On some nonlinear boundary value problems related to a Black-Scholes model with transaction costs

We deal with some generalizations on a Black-Scholes model arising in financial mathematics. As a novelty in this paper, we consider a variable volatility and abstract functional boundary conditions, which allow us to treat a very large class of problems involving Black-Scholes equation. Our main results involve the existence of extremal solutions in presence of lower and upper solutions. Some examples of applications are provided too.


Introduction
The Black-Scholes equation, introduced in 1973 [2], is a very well-known model arising in financial mathematics. Roughly speaking, it represents the value of an option V (S, t) which depends on time and on an stock price S. The BS model takes the following form: where r is the interest short rate and σ is the volatility of the stock returns.
If we include transaction costs in model (1.1) then we can obtain (see [6], [7]) the following nonlinear version of (1.1): whereσ is an adjusted volatility. Now, if we consider the stationary version of (1.2) then we obtain the following ordinary differential equation: In [7], the authors studied the existence and localisation of solutions for equation (1.3) in an interval [c, d], c, d > 0, with Dirichlet conditions, V (c) = V c , V (d) = V d . To do this, they deal with the following equation related to (1.3): where H(x, y, z) = p(x)x 2 − p(x) 2 x 4 + 4x 3 q(x)|xz − y| 2x 3 .
More concretely, they construct a pair of lower and upper solutions for the differential equation (1.4) with Dirichlet conditions in the case V c < V d and then they show that solutions of this equation correspond with solutions of (1.3). In the present paper, we generalize those results in the following ways: first, we let the volatility σ to depend on the stock price x, and so we replace constants p, q by two functions p(x), q(x); second, we replace the Dirichlet conditions by functional boundary conditions, which allow us to consider a very large class of problems involving equation (1.3). So, we deal with the following generalization of Black-Scholes equation with transaction costs: where p, q are nonnegative bounded functions in [c, d] and B i : are functions which satisfy some conditions that we will state later. Under this framework, a large class of boundary conditions are included, namely: 1. Dirichlet conditions: and so on.
This paper is organized as follows: in Section 2, we consider problem (1.5) with Dirichlet conditions. So, we start from paper [7] and we generalize it by considering variable volatility and by dropping the condition V c < V d . In Section 3 we deal with problem (1.5) on its full version, that is, with functional boundary conditions. Namely, we provide a result on the existence of extremal solutions between lower and upper solutions, by using a generalized iteration with Dirichlet problems. Some examples of application are provided too. Inspired by [1] and [7], the trick to obtain solutions for (1.5) is to consider the following related problem:

Problem with Dirichlet conditions
We begin by considering problem (1.6) with Dirichlet conditions, that is, Notice that problem (2.7) is a particular case of (1.6), with We begin by introducing the concept of lower and upper solutions for this problem. For technical reasons, in Section 3 we will need these functions to have "angles", and so we use the following weak definition, inspired by the idea of De Coster and Habets [4].
and for each x 0 ∈ (c, d) one of the following conditions holds: There exists an open interval J 0 such that x 0 ∈ J 0 ⊂ (c, d), α |J0 ∈ W 2,1 (J 0 ) and for almost all x ∈ J 0 we have We say that β ∈ AC([c, d]) is an upper solution for problem (2.7) if and for each x 0 ∈ (c, d) one of the following conditions holds:

There exists an open interval
Notice that if α, β are classical C 2 − lower and upper solutions for problem (2.7) (as defined in [7]) then they are also lower and upper solutions in the sense of Definition 2.1.

Proposition 2.1 The following holds:
is a lower solution for problem (2.7); (2) Take k > 0 such that Then the function is a lower solution for problem (2.7); (3) The function is an upper solution for problem (2.7). Proof.
(1) First, notice that α ′′ On the other hand, On the other hand, as (see [7,Lemma 1]) we obtain Then, the choice of k > 0 implies that Notice that we can choose such a k by virtue of being (3) It's easy to check, taking into account that has the extremal solutions, that is, the least and the greatest one, inside the interval Moreover, if k > 0 satisifies (2.9), then problem (2.7) has the extremal solutions inside the interval

