# Optimal Surrender Policy of Guaranteed Minimum Maturity Benefits in Variable Annuities with Regime-Switching Volatility

Authors: Xiankang Luo; Jie Xing
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This study investigates valuation of guaranteed minimum maturity benefits (GMMB) in variable annuity contract in the case where the guarantees can be surrendered at any time prior to the maturity. In the event of the option being exercised early, early surrender charges will be applied. We model the underlying mutual fund dynamics under regime-switching volatility. The valuation problem can be reduced to an American option pricing problem, which is essentially an optimal stopping problem. Then, we obtain the pricing partial differential equation by a standard Markovian argument. A detailed discussion shows that the solution of the problem involves an optimal surrender boundary. The properties of the optimal surrender boundary are given. The regime-switching Volterra-type integral equation of the optimal surrender boundary is derived by probabilistic methods. Furthermore, a sensitivity analysis is performed for the optimal surrender decision. In the end, we adopt the trinomial tree method to determine the optimal strategy.


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## Body

## 1. Introduction

A variable annuity (VA) contract is issued by an insurance company, which offers upside opportunities in financial market and also protects against downside risk by adding various types of benefit riders. In a VA contract, the policyholder agrees to pay either a single payment or a series of periodic premiums to his/her account. These payments are typically invested in risky financial assets in the market for better benefits. Due to the uncertainty of the market, the VA contract provides various benefits in the event of poor performance of financial risky assets. These guarantees embedded in VA contracts are similar to financial options. For instance, a guaranteed minimum accumulation benefit (GMAB) guarantees a lump sum amount at a predetermined future date, a guaranteed minimum death benefit (GMDB) guarantees a minimum lump sum upon death, a guaranteed minimum living benefit (GMLB) provides income guarantees to protect the policyholder’s income from declining, and a guaranteed minimum maturity benefit (GMMB) is a guarantee that provides the policyholder with a minimum benefit on maturity of the VA contract (see the works of Bernard et al. [1], Ignatieva et al. [2], Bauer et al. [3], and Shen et al. [4]).The following is a brief literature review. Bacinello [5] computed the value of participating life insurance policies with surrender options using a recursive binomial tree approach. Bernard et al. [1] split the value of VA contract into a European part and an early exercise premium and derived a Volterra-type integral equation with respect to the optimal surrender boundary under the assumption that the underlying risky asset follows the geometric Brownian motion (GBM) model. Alonso-Garcia et al. [6] considered VA contract with GMAB rider when policyholder’s proceeds are taxed. Shen et al. [4] considered a VA contract with a GMMB rider and proposed a numerical approach to the value function of the VA contract and the optimal surrender boundary. Kang and Ziveyi [7] analyzed how the policyholder surrender behavior is influenced by changes in various sources of risk impacting a VA contract under stochastic volatility and interest rate. Jeon and Kwak [8] studied VA contract with GMMB riders for path-dependent guarantees and applied Mellin transform to calculate the value function and the optimal surrender strategy. Jeon and Kim [9] derived an analytic pricing formula for a VA contract with GMMB rider by using the Laplace–Carson transform. MacKay and Ocejo [10] studied a portfolio optimization problem involving a VA contract with GMMB rider and solved constrained and unconstrained version of the optimization problem.The majority of the literature has been premised under the GBM framework. However, it is well known that the stochastic variability in the market parameters is not reflected in the GBM model. In filling this gap, we should extend the results presented in Shen et al. [4] to the stochastic volatility model. The GBM model has been extended in various ways. Among those generalizations, the regime-switching model has become a popular model. In this paper, we focus on a VA contract embedded with a GMMB rider by incorporating regime-switching volatility. This problem can be viewed as an optimal stopping problem with regime-switching. It is interesting to point out that our problem has a lot in common with the well-known American option in financial mathematics. However, the gain function in our problem is time dependent, which is different from the American option pricing problem. There are some papers that have dealt with the optimal stopping problem with regime-switching volatility. Buffington and Elliott [11] discussed American options in a regime-switching market and obtained an approximate valuation of American option. Jacka and Ocejo [12] studied the regularity of the value function for a finite-horizon optimal stopping problem in the presence of regime-switching uncertainty. Boyarchenkp and Levendorski [13] calculated the early exercise boundaries for American options by using a generalization of Carr’s randomization procedure for regime-switching models. In the insurance literature, Siu [14] considered the fair valuation of a participating life insurance policy with surrender options for regime-switching models and provided a decomposition result for the value of the policy. In a word, the above literature mainly concentrated on the value functions and none of them discussed the properties of the free boundary rigorously. Yang [15] studied the properties of free boundary arising from Russian option pricing problem with regime-switching volatility by the variational inequality approach.The main contributions of this paper are stated as follows. Firstly, we model the underlying mutual fund dynamics under regime-switching volatility. The related variational inequality is given by a standard Markovian argument. Secondly, compared with the method used in [15], we derive some basic properties of the corresponding value function and the optimal surrender boundary by probabilistic methods. Besides, to derive the regime-switching Volterra-type integral equations of the optimal surrender boundary, we use dynamic programming principle to convert the regime-switching optimal stopping problem into an optimal stopping problem with constant-coefficient geometric Brownian motion, which makes the proof significant in methodology. Furthermore, since there has been less focus on the development of a realistic modelling framework for analyzing the impact of various sources of risk in influencing the surrender behavior, it is crucial to make rigorous sensitivity analysis when we make risk management decision. Compared with the sensitivity analysis in Shen et al. [4], which was performed by numerical examples without proofs, we apply a rigorous variational inequality approach to explore the impacts of varying model parameters on the optimal surrender boundary. Finally, we calculate the optimal surrender boundary by trinomial tree method (see [16]) and perform numerical examples to analyze the policyholder’s surrender decisions.The remaining parts of this paper are organized as follows. In Section2, we state the valuation problem of a VA contract with GMMB rider under regime-switching volatility. In Section 3, we state the main results in this paper. In Section 4, we give some numerical experiments to verify our theoretical results. In Section 5, we give detailed proofs of the results in Section 3. In Section 6, we conclude the study.


