A fundamental goal of risk managers of insurance companies is to improve the solvency and stability of their companies. This goal can be reached through the choice of dividend-payout and reinsurance strategies. That is, the choice of how much of a company's reserves to be paid out as dividends and that of how much of claim risk to be shared with a ceded company.
The first strategy requires a trade off between the insurance company and its shareholders.
Shareholders would react positive when the dividend payouts increase, but, meanwhile, paying out dividends would also reduce the future reserves that are essential for the survival of the company. Due to the solvency capital requirement for large loss in emergency incidents, such as the "Black Swan" event, the outbreak of COVID-19, insurance companies tend to cede part of risk to reinsurer companies at the expense of reinsurance premium. So the second strategy comes in and it requires a trade off between the insurance company and the ceded company. The ceded company promises to cover certain part of the arising claims that the insurance company may face so as to increase the insurance company's survival chance; meanwhile, the latter has to reduce the its reserves to pay reinsurance premium to the former, decreasing the survival chance. This paper takes both strategies into consideration in a single model. Our objective is to find the optimal reinsurance and dividend-payout strategies for an insurer so as to maximize its expected cumulative discounted dividend payouts until its bankruptcy or a given maturity time which comes earlier.
In mathematical insurance, the optimal reinsurance and dividend-payout problem is well studied; see [29, 27, 28] . Because the dynamic surplus process must be random and depend on insurer's strategies, the problem usually boils down to a continuous-or discrete-time stochastic control problem. Two types of targets are widely considered: one is to minimize the ruin probability; see [6, 4, 23] , and the other one is to maximize dividend payouts until bankruptcy; see [1, 3, 19, 20] . Many existing studies on this topic show that it is optimal to cede risk and pay out dividends according to a band strategy; see Azcue and Muler [2] for a complete exposition on the topic. For instance, Taksar [27] considers a model in which the surplus of an insurance company follows a controlled diffusion process and the insurer pays out dividends in two different ways. In the first case, the dividend-payout rate is constrained to a bounded interval [0, l]. The optimal dividend-payout strategy turns out to be an "all or nothing" policy with respect to (w.r.t.) dividend rate. That is, paying out dividends at the minimum rate 0 if the surplus is lower than a threshold, and paying out at the maximum rate ℓ otherwise. In the second case, the dividend-payout rate is unbounded. The optimal strategy is to keep the surplus under some barrier, that is, paying out all reserves in excess of the barrier as dividends, and doing nothing under the barrier. The optimal reinsurance strategy, which clearly depends on surplus level, is shown to be that insurers should be exposed to higher risk as surplus increases. Along this line of research, other factors including, but not limit to, liability, regime switching, have also been taken into account in model formulation; see [29, 28] .
Insurance market has grown furious in recent years. Insurance companies, especially the global insurance companies, such as AIA Group Ltd., AIG, China Life Insurance Company, operate diverse businesses and offer insurance products on various term basis, such as property and casualty insurance, life insurance, health insurance. In this regard, it is natural for insurers to consider their total risk exposure as a combination of different types of risk. Mathematically speaking, the total risk exposure may take any, discrete or continuous, probability distributions. Similarly, but more importantly, the reinsurance strategies should also be based on different types of risk, resulting in non classical complicated reinsurance policies. Although the optimal reinsurance problem has been extensively investigated, most of existing literature analyzes only for typical reinsurance policies such as proportional and excess of loss reinsurance; see, e.g., [27, 22, 24, 18, 16, 17, 20] . This paper investigates an optimal reinsurance and dividend-payout problem, but to bring more practical features, we do not restrict ourself to these typical reinsurance policies. We consider a controlled diffusion surplus process, which is a good approximation of the classical Cramér-Lundberg process as well-justified by Grandell [12] . A closest model to this paper is considered by Guan, Yi and
Chen [14] , where the risk control model is relatively simple and the type of reinsurance policy is constrained to proportional ones. It turns out that the reinsurance scheme is restrictive.
