A NEW WEAK SOLUTION TO AN OPTIMAL STOPPING PROBLEM

. In this paper, we propose a new weak solution to an optimal stopping problem in ﬁnance and economics. The main advantage of this new deﬁni- tion is that we do not need the Dynamic Programming Principle, which is critical for both classical veriﬁcation argument and modern viscosity approach. Ad- ditionally, the classical methods in diﬀerential equations, e.g. penalty method, can be used to derive some useful results.


1.
Introduction. The theory of optimal stopping is very important due to its numerous and various applications in economics and finance. It provides a suitable modeling framework for the evaluation of optimal investment decisions, see e.g. [6]. In this paper, we consider the following optimal stopping problem: v(t, x) = sup where T t,T is the set of stopping time valued in [t, T ], f : [0, T ] × R → R and g : [0, T ] × R → R are continuous functions representing the running benefit and the final gain respectively, and the stock price {S t } t 0 satisfies dS ρ = S ρ αdρ + S ρ σdW ρ .
Here {W t } t 0 is a standard Brownian motion on a filtered probability space (Ω, F, {F ρ } ρ 0 , P) satisfying the usual conditions, constants α ∈ R, σ > 0 are expected return rate and volatility, respectively. A classic example is an American put option, which bestows the owner a right but not an obligation to sell a stock with a predetermined exercise price before a maturity. Mathematically, this put option's pricing model can be written as (1) and (2) with f ≡ 0 and g(t, s) = e −αt max{K −s, 0}, where α 0 represents the interest rate, and K > 0 is the exercise price, T is the maturity, and the expectation E is interpreted as a risk neutral measure; see e.g., [5] for more information. Motivated by this classic example and for simplicity, we make the following two assumptions throughout the paper: (A1) g ∈ C([0, ∞) × (0, ∞)) satisfies the growth condition: for each T > 0, there exist n > 0 and K(T ) > 0 such that |g(t, s)| K(T )[s −n + s n ] ∀ (t, s) ∈ D T := [0, T ] × (0, ∞). (3) (A2) f ∈ C([0, ∞) × (0, ∞)) and is uniformly bounded, i.e., there is a constant M f > 0 such that |f (t, s)| M f for all (t, s) ∈ D T .
We shall show that v is a solution of the following obstacle, or variational, problem: where m is some positive constant, and Before we proceed to prove our assertion, it is necessary to clarify that what a "solution" exactly means. One candidate is so called classical solution that admits sufficient regularity, i.e. the derivatives in the equation (4) are in classical sense. If the value function v defined as (1) is sufficiently smooth, one then can verify, typically with the help of Dynamic Programming Principle, that the value function v is the unique classical solution of equation (4). This is called verification theorem in the classical optimal stopping problem.
However, from the definition (1), it is very difficult (sometime impossible) to obtain the regularity of the function v. Therefore, in general, v is not a classical solution of equation (4). To avoid this difficulty, in modern mathematics, viscosity solutions are introduced as follows: Denote viscosity supersolution) of (4) if the following holds: ) for all (t, s) ∈ D T and some positive constants K and m; (ii) for any triple (t,s, ϕ) ∈ Q T × C 1,2 (Q T ) satisfying v(t,s) = φ(t,s) and v ϕ (resp. v ϕ) in Q T , there holds F (t,s, φ(t,s), φ t (t,s), φ s (t,s), φ ss (t,s)) 0 (resp. 0).
It is called a viscosity solution if it is both a viscosity supersolution and a subsolution.
The viscosity approach is very powerful, since the necessary condition of the function is locally bounded. Hence, the viscosity approach is widely used in applied mathematics, especially in mathematical finance. However, due to the relax requirements of the value function, the classical methods in differential equations cannot be used anymore. On the other hand, it is not trivial to prove that the value function is a viscosity solution of the equation. Besides, it is well known that the uniqueness of viscosity solution (i.e., the comparison principle) is very hard to prove. Therefore and for comparison with our new weak solution introduced later, the first reslut of this paper is the following: Because of the complexity of the proof of theorem 1.2, we shall find a simpler way to give a new definition of the solution. Following Perron's idea of constructing harmonic function by the infimum of superharmonic functions, we introduce the following new weak solution of equation (4), which is the main contribution of this paper: where K and m are some positive constants, and (iii) (2) v : D T → R is called a weak solution of (4) if v v v for every classical subsolution v and every classical supersolution v.
Under the above definition, we shall prove the following: Theorem 1.4. Assume (A1) and (A2). Then v defined in (1) is the unique weak solution of the variational inequality (4). In addition, the following holds: where the optimal stopping time τ * is given by One advantage of this definition is (as we will see in Section 3) that we do not need the Dynamic Programming Principle which is critical for both classical and viscosity approach. Secondly, since the sub-supersolution are defined in classical way, comparison principle can be applied directly. Additionally, the classical methods in differential equations, e.g. penalty method, can be used to derive some useful results, see e.g. [2,3]. As an illustration, with the modification of classical penalty method, we have a key estimate (see Theorem 3.2 in Section 3) which is pivotal for the uniqueness and approximation of the solution. Also, we expect that many other classical techniques can be used for further applications.
Thus, the rest of this paper is organized as follows: In Section 2, we shall prove Theorem 1.2 by a standard viscosity approach; Then, Section 3 is devoted to the proof of Theorem 1.4. Finally, we give a short conclusion in Section 4.