## 2. Formulation of the Problem

We consider a probability spaceΩ,ℱ,ℚ, where ℚ is a risk neutral measure. Also, we consider a standard Brownian motion W=Wt,t≥0. Assume that the continuous time Markov chain α=αt,t≥0 takes values in S=1,2 and the generator of αt is of the form(1)A=−λ1λ1λ2−λ2,where λ1 and λ2 are positive constants. Let F=ℱt,0≤t≤T be the natural filtration, which is generated by the standard Brownian motion W and Markov chain α and satisfies the usual condition. Let E⋅ denote the expectation under risk neutral measure ℚ. We consider the behavior of the policyholder who is involved in a VA contract embedded with a GMMB rider, where the contract can be surrendered anytime prior to maturity subject to penalty charges. The policyholder pays the premium at the contract initialization and then invests the money in a fund consisting of units of an underlying asset S=St,0≤t≤T, whose risk neutral dynamic evolves under the influence of regime-switching volatility, which is specified as follows:(2)dSt=rStdt+σαtStdWt,where r>0 is the risk-free interest rate and σαt∈σ1,σ2 is the volatility of the underlying asset. The resulting mutual fund’s value is given by(3)Ft=e−ctSt,where c is the continuously compounded insurance charge expressed in percentage terms by the annuity provider (see [4]). A direct computation shows that the dynamic of the fund value F=Ft,0≤t≤T satisfies(4)dFt=r−cFtdt+σαtFtdWt.We assume that the guaranteed amountGt≡G>0 is a constant throughout the life of the contract. Denote that maxa,0≔a+. Since the insurer will provide maxG,FT to the policyholder at maturity date, hence, under risk neutral argument, the initial value of the VA contract is given by(5)Ee−rTmaxG,FT=Ee−rTFT+Ee−rTG−FT+,where FT is the fund value at terminal time T, with G being the guaranteed value at maturity of the contract. In order to find the fair insurance charge c∗ during the time t∈0,T, we should set the initial fund value F0 to be equal to the value in (5). In (5), we decompose the value of the VA contract into two components, which was presented in [4] firstly.In view of (5), we may find it optimal to surrender the contract early for avoiding higher fees which are proportional to the fund value. In the event of the guarantee being exercised prior to the maturity, we assume that the surrender benefit equals to(6)1−κtFt,where κt is a penalty percentage charged for surrendering at time t∈0,T. As in Bernard et al. [1] and Shen et al. [4], κt is exponentially decreasing with time and is equal to(7)1−e−κT−t.Then, the surrender benefit becomes(8)e−κT−tFt.In this paper, we assume thatκ<c<r holds. Otherwise, it is optimal to surrender the option at maturity T.We notice that the policyholder will receive account valueFT from the mutual fund and the benefit G−FT+ from the embedded guarantee without a surrender option. More precisely, the value of the VA contract at time t in this case is given by(9)Ee−rT−tmaxFT,G|ℱt=Ee−rT−tFT|ℱt+Ee−rT−tG−FT+|ℱt.On the contrary, if there is a surrender option for the embedded guarantee, then the policyholder is allowed to surrender the guarantee at any timet before the maturity and the surrender benefit will be(10)G−e−κT−tFt+.Besides, the policyholder will also receive his/her account valueFT at maturity. Then, the value of the VA contract at time t can be written as follows:(11)Φt,f,i≔e−rT−tEFT|ℱt+supt≤τ≤TEe−rτ−tG−e−κT−τFτ+|ℱt.By the formula in [17] and put-call parity, we can derive a formula for European call option under regime-switching volatility. Then, by letting E⟶0, we can obtain an analytical formula for the first component in the above equation, where E is the strike price of the call option. Now, we focus on the second term and denote(12)Pt,f,i≔supt≤τ≤TEe−rτ−tG−e−κT−τFτ+|ℱt.We setYt=e−κT−tFt, y=e−κT−tf, and Pt,eκT−ty,i=Vt,y,i, where the dynamic of Y satisfies(13)dYt=r−c+κYtdt+σαtYtdWt.Solving (13) with initial condition Y0=y, we find that(14)Yt≔yMt=yexp∫0tr−c+κ−12σαs2ds+∫0tσαsdWs.Then, by Markovian property ofY, we derive that(15)Vt,y,i=supt≤τ≤TEe−rτ−thYτ|αt=i,Yt=y=sup0≤τ≤T−tEi,ye−rτhYτ,where Ei,y⋅ means E⋅|α0=i,Y0=y and hy=G−y+.


## 3. Main Results

Having stated main problem inSection 2, we formulate the main assertions of this paper, which are built on probabilistic methods of optimal stopping problem developed by Peskir and Shiryaev [18] and variational inequality approach by Yang [15]. Now, we show some useful properties of the value function defined in (15), which will be frequently used later. Firstly, we introduce the infinitesimal generator of Yt,αt as follows:(16)Aft,y,i≔ℒft,y,i+λift,y,j−ft,y,i,j≠i,where(17)ℒft,y,i≔12σi2y2fyyt,y,i+r−c+κyfyt,y,i,where fy denotes ∂ft,y,i/∂y and fyy is defined similarly.Lemma 1.
The value functionV satisfies the following variational inequality:(18)minrV−Vt−AVt,y,i,Vt,y,i−hy=0,t,y,i∈0,T×0,+∞×S,with terminal condition,(19)VT,y,i=hy.Proof.
The proof is easy, and we refer to Appendix A.Lemma 2.
The value functionVt,y,i is positive, nonincreasing in t and y, and convex in y.Proof.
The proof is easy, and we refer to Appendix B.Lemma 3.
Vt,⋅,i is Lipschitz continuous in ℝ+ and V⋅,y,i is 1/2-Hölder continuous uniformly over a neighborhood of y.Proof.
See Section5.To discuss the optimal exercise boundary, we introduce the stopping region and continuation region as follows:(20)D≔t,y,i∈0,T×0,+∞×S:Vt,y,i=hy,C≔t,y,i∈0,T×0,+∞×S:Vt,y,i>hy,Di≔t,y∈0,T×0,+∞:Vt,y,i=hy,i∈S,Ci≔t,y∈0,T×0,+∞:Vt,y,i>hy,i∈S.From [18], we know that the optimal stopping time for problem (15) can be represented as(21)τD=inf0≤s≤T−t:t+s,Ysy,αs∈D∧T−t.To investigate the shape of the stopping region, we need the following important lemma.Lemma 4.
The functiony↦Vt,y,i+y is increasing. Moreover, D is downward connected: if t,y′,i∈D, then t,y,i∈D, for y<y′.Proof.
See Section5.Now, we are prepared to describe the shape of the stopping region. The result is as follows.Theorem 1.
There exists a continuous functionbit such that(22)bit=supy>0:Vt,y,i=hy,i∈S.
Moreover,bi⋅ is nondecreasing and(23)biT=minG,rGc−κ.Proof.
See Section5.By the earlier assumption thatr>c>κ, it turns out that biT=G for i∈S. The next theorem shows that we can transform the original optimal stopping problem into a free boundary problem.Theorem 2.
The value functionV defined in (15) is the unique solution to the following free boundary problem:(24)Vtt,y,i+AVt,y,i−rVt,y,i=0inC=t,y,i:y>bit,(25)Vt,bit,i=hbit,t∈0,T,i∈S,(26)Vyt,bit,i=−1,t∈0,T,i∈S,(27)VT,y,i=hy,y∈0,+∞,i∈S.
In particular, the partial derivativesVyy, Vy, and Vt exist and are continuous in C.Proof.
See Section5.Next, we introduce the following notations to obtain the regime-switching Volterra-type integral equations:(28)Jit,y≔Ee−r+λiT−thYT−ti,(29)Kit,y,u,biu+t≔Ee−r+λiuΦiYuiχYui<biu+t−λiVt+u,Yui,jχYui>biu+t,j≠i,where(30)Φiy=c−r+λiy−r+λiG.Yti satisfies(31)dYti=Ytir−c+κdt+σidWt,with initial condition Y0i=y.Theorem 3.
The value function defined in (15) is given by(32)Vt,y,i=Jit,y−∫0T−tKit,y,u,biu+tdu,i∈S.
Furthermore, the free boundarybit satisfies the following nonlinear regime-switching Volterra integral equations:(33)hbit=Jit,bit−∫0T−tKit,bit,u,biu+tdu,i∈S.Proof.
See Section5.Remark 1.
In this remark, we shall calculateJi and Ki in Theorem 3. By (31), we have Yti=yexpr−c+κ−1/2σi2t+σiWt. We denote τ≔T−t and Z≔−Wτ/τ. Then, Z follows a standard normal distribution. We set d−τ,y;G≔r−c+κ−1/2σi2τ+logy/G/σiτ and d+τ,y;G≔d−τ,y;G+σiτ. By a direct computation,(34)Jit,y=Ee−r+λiτG−yexp−σiτZ+r−c+κ−12σi2τ+=12π∫−∞+∞e−r+λiτG−yexp−σiτz+r−c+κ−12σi2τ+e−z2/2dz=12π∫d−τ,y;G+∞e−r+λiτG−yexp−σiτz+r−c+κ−12σi2τe−z2/2dz=Ge−r+λiτ12π∫d−τ,y;G+∞e−z2/2dz−ye−c−κ+λiτ12π∫d−τ,y;G+∞e−1/2z+σiτ2dz=Ge−r+λiτN−d−τ,y;G−ye−c−κ+λiτN−d+τ,y;G,where Ny=1/2π∫−∞ye−z2/2dz.
To calculateKi, we denote y˜iz,u≔yexp−σiuz+r−c+κ−1/2σi2u. Then, we obtain that(35)−λiEe−r+λiuVt+u,Yui,jXYui>biu+t=−λie−r+λiuEVt+u,Yui,jXYui>biu+t=−λi2π∫−∞+∞Vt+u,y˜iz,u,jXy˜iz,u>biu+te−z2/2dz=−λi2π∫−∞d−u,y;biu+tVt+u,y˜iz,u,je−z2/2dz,and by definition of Φi (see (30)),(36)Ee−r+λiuΦiYuiXYui<biu+t=e−r+λiu2π∫−∞+∞Φiy˜iz,uXy˜iz,u<biu+te−z2/2dz=e−r+λiu2π∫d−u,y;biu+t+∞Φiy˜iz,ue−z2/2dz=c−κ+λiye−r+λiu2π∫d−u,y;biu+t+∞e−z2/2−σiuz+r−c+κ−1/2σi2udz−r+λiGN−d−u,y;biu+t=c−κ+λiye−c+κ−λiu2π∫d+u,y;biu+t+∞e−z2/2dz−r+λiGN−d−u,y;biu+t=c−κ+λiye−c−κ+λiuN−d+u,y;biu+t−r+λiGN−d−u,y;biu+t.
Hence,(37)Kit,y,u,biu+t=c−κ+λiye−c−κ+λiuN−d+u,y;biu+t−r+λiGN−d−u,y;biu+t−λi2π∫−∞d−u,y;biu+tVt+u,y˜iz,u,je−z2/2dz.At the end of this section, we present a sensitivity analysis for the impacts of model parameters on surrender strategies based on the variational inequality given in Lemma1.Theorem 4.
Supposeσ1>σ2. Let Yti be defined as (31). Then,(38)b10t≤b1t≤b2t≤b20t,0<t<T,where bi0 (i∈S) is the free boundary of the following optimal stopping problem:(39)Vi0t,y=sup0≤τ≤T−tEye−rτhYτi,i∈S.Proof.
See Section5.Theorem 5.
The free boundaryb1t,b2t of problem (15) is decreasing with respect to σii∈S and c and increasing with respect to G, r, and κ.Proof.
See Section5.