Tan, et al., [25] consider a similar surplus process, but their model is of infinite time horizon, so the corresponding Hamilton-Jacobi-Bellman (HJB) equation is an ordinary differential equation (ODE), which is much easier to handle than that of a partial differential equation (PDE) in our case.
In our model, both drift and volatility of the controlled surplus process depend on the reinsurance policy. The target of the insurer is to maximize the expected cumulative discounted dividend payouts until bankruptcy or a given maturity time which comes earlier.
The model has the following features: First, we do not confine the reinsurance contracts to be some particular ones such as proportional reinsurance or excess of loss reinsurance. The insurance company can chose its reinsurance policy freely at its will subject to the expected value premium principle. Second, the reinsurance contracts are chosen dynamically depending on the surplus level. It turns out that the optimal reinsurance policy is a feedback one that depends on surplus, time and ceded risk. Third, the insurance claims can admit any probability distribution having a finite third moment. For bounded claims, we have a better understanding of the optimal reinsurance strategy. Finally, we consider a finite time horizon problem, making the HJB equation being an extremely challenging fully nonlinear one with gradient constraint.
Like many existing studies, we adopt a PDE approach to study the problem. Because the problem is a singular control problem with respective to (w.r.t.) dividend payout, the corresponding HJB equation turns out be a variational inequality problem with gradient constraint. As an unavoidable consequence of the choice of the time-dependent reinsurance policy, there is a functional optimization problem appearing in the operator, making the problem become a fully nonlinear one. Using the method in [8, 9, 7] , we show that the gradient function itself satisfies an obstacle problem. Finally, a comparison principle, an optimal control and necessary properties (such as C 2,1 smoothness and uniqueness) for the full nonlinear problem are derived by penalty approximation method.
Our model provides numerous economic insights. We show that there is a smooth, timedependent, dividend-payout barrier that divides the surplus-time space into a non-dividendpayout region and a dividend-payout region. The insurer should pay out all reserves in excess of the dividend-payout barrier (that is, all reserves in the dividend-payout region). Furthermore, we find a ceded risk and time dependent smooth reinsurance barrier that divides the non-dividend-payout region into a reinsurance region and a non-reinsurance region, in an increasing order of surplus. As ceded risk increases, the non-reinsurance region is shrinking, while, the reinsurance region is expanding. The former never vanishes, but the latter disappears when the ceded risk is smaller than an explicitly given constant. The insurance company should be exposed to higher risk as surplus increases in the reinsurance region; exposed to all risk once its surplus goes into the non-reinsurance region. We also provide accurate explicitly estimations for the localities of these regions. This makes the relevant numerical solution scheme can be easily established. When the claims are bounded random variables, we show there is a uniform non-action region, in which the insurer should be exposed to all risk and not to pay out dividends.
The reminder of the paper is organised as follows. In Section 2, we formulate the optimal reinsurance and dividend-payout problem. Section 3 poses the corresponding HJB equation and gives a verification theorem. The properties such as the existence and uniqueness of a classical solution to the HJB problem are also provided. Section 4 is devoted to the study of the reinsurance, non-reinsurance and dividend-payout regions. The proofs of some results are given in the appendices.
This paper investigates an optimal reinsurance and dividend-payout problem. We first need to model the cash reserves for an insurance company (the insurer), let us start from the classical Cramér-Lundberg model.
In the classical Cramér-Lundberg model, there are two different components that affect the cash reserve (also called surplus) dynamics. The first one is the receiving payments of premium from the policyholders at a constant rate p continuously. The other one is the outgoing payments for claims. If we denote the total number of claims received until time t by N t and the size of the ith claim by Z i , then R t , the company's surplus at time t, is given by
where N t is a Poisson process with intensity 1, and all claims are independent and identically distributed random variables independent of N t .
