.16) and Lemma 6 under condition (101). Thus, the second part of this lemma holds.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant No. 11871275; No. 11371194).

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Copyright © 2019 Yan Zhang and Peibiao Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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