## 4. Numerical Examples

In this section, we shall give some numerical examples to verify our theoretical findings. We now outline numerical techniques which will be used for calculating the optimal surrender boundary. The numerical method that we adopt was developed by Yuen and Yang [16] who implemented the trinomial tree method to solve the regime-switching option pricing problem. We will briefly explain how to use the trinomial tree method to solve our problem.LetΔt=T/n be the time step size. We take the jump ratios of the lattice as(40)u=eσΔt,d=e−σΔt,where(41)σ=maxi∈Sσi+1.5−1σ˜,where σ˜ is the arithmetic mean or geometric mean of σi, i∈S. For regime i, let πui, πmi, and πdi be the risk neutral probabilities corresponding to the discounted fund value increases, remains the same, and decreases, respectively. By moment matching technique, the values of the probabilities are given by(42)πmi=1−σi2σ2,πui=er−c+κΔt−e−σΔt−1−σ2/σi21−e−σΔteσΔt−e−σΔt,πdi=eσΔt−er−c+κΔt−1−σ2/σi2eσΔt−1eσΔt−e−σΔt.Letyj+1=uyj and yj−1=dyj, and denote by Vkyj,i the trinomial approximation of the primal value function for regime i at asset price yj and time tk=kΔt. Then, the trinomial formula of the primal value function is given by, for k=0,1,…,n−1,(43)Vkyj,i=maxe−rΔt∑l=12pilπuiVk+1yj+1,l+πmiVk+1yj,l+πdiVk+1yj−1,l,hyj,with Vnyj,i=hyj for j=0,±1,…,±n, in which pil is the transition probability from state i to state l for the time interval with length Δt. It is given by(44)pili,l∈S=eAΔt=I+∑l=1∞ΔtlAll!,where I is the identity matrix and A is the generator matrix of the Markov chain process given by (1).Letbki≈bitk. The optimal surrender boundary can be calculated by(45)bki=maxyj:Vkyj,i=hyj,k=0,1,…,n−1,i∈S.Example 1.
In this example, we assume the generator of the Markov chainαt is taken as(46)A=−0.50.50.5−0.5,the terminal time T=1, and the time step of the trinomial tree model is equal to 10,000.(1)
In Figure1, we choose the penalty rate κ=0.01, the risk-free interest rate r=0.06, the insurance charge c=0.02, the volatility σ1,σ2=0.1,0.15 for Figure 1(a) and σ1,σ2=0.3,0.25 for Figure 1(b), the minimum guarantee amount G=100. From Figure 1, we know that, at terminal time T, the surrender boundary value is equal to the minimum guarantee. The graph of the surrender boundary is generally increasing. Furthermore, Figure 1 shows how the optimal surrender boundary is influenced by the economic state. When the economy is in the state that the volatility is lower, the optimal surrender boundary is higher.(2)
In Figure2, we assume that G=100, c=0.02, σ1,σ2=0.25,0.35, and κ=0.01. We depict the optimal surrender boundaries for r=0.03,0.04,0.05,0.06, respectively. This figure investigates how the interest rate affects the surrender decisions. It shows that increasing interest rate results in higher surrender boundary.(3)
In Figure3, the parameters of our problem are given by r=0.06, c=0.02, κ=0.01, and σ1,σ2=0.25,0.35. We plot the surrender boundaries in Figure 3. The minimum guarantee values are G=75,100,125,150, respectively. We note from this figure that the surrender boundaries are shifted upwards as the minimum guarantee value increases.(4)
In Figure4, we assume that G=100, r=0.06, σ1,σ2=0.25,0.35, and κ=0.01. We analyze the sensitivity of the surrender boundaries with respect to the fee c. For c=0.02,0.03,0.04,0.05, we compute the surrender boundaries in this figure. It shows that a higher fee results in a lower surrender boundaries.(5)
In Figure5, we take G=100, r=0.06, σ1,σ2=0.25,0.35, and c=0.04. This figure shows how the varied κ impacts on the surrender boundaries. We vary the penalty rate κ between 0 and 0.03 and compute the surrender boundaries. From this figure, we see that the surrender boundaries move up when κ increases and the surrender incentive becomes much lower. Intuitively, increasing levels of κ make it more expensive to surrender the guarantee early. Hence, it is rational to delay exercising the guarantee.(6)
In Figure6, the parameters of the problem are given by G=100, r=0.06, κ=0.01, and c=0.02. This figure presents the impact of the volatility on the surrender boundaries. We choose the volatility as σ1,σ2=0.1,0.15, 0.15,0.2, 0.2,0.25, 0.25,0.3, respectively. We note from this figure that the surrender boundaries move down as volatility increases.
We note that surrender happens only when the discounted fund value falls below the optimal surrender boundary. Hence, intuitively, for Figures1 and 6, because the embedded guarantee with surrender option is a kind of American put option, the higher volatility means that the option is more valuable, which leads to a lower surrender boundary; for Figures 2–5, a higher r or a higher κ or a lower c or a higher G will provide a greater payoff to the policyholder and enlarge the surrender region. In conclusion, all the above cases are consistent with the theoretical findings in Theorems 1, 4, and 5.Figure 1
(a) Optimal surrender boundary forσ1,σ2=0.1,0.15; (b) optimal surrender boundary for σ1,σ2=0.3,0.25.
(a)(b)Figure 2
(a) The impact of varying interest rates on the optimal surrender boundary for regime 1; (b) the impact of varying interest rates on the optimal surrender boundary for regime 2.
(a)(b)Figure 3
(a) The impact of varying minimum guarantee levels on the optimal surrender boundary for regime 1; (b) the impact of varying minimum guarantee levels on the optimal surrender boundary for regime 2.
(a)(b)Figure 4
(a) The impact of varying fee rates on the optimal surrender boundary for regime 1; (b) the impact of varying fee rates on the optimal surrender boundary for regime 2.
(a)(b)Figure 5
(a) The impact of varying penalty rates on the optimal surrender boundary for regime 1; (b) the impact of varying penalty rates on the optimal surrender boundary for regime 2.
(a)(b)Figure 6
(a) The impact of varying volatility on the optimal surrender boundary for regime 1; (b) the impact of varying volatility on the optimal surrender boundary for regime 2.
(a)(b)Example 2.
In this example, we depict sample path of the discounted mutual fund valueYt=e−κT−tFt and the corresponding optimal surrender boundary in Figure 7. The time step is set to be 10,000. Assume that the generator of the Markov chain is given by(47)A=−0.40.40.5−0.5.
We set the initial fund value to beF0=100 and the initial state of Markov chain α0=1. Also, we suppose the minimum guarantee amount G=100, the interest rate r=0.06, the insurance fee in percentage c=0.02, the penalty rate κ=0.01, and the volatility σ1,σ2=0.25,0.45. From this figure, we see that when the discounted fund value falls to Yt0, the state of the economy switches to αt0=2. The sample path of the discounted fund value hits the optimal surrender boundary in regime 2 for the first time. Hence, the policyholder should surrender to receive minimum benefit.Figure 7
(a) Sample path of continuous time Markov chain with initial stateα0=1; (b) optimal surrender boundary and sample path of fund value.
(a)(b)