Sometimes, the insurance company needs to protect itself by sharing its risk with a ceded company (the reinsurer). The insurer buys reinsurance contracts from the reinsurer. Given The presence of reinsurance modifies the risk exposure of the insurer. It distorts the incoming and outgoing cash flow of the insurer's surplus process (2.1). Under big portfolios, as well-justified by Grandell [12] , the surplus process R t can be approximated by the following diffusion process
where {W t } t 0 is a standard Brownian motion independent of the random claim Z 1 ,
, and p(I t ) denotes the reinsurance premium corresponding to the reinsurance contract I t . Remark that the contract I t may depend on the status of the surplus process up to t, so it is stochastic.
In this paper we consider the expected value premium principle for both insurance and reinsurance contracts, given by
where δ, ρ > 0 are called safety loadings of the insurance premium and reinsurance premium.
This is fundamental to the insurance pricing as it stipulates that the (re)insurer has a positive safety loading on the underwritten risk. In this case, we can rewrite (2.2) as
We impose that ρ > δ, i.e., γ > 0, to ensure that reinsurance is non-cheap. If reinsurance is too cheap, ρ < δ, then the insurer can simply eliminate all its risk exposure by ceding all arising claims to the reinsurer, reaping the certain profit of δ − ρ > 0 with zero ruin probability. After time scaling, we assume ρ = 1 from now on.
In this paper, we assume that the insurer will pay out part of its surplus as dividends to its shareholders. Let L t be the cumulative dividend extracted from the surplus process until t, which is a non-decreasing càdlàg (right continuous with left limits) process. It is chosen by the insurer according to its surplus status. Then the new surplus process {R s } s t beginning at time t with an initial value x satisfies, recalling that
This is also a càdlàg process. It jumps at the same time as L with the same jump size but
The insurer is not allowed to pay out dividends more than the existing surplus, so L s −L s− R s− at any time s. As a consequence, R s = R s− − (L s − L s− ) 0. So the surplus of the insurance company is R τ = 0 at the ruin time.
The objective of our optimal reinsurance and dividend-payout model is to find a retained loss policy H t = {H s } s t− (or equivalently, a reinsurance policy I t = {I s } s t− ) and a dividend-payout policy L t = {L s } s t− to maximize the expectation of discounted cumulative dividend payouts until bankruptcy or a given maturity T > 0 which arrives earlier. The value function of our problem is defined as
where c is a positive discount factor. This is a singular control problem.
In the rest of this paper, we will investigate the value function and provide the optimal reinsurance and dividend-payout strategies.
We now study Problem (2.5) by dynamic programming principle.
The claims are allowed to be discrete or continuous distributed. Let F (z) denote their common cumulative distribution function. Because the claims are nonnegative, F (0−) = 0.
For technical reason, we assume the third moment is finite and denote
Also the retained loss function (or equivalently, the ceded function) is subject to the con-
We introduce the following variational inequality
This variational inequality is linked to the optimal reinsurance and dividend-payout problem (2.5) by the following verification theorem.
where V , defined by (2.5), is the value function of optimal reinsurance and dividend-payout problem.
The proof is given in Appendix A. Clearly, this result tells us that Problem (3.1) admits at most one classical solution. For simplicity we call (3.1) the HJB equation for our problem (2.5), although it is a time-changed equation of the standard definition (see [31] ).
If γ µ 1 , then it is easy to check that v ≡ x is the solution to Problem (3.1). In this case one can see that the drift of the surplus process in (2.3) is either negative if E[I t (Z 1 )] > 0 or 0 otherwise. So the insurance company has no incentive to survive. Consequently, the optimal policy is to pay out all reserves as dividends and let the company be bankrupt immediately.
This gives the optimal value x for Problem (2.5). The problem is boring in this case. So in the rest of this paper, we assume 0 < γ < µ 1 .
To tackle Problem (3.1), one hopes to solve the supremum in the PDE, namely
that is,
where functions A(y) and B(y) are defined by
We note that A(y) and B(y) are both decreasing functions in C 1 (R), which satisfy
Define an operator
Then Problem (3.1) can be rewritten as
This is a variational inequality problem for a fully nonlinear elliptic operator with gradient constraint. A usual approach to study this kind of problem is to transform it into a variational inequality problem for its gradient. Then the gradient constraint becomes value constraint and the new variational inequality becomes a well-studied obstacle problem; see [8, 9, 14, 15] .
