FREE BOUNDARY PROBLEM OF BARENBLATT EQUATION IN STOCHASTIC CONTROL

. The following type of parabolic Barenblatt equations is studied, where L 1 and L 2 are diﬀerent elliptic operators of second order. The (unknown) free boundary of the problem is a divisional curve, which is the optimal insured boundary in our stochastic control problem. It will be proved that the free boundary is a diﬀerentiable curve. To the best of our knowledge, this is the ﬁrst result on free boundary for Barenblatt Equation. We will establish the model and veriﬁcation theorem by the use of stochastic analysis. The existence of classical solution to the HJB equation and the diﬀerentiability of free boundary are obtained by PDE techniques.


Introduction. A general form of parabolic Barenblatt equation is
where A is a finite or infinite set, L α is a set of elliptic operators of second order. In Lieberman's book [8] this fully nonlinear equation is called as parabolic Bellman equation. In physics and mechanics it is called as Barenblatt equation [3] and in mathematical finance it is called as Black-Scholes-Barenblatt equation [4,5,12]. Even if A is a infinite set, Krylov [6] proves this equation always has a classical solution in any bounded domain. In our problem α only takes two values, i.e., V (x, t) satisfies

XIAOSHAN CHEN AND FAHUAI YI
where 2) σ, µ, r, λ, L are positive constants with µ > λL. This problem arises naturally from mathematical finance [10]: where a firm's asset follows standard Ito process and may suffer from to a Poisson loss of given magnitude L (accident). However the firm can choose to insure part of this loss at a fair market premium each unit of time. The firm chooses the amount of coverage at each moment of time, thus the problem becomes a standard stochastic control problem. It has been shown that the value function for the problem satisfies the Barenblatt equation.
Explicit solution to the equation can be found in closed form when insurance contracts can be written with infinite maturity [10]. In this paper, we generalize the model by considering insurance contracts with finite maturity. We found that with finite maturity insurance contracts, the problem boils down to a free boundary problem of PDE instead of ODE. We justify our generalization as all insurance contracts in reality have finite maturity instead of infinite maturity.
The rest of the paper is organized as following, Section 2 formulates the model as an optimal exit time problem. Section 3 derives some properties of the classical solution to the Barenblatt equation. One of our main contributions lies in Section 4 which proves the differentiability of the free boundary. Finally, in Section 5, we present a verification theorem, derive the optimal insurance strategy and give an financial interpretation of the free boundary.
To the best of our knowledge, this is the first result on the free boundary for the Barenblatt equation. We establish monotonicity and concavity of value function, and the verification theorem through stochastic analysis. Then we prove existence of classical solution for HJB equation and differentiability of free boundary using PDE techniques.
2. Model formulation. We fix a filtered probability space (Ω, F, P) with a filtration (F t ) t≥0 that satisfies the usual conditions, the process W t is a standard Brownian motion, P t is a Poisson process with intensity λ.
Suppose µ is the expected return of the cash reserve, while the Poisson risk can be insured for a fair premium λL per unit of time, we assume that µ > λL (otherwise the technology would not be profitable [10]). The dynamics of the cash reserve X s is now given by The control variable of the firm is represented by an adapted process i s ∈ [0, 1], which is the fraction of the Poisson loss that is insured. σ is the volatility of W s . We denote by A t,x the set of all admissible control processes i s from time t and X t = x, where i s is a progressively measurable (with respect to the filtration (F t ) t≥0 ) process satisfying For all i s ∈ A t,x , the above condition ensures the existence and uniqueness of a strong solution to the SDE (2.1). Suppose that the company will distribute all the cash reserve as dividend at terminal time T . The objective is to choose the optimal control so as to maximize the expected discounted value of the cash reserve at time T if it is not bankrupt at [t, T ]. We define the bankruptcy time τ as The value function is defined as We formulate the problem as an exit time problem from (0, +∞). However, due to the absorbing nature of the boundary x = 0, it is equivalent to a state constraint problem. In this paper, only control process i s will be admitted such that the wealth constraint X s ≥ 0 holds for all s ≤ T . The boundary point x = 0 is absorbing, i.e., X τ = 0, then X s = 0 for all s ∈ [τ, T ]. Hence the bankruptcy time τ can be replaced by T . Take B t,x ⊆ A t,x the set of all admissible control i t ∈ A t,x with initial value X t = x, such that the solution to the stochastic differential equation Proof. First we prove the monotonicity of U (x, t) w.r.t. x. Fix some arbitrary 0 < x ≤ y, and i s ∈ B t,x ⊆ B t,y . Suppose X t,x s , X t,y s are two processes of SDE (2.1) related to the initial values X t = x and X t = y, respectively. By comparison principle of stochastic differential equation, we have By the arbitrary of i s ∈ B t,x we obtain U (x, t) ≤ U (y, t).
On the other hand for any 0 < x 1 < x 2 , i k s ∈ B t,x k , k = 1, 2, and X k s is the solution to the stochastic differential equation , from the linear dynamics of (2.1), we see that the solution to the following SDE