## 5. Proofs

In this section, we give detailed proofs of the results in the paper.


### 5.1. Proof of Lemma3

Proof.
For anyy1,y2>0, by the definition of V (see (15)) and moment estimates of Mt (see [19]), we have(48)Vt,y1,i−Vt,y2,i=sup0≤τ≤T−tEe−rτhYτy1−sup0≤τ≤T−tEe−rτhYτy2≤sup0≤τ≤T−tEe−rτhYτy1−hYτy2≤sup0≤τ≤T−tEYτy1−Yτy2≤sup0≤τ≤T−ty1−y2EMτ≤Cy1−y2,which means that V is Lipschitz continuous in y.
Next, we assume thatt1<t2 and τ1 is an ε-optimal stopping time for Vt1,y,i. Setting τ2=τ1∧T−t2, we obtain that(49)0≤Vt1,y,i−Vt2,y,i≤Ee−rτ1hYτ1y+ε−Ee−rτ2hYτ2y≤Ee−rτ2hYτ1y−hYτ2y+ε≤EYτ1y−Yτ2y+ε.
Integrating both sides of (13) and applying Burkholder–Davis–Gundy inequality, we derive that(50)EYτ1y−Yτ2y≤E∫τ2τ1r−c+κYsds+E∫τ2τ1σsYsdWs≤E∫0τ1−τ2r−c+κYs+τ2ds+E∫0τ1−τ2σs+τ2Ys+τ2dWs≤r−c+κEsup0≤t≤TYtt2−t1+Esup0≤t≤t2−t1∫0tσs+τ2Ys+τ2dWs≤r−c+κEsup0≤t≤TYtt2−t1+C′E∫0t2−t1σs+τ22Ys+τ22ds≤r−c+κEsup0≤t≤TYtt2−t1+C′σ1+σ2Esup0≤t≤TYtt2−t1,where σs means σαs. By moment estimates (see [19]), we have Esup0≤t≤TYt≤C″. Letting ε⟶0, we conclude that V is 1/2-Hölder continuous in t.


### 5.2. Proof of Lemma4

Proof.
Assume thaty1<y2 and let τ be the optimal stopping time for Vt,y1,i. By the definition of V (see (15)) and the assumption that κ<c, we derive that(51)Vt,y2,i−Vt,y1,i≥Ee−rτhYτy2−Ee−rτhYτy1≥Ee−rτYτy1−Yτy2=y1−y2Eeκ−cτNτ≥y1−y2,where Nt=exp∫0t−1/2σs2ds+∫0tσsdWs is a martingale. This implies the monotonicity of the function y↦Vt,y,i+y. If t,y′,i∈D, then the above inequality leads to(52)0≤Vt,y,i−hy≤Vt,y,i−G−y≤Vt,y′,i−G−y′=0,fory<y′,that is, t,y,i∈D.