In this paper, we adopt the same approach.
Below we use intuitive argument to find a variational inequality for the gradient of v. We first note that
where we used (3.6) and (3.7) to get the last equation. By this, one can easily deduce that
where the operator T is defined as
Next, we deduce a boundary condition for v x . Define a continuous function
Hence f is strictly decreasing. Also
so f has a unique root, which is positive, denote by λ in the rest of this paper. See Figure   1 for an illustration of the function f . Figure 1 : The function f is continuous and strictly deceasing, and admits a unique positive root λ.
On the other hand, owing to the boundary condition v(0, t) = 0 and that v t − Lv = 0 Combining the above results, we suggest the following variational inequality for
This is an obstacle problem for a quasilinear elliptic operator with mixed boundary conditions, so we can study it by penalty approximation method. This method gives the following existence and uniqueness results.
Theorem 3.2 Problem (3.1) (or equivalently (3.8)) has a unique solution v ∈ C 2,1 Q T \
Proof: The proof is given in Appendix B.
From now on, we fix v as in Theorem 3.2. By Verification Theorem 3.1, it completely characterizes the value function of the optimal reinsurance and dividend-payout problem (2.5) . In the rest part of this paper, we express the optimal reinsurance and dividend-payout strategies by this function and study their properties.
In the previous section, we have solved the existence and uniqueness issues for the HJB equation (3.1). In the following sections we investigate the optimal strategies for Problem (2.5) . We first study the optimal dividend-payout strategy and then the optimal reinsurance strategy.
To investigate the optimal dividend-payout strategy, we divide the whole domain Q T :=
and a non-dividend-payout region
Here and hereafter we use the notation t := T − t.
Because v x 1 and v xx 0, we can express them as
where d(·) is the dividend-payout boundary, defined by
In the following, we come to show the boundary d(·) is uniformly upper bounded by an explicit given constant. To this end, we will construct a function u(x) such that u(x) v x (x, t), then clearly inf{x 0 | u(x) = 1} provides a uniformly upper bound for d(·).
First, we construct a function to bound v x (0, t). For this, we let v(x) :=
It is easily seen that v x is continuous and decreasing, so v is a concave function.
Now, we are ready to construct a supper solution to Problem (3.13). Define
Clearly, u is convex and u ∈ W 2,1 p (Q T ). If 0 < x < x 2 , then u x 0. By (3.5) and using the elementary inequality x 2 − 2xy −y 2 , we obtain
Summarizing the above results, the dividend-payout boundary is completely characterized in the following theorem.
See Figure 2 for an illustration of the dividend-payout barrier d( t) as well as dividend-payout region D and non-dividend-payout region N D. Proof: The monotone property of d(·) is due to (3.18) . We now prove the continuity. Suppose on the contrary there exists t 0 such that d( t 0 −) < d( t 0 +). By the continuity of v x ,
Since v x is decreasing in x, we have v x ≡ 1 and thus v xx ≡ 0 on (d( t 0 −), d( t 0 +)). By (3.8), we further have v t (x, t 0 ) = −µ 1 +γ −cv(x, t 0 ) for
x ∈ (d( t 0 −), d( t 0 +)); and consequently, v tx (x, t 0 ) = −cv x (x, t 0 ) = −c < 0, which contradicts (3.18) . In a similar way, we can prove lim t →0+ d( t ) = 0. Now we prove d(·) > 0. Suppose not, then, by monotonicity, there exists t 0 > 0 such
, then both v and v satisfy (3.8). By the uniqueness, we get v ≡ v.
This implies v ≡ x for all t > 0, but this contradicts v t − Lv 0 by recalling our assumption that γ < µ 1 .
It is only left to show the smoothness of d(·) when Z 1 is bounded. Suppose F (ẑ) = 1 for someẑ > 0. Then
For any t 0 , because v x and v xx are continuous, we have v x = 1 and v xx = 0 at (d( t 0 ), t 0 ).