XIAOSHAN CHEN AND FAHUAI YI
The above equality also shows X α s > 0 by X 1 s > 0 and X 2 s > 0, thus i α s ∈ B t,xα . Hence by the equivalent definition of the value function (2.3), Since the above inequality holds true for any i 1 So we obtain the concavity of the value function.
Now we will establish HJB equation of the problem. Suppose U (x, t) ∈ C 2,1 ((0, +∞) × (0, T )) (we will prove this in Theorem 3.2), applying dynamic programming principle [9] to (2.3), for any t < δ < T , Applying Itô's formula with jump, Suppose ∂ x U (x, t) is uniformly bounded (it will be proved, see (3.13)), then the last term in the above equality is a square integral martingale. Taking the expectation in above equality, where compensated Poisson process P s − λs is a martingale ( [11], Th. 11.2.4). So (2.6) becomes Plugging it back into (2.5), dividing the equation in (2.5) by δ − t and letting i.e., Notice that, when x ≤ 0, then the firm is liquidated, thus • When x − L ≥ 0, with the concavity of U (x, t) in (0, +∞), we obtain which implies sup (1); (2.10) • when x − L < 0, from (2.9) and the monotonicity and concavity of U (x, t) in (0, +∞) to obtain Thus the control is the bang-bang type, and In fact, if x − L ≥ 0, applying (2.10), At the terminal time T , since there is no time left for the firm to become liquidated, and we want to maximize the expected discounted value of dividends, then the firm would distribute X T as dividends, hence (2.16) Especially, (2.10) shows that We summarize the above facts as the following lemma.
Lemma 2.2. The value function U (x, t) defined in (2.2) satisfies the following HJB equation with boundary and terminal conditions
In order to prove (3.8), we first show In terms of comparison principle, we obtain (3.9). Hence owing to (3.9), we have Thanks to the left hand side of (3.7), we also have It is obviously k 1 k 2 is a supersolution of (3.10), hence v(x, t) ≤ k 1 k 2 .
Set v 4 (x, t) := e −(r+λ)t , then v 4 (x, t) satisfies combining with the initial and boundary conditions, applying comparison principle to v 4 and v we get the left hand side of (3.8).
where k 1 , k 2 are defined in Lemma 3.1. Moreover, the classical solution of problem (3.5) satisfying estimate (3.12) is unique.
Proof. In view of (3.7) and (3.8) we have hence for any m < n where C m is independent of n. Thus for any p > 2, where C m is also independent of n. Letting n → ∞, there exists V (m) (x, t) ∈ W 2,1 p (Q m ) and a subsequence of {V n } (still denoted by {V n }), such that then V (x, t) is well defined on Q and for each fixed n, V (x, t) ∈ W 2,1 p (Q n ). Sending n → ∞ in (3.6), we see that V (x, t) is a solution of problem (3.5).

XIAOSHAN CHEN AND FAHUAI YI
Finally we prove the uniqueness. Suppose that V 1 , V 2 are two classical solutions satisfying (3.12) to the problem (3.5). Then where ∂ p Q is parabolic boundary of Q, thus 14) note that Since V 1 and V 2 satisfy estimate (3.12), we can apply the maximum principle [1] to know Similarly, we can deduce from Hence we proved the uniqueness of the problem.

Differentiability of the free boundary.
Denote The equation in (3.5) can be rewritten as Differentiating (4.2) with respect to x yields where H(·) was defined in (3.11).
Multiplying (4.3) by L and abstracting (4.2) from the resulting equation, we obtain the initial and boundary values of w are Proof. According to (4.6) and (3.13), we have by (3.13), so (4.8) holds.
Now we prove main theorem.
(4.8) means that w(x, t) is strictly monotonic decreasing, so (4.9) and (4.10) hold. S(t) > 0 follows from (4.7). On the other hand from (2.19), Applying (3.4), we see that Comparing this with the equation in (3.5), we have that From the definition (3.2), It means that S(t) ≤ L. At last we prove S(t) ∈ C 1 (0, T ]. If fact if ∂ t w(S(t), t) is continuous in Q, then the implicit function Theorem claims that S(t) is continuous differentiable in (0, T ]. In fact, differentiating (3.1) with respect to t, then ∂ t V (x, t) satisfies Applying W 2,1 p interior estimate, we know ∂ t V (x, t) ∈ W 2,1 p,loc (Q) for any p > 3, where And then the embedding theorem tells us that ∂ xt V (x, t) ∈ C(Q), therefore Hence S(t) is continuous differentiable and 5. Verification theorem and optimal strategy. First we present a version of the verification theorem for the value function U (x, t) defined in (2.3) in this section.
If, in addition, let i * ∈ A t,x be the measurable function defined by where X * s be the solution to the following SDE Proof. For any fixed x > 0, an admissible control process i s ∈ A t,x . By the general Itô's formula, it yields Since ∂ x u(x, t) is uniformly bounded, the last term in the above formula is a square integrable martingale. Observe that ∆X s := X s − X s where L i u = ∂ t u + 1 2 σ 2 ∂ xx u + (µ − λLi s )∂ x u − ru. ≤ max ∂ t u + L 1 u, ∂ t u + L 2 u = 0.
Hence, we have u(x, t) ≥ E tx e −r(T −t) u(X T , T ) = E tx e −r(T −t) X T .
By the arbitrary of i s ∈ A t,x , we know u(x, t) ≥ U (x, t).
On the other hand, from the definitions of i * s and X * s , (L i * u)(X * s , s) + λ[u(X * s− − (1 − i * s )L, s) − u(X * s− , s)] = 0, t ≤ s ≤ T, then u(x, t) =E tx e −r(T −t) X * T ≤ U (x, t). Hence we obtain u(x, t) = E tx e −r(T −t) X * T = U (x, t). Recalling U (x, t) = V (x, T − t), denote f (t) = S(T − t), definition (5.1) is equivalent to i * s = χ {X * s >f (s)} , and x = f (t) is the optimal insurance boundary (Fig. 1). When x lies on the right hand side of the boundary x = f (t), the optimal insurance of the Poisson loss is to have full coverage, when x lies on the left hand side of the boundary, the optimal strategy is to have no coverage at all. Fig. 1. Optimal strategy At any given time t, if the cash reserves of the firm X t falls in the region {X t > f (t)}, then the firm should fully insure the Poisson risk at once; otherwise, the firm should not insure the Poisson risk.