### 5.3. Proof of Theorem1

Proof.
We make an assertion that(53)Di⊂t,y:y≤G∩t,y:y≤rGc−κ.
For anyt,y∈Di, since V>0 (Lemma 2) and Vt,y,i=G−y+, we derive that y<G.
We denotegy≔G−y. By Dynkin’s formula, we obtain that(54)Ee−rsgYsy=gy+E∫0se−rv⋅−rg+ℒgt+v,Yvy,αvdv.
For anyt,y∈t,y:−rgy+ℒgt,y,i>0, we can choose an open neighborhood U⊂0,T×0,+∞ of t,y such that(55)−rgy+ℒgt,y,i>0,for any t,y∈U. Let τU be the first exist time of U when t+s,Ysy started at t,y at time 0. Denote by Ti the first transition time of α with α0=i. Let τ∗≔τU∧Ti∧T−t. Since −rgYvy+ℒgt+v,Yvy,αv>0 for v∈0,τ∗ a.s., setting s=τ∗ in (54), by (15), we derive that(56)Vt,y,i≥Ee−rτ∗hYτ∗y≥Ee−rτ∗gYτ∗y=gy+E∫0τ∗e−rv⋅−rg+ℒgt+v,Yvy,αvdv>gy.
This implies thatt,y∈Ci, which is equivalent to(57)Di⊂t,y:−rgy+ℒgt,y,i≤0=t,y:y≤rGc−κ.
Moreover,Di≠∅. In fact, by Lemma 1,(58)Vtt,y,i+AVt,y,i−rVt,y,i=0inCi.
IfDi=∅, then Ci=0,T×0,+∞. Letting t⟶T,(59)limt⟶TVt,y,i=rhy−ℒht,y=rG−c−κy>0,when y<rG/c−κ. This contradicts with the fact that V is nonincreasing in t (Lemma 2). Now, the downward connectedness implies (22).
Then, we shall discuss the monotonicity ofbit. Assume that there exists some 0≤t1≤t2≤T such that bit1>bit2, by the definition of bit (see (22)), we derive that(60)Vt2,bit1,i>hbit1=Vt1,bit1,i,which is a contradiction (Lemma 2).
Next, we consider the continuity ofbit. Suppose that there exists some t0∈0,T such that(61)limt⟶t0−bit=y1<limt⟶t0+bit=y2.
Lettingtn≥t0 and tn⟶t0 as n⟶+∞, obviously, tn,bitn∈Di. Since Di is closed, we have t0,bit0+∈Di. By the definition of bi, bit0≥bit0+. On the contrary, since bi is increasing, bit0+≥bit0. Hence, bit is right continuous, i.e., t0,y∈Di, for y1<y<y2.
By Lemma1, for any t,y,i∈0,T×0,+∞×S,(62)Vt+AV−rVt,y,i≤0.
Lettingt⟶t0+, then, for any y∈y1,y2,(63)rVt0,y,j≤1λir+λiG−c−κ+λiy.
On the contrary, lettingt⟶t0, in (58), we derive that(64)Vt0,y,j=1λir+λiVt0,y,i−Vt+ℒVt0,y,i≥1λir+λiG−c−κ+λiy,where the last inequality follows from Lemma 2. Therefore,(65)Vt0,y,j=1λir+λiG−c−κ+λiy.
This leads to(66)Vyt0,y,j=−c−κ+λiλi<−1,where the last inequality follows from the fact that c>κ. This contradicts with Lemma 4. Hence, bit is continuous.
Finally, we prove (23) holds. As bit is nondecreasing and (53) holds, we have biT is well defined and biT≔limt⟶T−bit≤minG,rG/c−κ≔y∗. Suppose that biT<y∗. For any y¯∈biT,y∗, since VT,y,i=hy=gy, letting t⟶T− in (58), we have(67)VtT,y¯,1=rVT,y¯,1−AVT,y¯,1=rG−c−κy¯>0,which contradicts with Vt≤0 (Lemma 2).


### 5.4. Proof of Theorem2

Proof.
(24) follows directly from (58). (25) follows from the continuity of V (Lemma 3). (27) follows from (15) directly.
We shall prove the smooth pasting condition (26). We set y=bit. As t,y+ε∈Ci, we derive that(68)Vt,y+ε,i−Vt,y,iε≥G−y−ε−G−yε=−1,where the inequality follows from (15) and (25) and V is positive. Letting ε⟶0,(69)Vy+t,y,i≥−1,where Vy+ is the right derivative of V in y. For any ε>0, let τε′ be the optimal stopping time for Vt,y+ε,i and Ti be the first transition time of α with α0=i. Denote(70)τε≔infs∈0,Ti|Ysy+ε≤bit+s.
Obviously, ifTi≥T−t, then τε′=τε; otherwise, we have(71)τε′t,y,i=infs∈0,T−t|Ysy+ε≤bαst+s=τε∧infs∈Ti,T−t|Ysy+ε≤bαst+s.
By (14), we have(72)τε=infs∈0,Ti:∫0sr−c+κ−12σ2αvdv+∫0sσαvdWv≤logbit+sy+ε≤infs∈0,Ti:r−c+κ−12σi2s+σiWs≤logyy+ε,where the inequality follows from bi which is nondecreasing.
By the law of the iterated logarithm,ℚmin0≤t≤ar−c+κ−1/2σi2s+σiWs<0=1. This means that, for any a>0, there exists some ε>0 small enough such that τε<a, i.e., τε⟶0 as ε⟶0. Hence, (71) leads to τε′⟶0 as ε⟶0.
Now, we prepare to proveVy+t,y,i≤−1. As τε′ is the first time that Yty+ε hits the free boundary and bi≤G, we have G−Yτε′y+ε>0 and G−Yτε′y>0. Hence,(73)Vt,y+ε,i−Vt,y,iε≤Ee−rτε′G−Yτε′y+ε−Ee−rτε′G−Yτε′yε=Ee−rτε′Yτε′y−Yτε′y+εε=−EMτε′,where Mt is defined in (14). Letting ε⟶0, by Fatou’s lemma, we conclude that Vy+t,y,i=−1. As Vt,y,i=G−y for y<bit, it is easy to see that Vy−t,y,i=−1. Here, Vy− is the left derivative of V in y. We conclude that the smooth pasting condition (26) holds.
For uniqueness, letft,y,i defined on C¯ be a solution of the free boundary problem. We define Ns≔e−rsfs+t,Ys,αs and set τD≔infs∈0,T−t:Ysy≤bαss+t. By It o^’s lemma,(74)Ns∧τD=ft,y,i+∫0s∧τDe−rvft+Af−rft+v,Yvy,αvdv+N˜s∧τD,where N˜s∧τD,s≥0 is a martingale. Hence, by (24), Ns∧τD,s≥0 is a martingale. We know from the optional sampling theorem that(75)ft,y,i=N0=ENτD=Ee−rτDfτD+t,YτD,ατD=Ee−rτDhYτD=Vt,y,i,where the second equality follows from that Ns∧τD,s≥0 is a bounded martingale and dominated convergent theorem, the fourth equality follows from (25) and definition of τD, and the last equality follows from that τD is an optimal stopping time of Vt,y,i. This implies the uniqueness.
By Lemma3 and regularity of parabolic equation (see [20]), it is obvious that Vt, Vy, and Vyy exist and are continuous in C.


### 5.5. Proof of Theorem3

Proof.
Firstly, we consider the following optimal stopping problem:(76)Vit,y=sup0≤τ≤T−tEe−r+λiτhYτi+λi∫0τe−r+λiuVt+u,Yui,j,j≠i,where Vt,y,i is defined in (15) and Yti satisfies (31). Fixing i∈S, by a similar Markovian argument, we conclude that Vit,y is a solution to the free boundary problems (24)–(27). The uniqueness implies that Vit,y=Vt,y,i for t,y,i∈0,T×0,+∞×S.
By the regularity ofV (see Theorem 2), we can use the local time-space formula given in Peskir and Shiryaev [18] to obtain that(77)e−r+λisVis+t,Ysi=Vit,y+∫0se−r+λiuVit+12σi2y2Viyy+r−c+κyViy−r+λiVit+u,YuiχYui≠biu+tdu+∫0sσiYuiViyt+u,YuiχYui≠biu+tdWu+12∫0sViyt+u,Yui+−Viyt+u,Yui−⋅χYui=biu+tdℓubiYi,where ℓubiYi is the local time of Yi at the curve bi given by(78)ℓubiYi≔limε⟶012ε∫0uχbiv−ε<Yvi<biv+εdYiv,and dℓubiYi refers to integration with respect to the continuous increasing function u↦ℓubiYi.
We notice thatVit,y=G−y for y<bit and Vi satisfies (58) for y>bit. Hence,(79)Vit+12σi2y2Viyy+r−c+κyViy−r+λiVit+u,Yui=ΦiYui,ifYui<biu+t,−λiVu,Yui,j,ifYui>biu+t,where j≠i. Setting s=T−t together with (26) and (27), taking expectation in (77), and using notations (28) and (29), we derive that(80)Vit,y=Ee−r+λiT−thYT−ti−E∫0T−te−r+λiuΦiYuiχYui<biu+t−λiVt+u,Yui,jχYui>biu+tdu=Jit,y−∫0T−tKit,y,u,biu+tdu=Vt,y,i.
Finally, by condition (25), we derive that(81)Jit,bit−∫0T−tKit,bit,u,biu+tdu=hbit,i∈S.