This implies v x +ẑv xx > 0 in some neighborhood B of (d( t 0 ), t 0 ). By (3.16) and (4.2), we
Because the coefficients are constants in the above equation, using the method in [11] , we can prove d( t ) ∈ C ∞ at a neighborhood of t 0 . Since t 0 is arbitrarily chosen, we conclude that d(t) ∈ C ∞ (0, T ).
In view of the original optimal reinsurance and dividend-payout problem (2.5), by the proof of Theorem 3.1, the optimal dividend-payout policy L * s is the local time of corresponding reserve process R * s at the level d( s ), namely
The financial meaning is that, when the insurance surplus R * s− is above the threshold d( s ), the insurance company should pay out the reserves of an amount R * s− − d( s ) as dividends to its shareholders at time s; otherwise, not. We remark that the reserve process R * s is always continuous and no more than the d( s ), except for the initial time.
In this section we study the optimal reinsurance strategy. Recall that we have ascertained the value function V (x, t) = v(x, t ) by Theorem 3.1.
For the insurer, if its current status is (x, t), by (3.4) , the corresponding optimal retained function is
z, otherwise, (4.4) and the optimal reinsurance function is
These strategies not only depend on the claim z, but also depend on the current insurance surplus x and time t. In this section, we will discuss the behavior of them.
One immediate consequence of (4.4) is that H(z, x, t ) > 0 for z > 0, so
In any case, the insurance company should not cede all risks; it has to bear some risks by itself. This is due to our assumption that the reinsurance is non-cheap, γ > 0. On the other hand, because v x (x, t ) = 1 for (x, t) ∈ D, it yields H(z, x, t ) = z and I(z, x, t ) = 0.
Therefore, the insurance company should not cede risk in the dividend-payout region. In other words, the insurance company shares its risk with a ceded company only when the surplus is lower than the dividend-payout boundary d( t ).
For each z > 0, depending on whether I is zero, we divide the surplus-time space into a reinsurance region R z = (x, t) ∈ Q T I(z, x, t ) > 0 and a non-reinsurance region
By (4.4), they can also expressed as
By our above discussion, D ⊆ N R z and R z ⊆ N D. Proof: The proof is given in Appendix C. This results implies that I is an increasing function w.r.t z, which of course makes perfect financial meaning that the insurer should get more compensation from the reinsurance company when a larger claim arises.
Proof: The proof is given in Appendix D. From (4.7) we see that, if (x, t) ∈ N D, then 
where K(z, t ) is the reinsurance boundary, given by
. Moreover, when z 1/λ, we have by (4.9) R z = ∅, so K(z, t ) = 0.
For each z > 0, define the overlapping of the non-reinsurance and non-dividend regions as non-action region
In this region the insurer should not cede risk z or pay out dividends. We are now show this region is always non-empty. This is equivalent to showing K(z, t) = d( t) by recaling K(z, t)
Combining the above results, we conclude that, for each risk z > 0, the reinsurance and dividend-payout barriers divide the surplus-time space into three non-overlapping regions: a (possible empty) reinsurance region, a non-action region and a dividend-payout region, in an increasing order of surplus. This is illustrated in Figure 3 . Figure 3 : The surplus-time space is divided into three non-overlapping regions: R z , N A z and D by the reinsurance barrier x = K(z, t ) and dividend-payout barrier x = d( t ). Moreover, N D = R z N A z and the t-axis can be regarded as
Economically speaking, if the surplus level of an insurance company is low (x < K(z, t )), then the company must cede risk z with a reinsurance company to avoid bankruptcy. If the surplus level is medium (K(z, t ) x d( t )), then the company can cover the claim z by itself, but has not enough reserves to pay out as dividend, so no actions will be taken.
If the surplus level is very high (x > d( t )), the company should pay out its extra reserves as dividends to its shareholders. It is never optimal for the company to buy reinsurance contracts and to pay out reserves as dividend, simultaneously.
By Lemma 4.3, K(z, t ) is increasing and the reinsurance region R z is getting larger as cede risk z increases. Economically speaking, if a risk is covered by a reinsurance contract, then any larger risks should be covered as well.