Proof.
Put and, if conditions referred above hold, Thus defined, H is continuous in E 1 and E 2 . Moreover, we have that Then, function H satisfies the Nagumo condition both in E 1 and E 2 . Using the fact that α 1 , α 2 are lower solutions and β is an upper solution for problem (2.7), the conclusion holds by application of a well-known result (see [3] Proof.
The proof is the same to that done in [7]. Convexity of solutions provides from the fact On the other hand, the second assertion follows from the fact that function is nondecreasing and ⊓ ⊔ Theorem 2.3 Consider problem (1.5) with standard Dirichlet conditions, that is, (2) If k > 0 satisfies (2.9) then problem (2.14) has the extremal convex solutions in the functional interval [α, β], where α(x) = max{α 1 (x), α 2 (x)} and α 1 , α 2 , β are provided, respectively, by (2.8), (2.10) and (2.11); a solution of problem (2.14). On the other hand, in the periodic a solution of (2.14).

Proof.
It suffices to check that ifV is a solution of (2.7) thenV solves (2.14). Indeed, by virtue of Then,V solves (2.14).
On the other hand, notice that if V is a convex solution of (2.14) then V is a solution of (2.7) too. If this is false then we have and then V must be concave.
To see part (3), notice that in the case The periodic case is trivial too.
is essential and it cannot be removed at all, as we show in the following proposition.

Proposition 2.2
The following conditions are equivalent: x is a lower solution for problem (2.7); (3) Each solution of (2.7) is a convex solution of (2.14).

Proof.
We have already shown that (1) ⇒ (2) ⇒ (3). On the other hand, if V is a solution of (2.7) then V solves (2.14) if and only if |xV In particular, α 1 (c) ≤ V c , and so α 1 is a lower solution for problem (2.7).
Finally, if (2) is an upper solution.

Problem with functional boundary conditions
In this section we deal with problem (1.6) on its full expression. We begin by extending Definition 2.1 to this case.

Definition 3.1 We say that α ∈ AC([c, d]) is a lower solution for problem (1.6) if
and for each x 0 ∈ (c, d) one of the following conditions holds: There exists an open interval J 0 such that x 0 ∈ J 0 ⊂ (c, d), α |J0 ∈ W 2,1 (J 0 ) and for almost all x ∈ J 0 we have

We say that β ∈ AC([c, d]) is an upper solution for problem (1.6) if
and for each x 0 ∈ (c, d) one of the following conditions holds: There exists an open interval J 0 such that x 0 ∈ J 0 ⊂ (c, d), β |J0 ∈ W 2,1 (J 0 ) and for almost all x ∈ J 0 we have In the construction of our generalized iterative method, we will use two technical lemmas. First of them is the following generalization of Bolzano's theorem. The second auxiliar result we need deals with the existence of extremal fixed points for nondecreasing operators defined in the space of absolutely continuous functions. Then, G has in [α, β] the greatest, V * , and the least, V * , fixed points. Moreover, they satisfy  (1.6) and assume, moreover, that the following conditions hold: In these conditions, problem (1.6) has the extremal convex solutions in [α, β].

Proof.
We define a mapping G : [α, β] −→ [α, β] as follows: for all γ ∈ [α, β], Gγ is the greatest convex solution in [α, β] for the Dirichlet problem Step 1: The mapping G is well-defined. First, by virtue of being α and β a lower and an upper solution for problem (1.6) and by condition (H 2 ), we have for all γ ∈ [α, β]: and so condition (H 1 ) implies that the numbers γ c and γ d are well-defined, by application of Lemma 3.1.
Step 3: G has the extremal fixed points.
and so whereÂ,B are as in (2.12). Then, by application of Lemma 3.2 we obtain that G has the extremal fixed points in [α, β], say V * , V * , which moreover satisfy (3.16).
Step 4: V * is the greatest convex solution of problem (1.6) in [α, β]. First, it is clear, as GV * = V * , that V * is a solution of problem (1.6). Now, if V is another solution of (1.6) then we have that V ≤ GV and so (3.16 To obtain the least convex solution of (1. 6 is a lower solution for problem (1.6) such that α ≤ α 1 . Then problem (1.5)