### 5.6. Proof of Theorem4

Proof.
We setVt,y,i≔Vit,y, for i∈S. We denote N≔t,y:V1t,y<V2t,y. We shall prove N=∅. Otherwise, in N,(82)V2t,y>V1t,y≥hy,V2t+12σ22y2V2yy+r−c+κyV2y−rV2=−λ2V1−V2≥0,V1t+12σ12y2V1yy+r−c+κyV1y−rV1=−λ1V2−V1≤0,where the last two inequalities follow from (58). By Lemma 2, it follows that(83)V1t+12σ22y2V1yy+r−c+κyV1y−rV1=V1t+12σ12y2V1yy+r−c+κyV1y−rV1+12y2σ22−σ12V1yy≤0.
Furthermore,V1T,y=V2T,y and V1=V2 on the boundary of N. Hence, the comparison principle of variational inequality implies that V1≥V2 in N, which contradicts with definition of N. This leads to V1≥V2 for t,y∈0,T×0,+∞. That means C2⊂C1, i.e., b2t≥b1t.
By Markovian argument,Vi0 satisfies(84)min−Vit0−12σi2y2Viyy0−r−c+κyViy0+rVi0,Vi0−hy=0,Vi0T,y=hy.
By Lemma1, it follows that(85)−V1t0−12σ12y2V1yy0−r−c+κyV1y0+rV10−λ1V2−V1≥−λ1V2−V1≥0,−V2t−12σ22y2V2yy−r−c+κyV2y+rV2≥λ2V1−V2≥0.
Applying comparison principle of variational inequality, we deduce thatV10≥V1 and V2≥V20, i.e., b10t≤b1t and b2t≤b20t.


### 5.7. Proof of Theorem5

Proof.
Assume thatV^1,V^2 is the solution to variational inequality (18), where σi is σ^i, V¯1,V¯2 is the solution to variational inequality (18), where σi is σ¯i, and σ^i≥σ¯i for each i∈S. By Lemmas 1 and 2, it follows that(86)V^it+12σ¯i2y2V^iyy+r−c+κyV^iy−rV^i+λiV^j−V^i=12y2σ¯i2−σ^i2V^iyy≤0if V^it,y>hy and(87)V^it+12σ¯i2y2V^iyy+r−c+κyV^iy−rV^i+λiV^j−V^i≤12y2σ¯i2−σ^i2V^iyy≤0if V^it,y=hy. Employing comparison principle of the variational system (see [15]), we conclude that V^i≥V¯i, i.e., b¯it≥b^it, for i∈S. Hence, b1t,b2t is decreasing with respect to σi for each i∈S.
Next, assume thatV^1,V^2 is the solution to variational inequality (18), where G is G1, V¯1,V¯2 is the solution to variational inequality (18), where G is G2, and G1<G2. Then, we have(88)V¯it,y=sup0≤τ≤T−tEi,ye−rτG2−Yτ+≤G2−G1+sup0≤τ≤T−tEi,ye−rτG1−Yτ+=G2−G1+V^it,y.
This implies that ifV¯it,y>G2−y, then V^it,y≥V¯it,y+G1−G2>G1−y. If we denote by C^i the stopping region with respect to G1 and C¯i the stopping region with respect to G2, then C¯i⊂C^i, i.e., b¯i≥b^i. Hence, b1t,b2t is increasing with respect to G.
In the following, assume thatV^1,V^2 is the solution to variational inequality (18), where κ is κ1, V¯1,V¯2 is the solution to variational inequality (18), where κ is κ2, and κ1<κ2. By Lemma 2, it follows that(89)V^it+12σi2y2V^iyy+r−c+κ2yV^iy−rV^i+λiV^j−V^i=κ2−κ1yV^iy≤0if V^it,y>hy and(90)V^it+12σ¯i2y2V^iyy+r−c+κ2yV^iy−rV^i+λiV^j−V^i≤κ2−κ1yV^iy≤0if V^it,y=hy. Employing comparison principle of the variational system (see [15]), we conclude that V^i≥V¯i, i.e., b¯it≥b^it, for i∈S. By a similar argument, we can prove b1t,b2t is increasing with respect to r and decreasing with respect to c.


## 5.1. Proof of Lemma3

Proof.
For anyy1,y2>0, by the definition of V (see (15)) and moment estimates of Mt (see [19]), we have(48)Vt,y1,i−Vt,y2,i=sup0≤τ≤T−tEe−rτhYτy1−sup0≤τ≤T−tEe−rτhYτy2≤sup0≤τ≤T−tEe−rτhYτy1−hYτy2≤sup0≤τ≤T−tEYτy1−Yτy2≤sup0≤τ≤T−ty1−y2EMτ≤Cy1−y2,which means that V is Lipschitz continuous in y.
Next, we assume thatt1<t2 and τ1 is an ε-optimal stopping time for Vt1,y,i. Setting τ2=τ1∧T−t2, we obtain that(49)0≤Vt1,y,i−Vt2,y,i≤Ee−rτ1hYτ1y+ε−Ee−rτ2hYτ2y≤Ee−rτ2hYτ1y−hYτ2y+ε≤EYτ1y−Yτ2y+ε.
Integrating both sides of (13) and applying Burkholder–Davis–Gundy inequality, we derive that(50)EYτ1y−Yτ2y≤E∫τ2τ1r−c+κYsds+E∫τ2τ1σsYsdWs≤E∫0τ1−τ2r−c+κYs+τ2ds+E∫0τ1−τ2σs+τ2Ys+τ2dWs≤r−c+κEsup0≤t≤TYtt2−t1+Esup0≤t≤t2−t1∫0tσs+τ2Ys+τ2dWs≤r−c+κEsup0≤t≤TYtt2−t1+C′E∫0t2−t1σs+τ22Ys+τ22ds≤r−c+κEsup0≤t≤TYtt2−t1+C′σ1+σ2Esup0≤t≤TYtt2−t1,where σs means σαs. By moment estimates (see [19]), we have Esup0≤t≤TYt≤C″. Letting ε⟶0, we conclude that V is 1/2-Hölder continuous in t.


## 5.2. Proof of Lemma4

Proof.
Assume thaty1<y2 and let τ be the optimal stopping time for Vt,y1,i. By the definition of V (see (15)) and the assumption that κ<c, we derive that(51)Vt,y2,i−Vt,y1,i≥Ee−rτhYτy2−Ee−rτhYτy1≥Ee−rτYτy1−Yτy2=y1−y2Eeκ−cτNτ≥y1−y2,where Nt=exp∫0t−1/2σs2ds+∫0tσsdWs is a martingale. This implies the monotonicity of the function y↦Vt,y,i+y. If t,y′,i∈D, then the above inequality leads to(52)0≤Vt,y,i−hy≤Vt,y,i−G−y≤Vt,y′,i−G−y′=0,fory<y′,that is, t,y,i∈D.