Define a uniform non-action region
In this region the insurer should not cede any risk or pay out dividends. Ifẑ = ess sup Z 1 < ∞, then K(z, t ) K(ẑ, t ). Therefore,
As K(ẑ, t ) < d( t ), we see that
This is illustrated in Figure 4 . Figure 4 : As z is increasing, the region R z is extending to Rẑ and the curve x = K(z, t ) is moving right up to x = K(ẑ, t ). The region N A 0 = N Aẑ is non-empty when Z 1 is bounded.
Economically speaking, when the potential claims are bounded ( ẑ) and the surplus level is relatively high (x K(ẑ, t )), then the insurance company can bear them by itself without asking help from a ceded company.
Next, by (4.8), we have
This means the insurance company should significantly reduce its purchase of reinsurance contracts as surplus increases. If its surplus level is very high x z 2c − 1 2cλ , then we claim there is not need to buy reinsurance for claim z, that is, (x, t) / ∈ R z . In fact, if z 1/λ, then there is nothing to prove since R z = ∅. Otherwise, suppose (x, t) ∈ R z . As (x, t) ∈ N D, by (4.8), (4.9) and the mean value theorem, we have
contradicting that (x, t) ∈ R z . Therefore, we conclude that
and consequently,
This is illustrated in Figure 5 . Figure 5 : The line x = z 2c − 1 2cλ gives a universal upper bound for the reinsurance barrier K(z, t ) and the reinsurance region R z when z > 1/λ.
Economically speaking, an insurance company only shares sufficiently large claims with a ceded company.
In this part we assume the claims follows a discrete probability distribution given by P(Z 1 = z j ) = p j > 0, j = 1, 2, ..., N, with 0 < z 1 < z 2 < ... < z N and N j=1 p j = 1.
Define the j th reinsurance boundary as
And define the j th reinsurance region as
Thanks to Lemma 4.3, the properties of these reinsurance boundaries are given in the following result. We leave its proof to the interested readers.
Theorem 4.4 The reinsurance boundaries K j (·), j = 1, 2, · · · , N, are all continuously differentiable in time. Moreover,
for each j = 2, ..., N;
K j ( t ) = 0. Figure 6 illustrates this result. Figure 6 : The reinsurance boundaries K j and reinsurance regions R j z are increasing in j. The uniform non-action region N A 0 is N A z N .
Economically speaking, the insurer should only cede the claims z j , z j+1 , · · · z N , if the surplus level is in R j z .
Appendix A Proof of Theorem 3.1. 
where L c s is the continuous part of L s . The first two expectations are non-negative since
and as R s R s− ,
Since the policies are arbitrary chosen, it implies V (x, t) V (x, t).
We now prove the reverse inequality V (x, t) V (x, t). Define a boundary
Because V is concave in x by hypothesis, it yields
By the equation in (3.1), it follows
where the C 2,1 continuity of V ensures the above equation holds at (d * (s), s). Assume R * s is the solution to problem (2.3) with the feedback controls
where h * is defined by (3.4) . Under these controls, we see that R * s is continuous and R * s d * (s) for s > t. Denote τ * be the corresponding ruin time.
Now we show the controls defined above are indeed the optimal controls. By Itô's formula,
If τ * < T , then R * T ∧τ * = R * τ * = 0, so V (R * T ∧τ * , T ∧ τ * ) = V (0, τ * ) = 0 by the boundary condition in (3.1). Otherwise V (R * T ∧τ * , T ∧ τ * ) = V (R * T , T ) = 0, again by the boundary condition in (3.1). Therefore, the first expectation in (A.2) is zero. By our choice of H * , we see that
Because R * s d * (s) for s > t and noticing (A.1), we conclude that the second expectation in (A.2) is also zero.
Notice dL * c s = 0 unless R * s = d * (s), so
Recall that R * s is continuous for s > t, so See Figure 7 for an illustration of the penalty function β ε .