CONG QIN AND XINFU CHEN
2. The modern viscosity solution approach.

The idea of the proof.
We shall prove Theorem 1.2 by the following steps: 1. |v| is locally bounded and has at most polynomial growth (i.e. bounded by K[s n + s −n ] for some positive constants K and n); The above five properties, especially the last one, called comparison principle, imply that v * v * . Hence, v = v * = v * is a viscosity solution, which is unique in the class of continuous functions with polynomial growth.
In the sequel, T > 0 is fixed.

Dynamic programming principle.
The key to link optimal stopping problem and viscosity solution is the following: Proof. Let (t, s) ∈ D T and θ ∈ T t,T be fixed.
2.3. Subsolution property. Before we prove the subsolution property, we need the following: Hence, setting m = max{M ε , M 1ε } we see that ζ m is a classical supersolution, so v ζ m . This implies that Sending ε 0 we obtain v * (T, s 0 ) g(T, s 0 ). Since v g, the assertion of the lemma thus follows. Lemma 2.3. v * is a viscosity subsolution of (4). 0, ∞). Next, we have v * (T, ·) = g(T, ·). Also, by the classical supersolution, v * K[s n + s −n ]e k(T −t) for some constants n, k and K. 2 We want to show that min{−ϕ t −L ϕ−f, ϕ−g}| (t,s) 0. We argue by contradiction by assuming which converges to (t,s, v * (t,s)), as m → ∞. Without loss of generality, we can assume By the continuity of ϕ we have For each τ ∈ T tm,T we can apply Itô's formula to ϕ between (t m , s m ) and (τ ∧ τ m , S τ ∧τm ) to obtain where we used the fact that the stochastic integral on the right-hand side of second equation is a martingale, that ϕ v in Q T and ϕ g + δ 0 , −ϕ t − L ϕ − f δ 0 in Ω in the first inequality. Take the supreme over all τ ∈ T tm,T and use the principle of dynamic programming, we have Hence, Let m go to infinity and use S t = s m e (α−σ 2 /2)(t−tm)+σ[Wt−Wt m ] for t > t m . We derive that where τ = inf{t >t | (t, S t ) ∈ Ω}. Subject to St =s, we know that τ >t a.s., i.e., E[min{1, τ −t}|St =s] > 0. Hence we obtain a contradiction. This completes the proof.

Comparison principle.
In this subsection, we shall discuss the comparison principle which is critical to the uniqueness. First of all, let us give an alternative definition of viscosity solution based on jets whose proof can be found in [7].
The following Ishii's Lemma plays a pivotal role in the proof of comparison principle. One can find the proof in [4].
Now we can present the following comparison principle: Lemma 2.7. Suppose u is a viscosity subsolution and v is a viscosity supersolution of (4). Proof.
On the other hand, (8) implies By substracting the above two inequalities, we have where C = max{2α, 3σ 2 }. By the property of φ and continuity of f m , let go to zero, we obtain mM 0, a contradiction with M > 0.
(ii) Otherwise, up to a subsequence, U (t , s ) − g m (t , s ) 0 for all m, and since V (τ , x ) − g m (τ , x ) 0 by (8), we obtain that By sending m to infinity, and from the continuity of g m , we also have required contradiction M 0. This completes the proof.
Proof of Theorem 1.2.
Proof. We know that v * is a viscosity supersolution and v * is a viscosity subsolution. By comparison, obtain v * v * . On the other hand, v * v v * . Hence v = v * = v * is a viscosity solution. By comparison, viscosity solutions, if it exists, is unique. The assertion of the theorem thus follows.

A classical approach with new ingredients.
3.1. v is a weak solution. First we show that v is a weak solution and is locally bounded. The lemma implies that any classical supersolution is always bigger than or equal to any classical subsolution, a fact that can be shown directly by a standard PDE comparison technique.

3.2.
A new usage of the old penalty method.
In this subsection, we modify the penalty method to get a priori estimates of the weak solution. Then using the estimates we can derive the uniqueness of the solution.