## 5.3. Proof of Theorem1

Proof.
We make an assertion that(53)Di⊂t,y:y≤G∩t,y:y≤rGc−κ.
For anyt,y∈Di, since V>0 (Lemma 2) and Vt,y,i=G−y+, we derive that y<G.
We denotegy≔G−y. By Dynkin’s formula, we obtain that(54)Ee−rsgYsy=gy+E∫0se−rv⋅−rg+ℒgt+v,Yvy,αvdv.
For anyt,y∈t,y:−rgy+ℒgt,y,i>0, we can choose an open neighborhood U⊂0,T×0,+∞ of t,y such that(55)−rgy+ℒgt,y,i>0,for any t,y∈U. Let τU be the first exist time of U when t+s,Ysy started at t,y at time 0. Denote by Ti the first transition time of α with α0=i. Let τ∗≔τU∧Ti∧T−t. Since −rgYvy+ℒgt+v,Yvy,αv>0 for v∈0,τ∗ a.s., setting s=τ∗ in (54), by (15), we derive that(56)Vt,y,i≥Ee−rτ∗hYτ∗y≥Ee−rτ∗gYτ∗y=gy+E∫0τ∗e−rv⋅−rg+ℒgt+v,Yvy,αvdv>gy.
This implies thatt,y∈Ci, which is equivalent to(57)Di⊂t,y:−rgy+ℒgt,y,i≤0=t,y:y≤rGc−κ.
Moreover,Di≠∅. In fact, by Lemma 1,(58)Vtt,y,i+AVt,y,i−rVt,y,i=0inCi.
IfDi=∅, then Ci=0,T×0,+∞. Letting t⟶T,(59)limt⟶TVt,y,i=rhy−ℒht,y=rG−c−κy>0,when y<rG/c−κ. This contradicts with the fact that V is nonincreasing in t (Lemma 2). Now, the downward connectedness implies (22).
Then, we shall discuss the monotonicity ofbit. Assume that there exists some 0≤t1≤t2≤T such that bit1>bit2, by the definition of bit (see (22)), we derive that(60)Vt2,bit1,i>hbit1=Vt1,bit1,i,which is a contradiction (Lemma 2).
Next, we consider the continuity ofbit. Suppose that there exists some t0∈0,T such that(61)limt⟶t0−bit=y1<limt⟶t0+bit=y2.
Lettingtn≥t0 and tn⟶t0 as n⟶+∞, obviously, tn,bitn∈Di. Since Di is closed, we have t0,bit0+∈Di. By the definition of bi, bit0≥bit0+. On the contrary, since bi is increasing, bit0+≥bit0. Hence, bit is right continuous, i.e., t0,y∈Di, for y1<y<y2.
By Lemma1, for any t,y,i∈0,T×0,+∞×S,(62)Vt+AV−rVt,y,i≤0.
Lettingt⟶t0+, then, for any y∈y1,y2,(63)rVt0,y,j≤1λir+λiG−c−κ+λiy.
On the contrary, lettingt⟶t0, in (58), we derive that(64)Vt0,y,j=1λir+λiVt0,y,i−Vt+ℒVt0,y,i≥1λir+λiG−c−κ+λiy,where the last inequality follows from Lemma 2. Therefore,(65)Vt0,y,j=1λir+λiG−c−κ+λiy.
This leads to(66)Vyt0,y,j=−c−κ+λiλi<−1,where the last inequality follows from the fact that c>κ. This contradicts with Lemma 4. Hence, bit is continuous.
Finally, we prove (23) holds. As bit is nondecreasing and (53) holds, we have biT is well defined and biT≔limt⟶T−bit≤minG,rG/c−κ≔y∗. Suppose that biT<y∗. For any y¯∈biT,y∗, since VT,y,i=hy=gy, letting t⟶T− in (58), we have(67)VtT,y¯,1=rVT,y¯,1−AVT,y¯,1=rG−c−κy¯>0,which contradicts with Vt≤0 (Lemma 2).


## 5.4. Proof of Theorem2

Proof.
(24) follows directly from (58). (25) follows from the continuity of V (Lemma 3). (27) follows from (15) directly.
We shall prove the smooth pasting condition (26). We set y=bit. As t,y+ε∈Ci, we derive that(68)Vt,y+ε,i−Vt,y,iε≥G−y−ε−G−yε=−1,where the inequality follows from (15) and (25) and V is positive. Letting ε⟶0,(69)Vy+t,y,i≥−1,where Vy+ is the right derivative of V in y. For any ε>0, let τε′ be the optimal stopping time for Vt,y+ε,i and Ti be the first transition time of α with α0=i. Denote(70)τε≔infs∈0,Ti|Ysy+ε≤bit+s.
Obviously, ifTi≥T−t, then τε′=τε; otherwise, we have(71)τε′t,y,i=infs∈0,T−t|Ysy+ε≤bαst+s=τε∧infs∈Ti,T−t|Ysy+ε≤bαst+s.
By (14), we have(72)τε=infs∈0,Ti:∫0sr−c+κ−12σ2αvdv+∫0sσαvdWv≤logbit+sy+ε≤infs∈0,Ti:r−c+κ−12σi2s+σiWs≤logyy+ε,where the inequality follows from bi which is nondecreasing.
By the law of the iterated logarithm,ℚmin0≤t≤ar−c+κ−1/2σi2s+σiWs<0=1. This means that, for any a>0, there exists some ε>0 small enough such that τε<a, i.e., τε⟶0 as ε⟶0. Hence, (71) leads to τε′⟶0 as ε⟶0.
Now, we prepare to proveVy+t,y,i≤−1. As τε′ is the first time that Yty+ε hits the free boundary and bi≤G, we have G−Yτε′y+ε>0 and G−Yτε′y>0. Hence,(73)Vt,y+ε,i−Vt,y,iε≤Ee−rτε′G−Yτε′y+ε−Ee−rτε′G−Yτε′yε=Ee−rτε′Yτε′y−Yτε′y+εε=−EMτε′,where Mt is defined in (14). Letting ε⟶0, by Fatou’s lemma, we conclude that Vy+t,y,i=−1. As Vt,y,i=G−y for y<bit, it is easy to see that Vy−t,y,i=−1. Here, Vy− is the left derivative of V in y. We conclude that the smooth pasting condition (26) holds.
For uniqueness, letft,y,i defined on C¯ be a solution of the free boundary problem. We define Ns≔e−rsfs+t,Ys,αs and set τD≔infs∈0,T−t:Ysy≤bαss+t. By It o^’s lemma,(74)Ns∧τD=ft,y,i+∫0s∧τDe−rvft+Af−rft+v,Yvy,αvdv+N˜s∧τD,where N˜s∧τD,s≥0 is a martingale. Hence, by (24), Ns∧τD,s≥0 is a martingale. We know from the optional sampling theorem that(75)ft,y,i=N0=ENτD=Ee−rτDfτD+t,YτD,ατD=Ee−rτDhYτD=Vt,y,i,where the second equality follows from that Ns∧τD,s≥0 is a bounded martingale and dominated convergent theorem, the fourth equality follows from (25) and definition of τD, and the last equality follows from that τD is an optimal stopping time of Vt,y,i. This implies the uniqueness.
By Lemma3 and regularity of parabolic equation (see [20]), it is obvious that Vt, Vy, and Vyy exist and are continuous in C.