✲ ✻ x y ε −c r r Figure 7 : The penalty function β ε .
Because the left boundary condition and initial condition in (3.13) are not consistent at (0, 0), in order to show the existence of a solution, we choose f ε (t) ∈ C 2 ([0, +∞)) to satisfy
and consider the following penalty approximation problem,
u ε (L, t) = 1, 0 < t T, u ε (x, 0) = 1, 0 < x < L.
where L is any fixed positive constant.
The following result is useful for our handling the above problem. Moreover, by (3.6), for any z 1,
The proof for G is similar.
We first establish a comparison principle for Problem (B.1).
If u 1 , u 2 1, then
Then
By Remark B.1, the assumption that u 1 , u 2 1 is sufficient to guarantee F (·, ·) and G(·, ·)
in T (e −x/λ w i ) be Lipschitz continuous on w i , w ix , i = 1, 2. By the comparison principle for non linear equations (see [21] ), we obtain w 1 w 2 in Q L,T .
There exists a solution u ε ∈ C 2,1 Q L,T to Problem (B.1). Moreover, for sufficiently small ε > 0, u ε satisfies
where K = L + 1/λ + 1, Λ = µ 2 γ 2 + γλ > 0.
Proof: Using the Leray-Schauder fixed point theorem (see [13] ) and the embedding theorem (see [10] ), we get the existence of a C 1+α, 1+α 2 Q L,T (0 < α < 1) solution u ε to Problem (B.1).
By the Schauder estimation (see [26] ), we also have u ε ∈ C 2+α,1+ α 2 Q L,T . It is left to establish (B.3). Let φ ≡ 1, then
By a similar argument to prove Lemma B.2 and using the comparison principle for linear equations, we have u ε φ = 1.
Let Φ = Ke Λt /(x + 1/λ). Then, in Q L,T , Φ 1 + ε for sufficiently small ε > 0, so we
Together with boundary conditions
applying Lemma B.2, we obtain u ε Φ.
Before passing to the limit, we give some properties of u ε . 
u ε (L, t) = u ε (L, t) = 1, 0 < t T − ∆, u ε (x, 0) = u ε (x, ∆) 1 = u ε (x, 0), 0 < x < L.
u ε x (x, 0) = 0, by the maximum principle for weak solution (see [21] ), we deduce that u ε x 0. This gives (B.14).
Now we give some uniform norm estimates for u ε . Applying C α, α 2 estimate (see [5] for non-divergence form) to (B.1), we have |u ε | α,Q L,T C(|u ε | 0,Q L,T + |β ε (·)| Lp(Q L,T ) + 1) C.
Applying C α, α 2 interior estimate (see [21] for divergence form), we obtain |u ε x | α,Q r/2 C(|u ε x | 0,Q r/4 + |β ε (·)| Lp(Q r/4 ) + 1) C.
where Q r := Q L,T \ {(x, t) | x 2 + t 2 r}.
According to Remark B.1, A (−u ε x /u ε ) and B (−u ε x /u ε ) are uniform C α, α 2 in Q r/2 . So we can apply W 2,1 p interior estimate (see [21] ) to (B.1) in Q L,T to obtain |u ε | W 2,1 p (Q r ) C(|u ε | Lp(Q r/2 ) + |β ε (u ε − 1)| Lp(Q r/2 ) + 1) C.
We emphasize that Cs in the above estimates are independent of ε, so there exits u ∈ W 2,1 p, loc (Q L,T ) C(Q L,T ) and a subsequence of u ε (still denoted by u ε ) such that u ε −→ u weakly in W 2,1 p (Q r ) and uniformly in C(Q L,T ).
v(x, t) = We come to ascertain the order of smoothness of v. Now, we already proved v x = u ∈ W 2,1 p (Q r ) C Q L,T . The Sobolev embedding theorem implies that v xx = u x ∈ C Q r . Moreover, using the method in [11] , we can prove v xt = u t is continuous passing through the free boundary.
Next, we prove the uniqueness. Suppose v 1 , v 2 are two solutions to (3.8) 