## 5.5. Proof of Theorem3

Proof.
Firstly, we consider the following optimal stopping problem:(76)Vit,y=sup0≤τ≤T−tEe−r+λiτhYτi+λi∫0τe−r+λiuVt+u,Yui,j,j≠i,where Vt,y,i is defined in (15) and Yti satisfies (31). Fixing i∈S, by a similar Markovian argument, we conclude that Vit,y is a solution to the free boundary problems (24)–(27). The uniqueness implies that Vit,y=Vt,y,i for t,y,i∈0,T×0,+∞×S.
By the regularity ofV (see Theorem 2), we can use the local time-space formula given in Peskir and Shiryaev [18] to obtain that(77)e−r+λisVis+t,Ysi=Vit,y+∫0se−r+λiuVit+12σi2y2Viyy+r−c+κyViy−r+λiVit+u,YuiχYui≠biu+tdu+∫0sσiYuiViyt+u,YuiχYui≠biu+tdWu+12∫0sViyt+u,Yui+−Viyt+u,Yui−⋅χYui=biu+tdℓubiYi,where ℓubiYi is the local time of Yi at the curve bi given by(78)ℓubiYi≔limε⟶012ε∫0uχbiv−ε<Yvi<biv+εdYiv,and dℓubiYi refers to integration with respect to the continuous increasing function u↦ℓubiYi.
We notice thatVit,y=G−y for y<bit and Vi satisfies (58) for y>bit. Hence,(79)Vit+12σi2y2Viyy+r−c+κyViy−r+λiVit+u,Yui=ΦiYui,ifYui<biu+t,−λiVu,Yui,j,ifYui>biu+t,where j≠i. Setting s=T−t together with (26) and (27), taking expectation in (77), and using notations (28) and (29), we derive that(80)Vit,y=Ee−r+λiT−thYT−ti−E∫0T−te−r+λiuΦiYuiχYui<biu+t−λiVt+u,Yui,jχYui>biu+tdu=Jit,y−∫0T−tKit,y,u,biu+tdu=Vt,y,i.
Finally, by condition (25), we derive that(81)Jit,bit−∫0T−tKit,bit,u,biu+tdu=hbit,i∈S.


## 5.6. Proof of Theorem4

Proof.
We setVt,y,i≔Vit,y, for i∈S. We denote N≔t,y:V1t,y<V2t,y. We shall prove N=∅. Otherwise, in N,(82)V2t,y>V1t,y≥hy,V2t+12σ22y2V2yy+r−c+κyV2y−rV2=−λ2V1−V2≥0,V1t+12σ12y2V1yy+r−c+κyV1y−rV1=−λ1V2−V1≤0,where the last two inequalities follow from (58). By Lemma 2, it follows that(83)V1t+12σ22y2V1yy+r−c+κyV1y−rV1=V1t+12σ12y2V1yy+r−c+κyV1y−rV1+12y2σ22−σ12V1yy≤0.
Furthermore,V1T,y=V2T,y and V1=V2 on the boundary of N. Hence, the comparison principle of variational inequality implies that V1≥V2 in N, which contradicts with definition of N. This leads to V1≥V2 for t,y∈0,T×0,+∞. That means C2⊂C1, i.e., b2t≥b1t.
By Markovian argument,Vi0 satisfies(84)min−Vit0−12σi2y2Viyy0−r−c+κyViy0+rVi0,Vi0−hy=0,Vi0T,y=hy.
By Lemma1, it follows that(85)−V1t0−12σ12y2V1yy0−r−c+κyV1y0+rV10−λ1V2−V1≥−λ1V2−V1≥0,−V2t−12σ22y2V2yy−r−c+κyV2y+rV2≥λ2V1−V2≥0.
Applying comparison principle of variational inequality, we deduce thatV10≥V1 and V2≥V20, i.e., b10t≤b1t and b2t≤b20t.


## 5.7. Proof of Theorem5

Proof.
Assume thatV^1,V^2 is the solution to variational inequality (18), where σi is σ^i, V¯1,V¯2 is the solution to variational inequality (18), where σi is σ¯i, and σ^i≥σ¯i for each i∈S. By Lemmas 1 and 2, it follows that(86)V^it+12σ¯i2y2V^iyy+r−c+κyV^iy−rV^i+λiV^j−V^i=12y2σ¯i2−σ^i2V^iyy≤0if V^it,y>hy and(87)V^it+12σ¯i2y2V^iyy+r−c+κyV^iy−rV^i+λiV^j−V^i≤12y2σ¯i2−σ^i2V^iyy≤0if V^it,y=hy. Employing comparison principle of the variational system (see [15]), we conclude that V^i≥V¯i, i.e., b¯it≥b^it, for i∈S. Hence, b1t,b2t is decreasing with respect to σi for each i∈S.
Next, assume thatV^1,V^2 is the solution to variational inequality (18), where G is G1, V¯1,V¯2 is the solution to variational inequality (18), where G is G2, and G1<G2. Then, we have(88)V¯it,y=sup0≤τ≤T−tEi,ye−rτG2−Yτ+≤G2−G1+sup0≤τ≤T−tEi,ye−rτG1−Yτ+=G2−G1+V^it,y.
This implies that ifV¯it,y>G2−y, then V^it,y≥V¯it,y+G1−G2>G1−y. If we denote by C^i the stopping region with respect to G1 and C¯i the stopping region with respect to G2, then C¯i⊂C^i, i.e., b¯i≥b^i. Hence, b1t,b2t is increasing with respect to G.
In the following, assume thatV^1,V^2 is the solution to variational inequality (18), where κ is κ1, V¯1,V¯2 is the solution to variational inequality (18), where κ is κ2, and κ1<κ2. By Lemma 2, it follows that(89)V^it+12σi2y2V^iyy+r−c+κ2yV^iy−rV^i+λiV^j−V^i=κ2−κ1yV^iy≤0if V^it,y>hy and(90)V^it+12σ¯i2y2V^iyy+r−c+κ2yV^iy−rV^i+λiV^j−V^i≤κ2−κ1yV^iy≤0if V^it,y=hy. Employing comparison principle of the variational system (see [15]), we conclude that V^i≥V¯i, i.e., b¯it≥b^it, for i∈S. By a similar argument, we can prove b1t,b2t is increasing with respect to r and decreasing with respect to c.


## 6. Conclusions

In this paper, we study the valuation of the GMMB rider embedded in a VA contract. We focus on the optimal surrender boundary and use partial differential equation approach together with probabilistic approach to analyze the properties of it. Moreover, the impacts of the varying model parameters on the optimal surrender boundary are studied rigorously. The regime-switching Volterra-type integral equations are derived by the local time-space formula. However, the numerical methods to solve the integral equations remain unknown, which will be a challenging problem for future research. The numerical evidence supports our theoretical findings. Besides, it is important to extend our results by incorporating interest rates that depend on the Markov chain. We will consider this case in the future.



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*Source: 9969937-2021-07-14.xml*