DEGENERATE SECOND ORDER DIFFERENTIAL OPERATORS WITH GENERALIZED REFLECTING BARRIERS BOUNDARY CONDITIONS

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1 Dedicated to Professor Gheorghe Bucur on the occasion of his 70th birthday DEGENERATE SECOND ORDER DIFFERENTIAL OPERATORS WITH GENERALIZED REFLECTING BARRIERS BOUNDARY CONDITIONS FRANCESCO ALTOMARE GRAZIANA MUSCEO Continuing some previous investigations, in the present paper we find some new boundary conditions under which degenerate second-order differential operators of the form Lu = αu + βu + γu generate strongly continuous positive semigroups, in the setting of weighted continuous function spaces on real intervals. These conditions generalize the so-called reflecting barriers boundary conditions. Several applications are also shown. AMS 2000 Subject Classification: 47D06, 47D07, 35K5, 35J70. Key words: degenerate second-order differential operator, degenerate evolution equation, weighted continuous function space, strongly continuous positive semigroup, generalized Reflecting Barriers boundary condition.. INTRODUCTION Recently in [7] we started to study the generation properties of degenerate second-order differential operators on weighted continuous function spaces on a real interval. The interest for such operators arise from their relationship with degenerate evolution equations of the form u (.) t (x, t) 2 u u + β(x) (x, t) + γ(x) u(x, t), x J, t > 0 x2 x which are involved in several problems of the modern applied Mathematics: from the Genetics to the Transport Theory, from Stochastic Process Theory to the Mathematical Finance so on. Generally speaking, the main aim is to establish some general conditions under which degenerate second-order differential operators with continuous coefficients of the form (.2) Lu = αu + βu + γu, MATH. REPORTS 2(62), 2 (200), 0 7

2 02 Francesco Altomare Graziana Musceo 2 acting on suitable Banach spaces of continuous functions on a real interval, generate strongly continuous positive semigroups, which, in turn, represent the solution to the Cauchy problem associated with (.) coupled with suitable boundary conditions. Actually, several authors have been interested in generation results for differential operators of type (.2). In this respect, we refer, e.g., to [2], [3], [4], [5], [9], [4], [5]. In [7], among other things, we showed a generation result for the operator L defined by (.2) in the framework of weighted continuous function spaces when the domain incorporates maximal type boundary conditions or Wentzel type boundary conditions. In this paper we continue our investigation in these function spaces we consider other boundary conditions which guarantee further generation results. The starting point is the paper [9] where the authors study the generation property of second-order degenerate differential operators in spaces of bounded continuous functions on a real open interval, under Reflecting Barriers boundary conditions. In the present paper we consider generalized Reflecting Barriers boundary conditions which are incorporated into the domain (u w) (x) D N (L) = u D M (L) : x ri w 2 (x)w (x) = 0 for every i =, 2 of the weighted continuous function spaces E w (J) := f C(J) : wf E(J), where J = (r, r 2 ) is an open interval, W is the Wronskian function, D M (L) is the maximal domain (see Section 2 for more details), w is a weight function on J, i.e., w C(J) w(x) > 0 for every x J, E(J) := f C(J) : there exists f(x) R for every i =, 2 x ri. Under suitable assumptions on α, β, γ w, we then show that the operator (L, D N (L)) generates a strongly continuous (quasicontractive) positive semigroup on E w (J). As in [7] our method consists in identifying the space E w (J) with E(J) by an isometric isomorphism then in studying the differential operator on E(J) obtained from L by the similarity associated with this isomorphism. In Section 3 we apply the main generation result of Section 2 to some differential operators acting on weighted continuous function spaces on the real interval J = ]0, [ on the real interval J = ]0, + [, considering in some cases a bounded continuous weight function in other cases a continuous not bounded weight function. In particular, the last example concerns the

3 3 Degenerate differential operators with reflecting barriers 03 widely famous Black-Scholes equation which arise from the theory of option pricing. 2. POSITIVE SEMIGROUPS WITH GENERALIZED REFLECTING BARRIERS ON WEIGHTED CONTINUOUS FUNCTION SPACES Before stating our main result, we begin by recalling the Feller characterization of quasicontractive positive semigroup on the Banach lattice C(X) of all real-valued continuous functions defined on a compact space X. This characterization is essentially based on the generalized positive maximum principle. We recall that a linear operator A : D(A) C(X) defined on a linear subspace D(A) of C(X) is said to verify the generalized positive maximum principle with respect to ω R if Au(x 0 ) ω u(x 0 ) for every u D(A) x 0 R satisfying sup u(x) = u(x 0 ) > 0. x X For the proof of the next result we refer, e.g., to [, Theorem 2.2] [8, Theorem 9.3.3]. Here in the sequel the symbol I sts for the identity operator. Theorem 2.. Consider a linear operator A : D(A) C(X) defined on a linear subspace D(A) of C(X) ω R. The following statements are equivalent: a) (A, D(A)) is the generator of a strongly continuous positive semigroup (T (t)) t 0 on C(X) satisfying T (t) e ωt for every t 0. b) (i) There exists λ > ω such that (λi A)(D(A)) = C(X). (ii) A verifies the generalized positive maximum principle with respect to ω. As a consequence we have the following generation result for an additive perturbation of (A, D(A)). Corollary 2.2. Let (A, D(A)) be the generator of a strongly continuous positive semigroup (T (t)) t 0 on C(X) such that T (t) e ωt for every t 0 for some ω R. If B is a bounded linear operator on C(X) satisfying a generalized positive maximum principle with respect to some ω R, then (A+B, D(A)) generates a strongly continuous positive semigroup (S(t)) t 0 on C(X) satisfying (2.) S(t) e (ω+ω )t, t 0.

4 04 Francesco Altomare Graziana Musceo 4 In particular, if γ C(X), then the operator (A + γi, D(A)) is the generator of a strongly continuous positive semigroup (S(t)) t 0 on C(X) satisfying (2.2) S(t) e (ω+γ )t, t 0, where γ := sup γ(x). x X Proof. Since B is bounded, it is well-known that (A+B, D(A)) generates a strongly continuous semigroup on C(X) (see, e.g., [, Theorem.3, p. 58]). Therefore, for sufficiently large λ > 0, we get (λi (A + B))(D(A)) = C(X). On the other h, A + B satisfies the generalized positive maximum principle with respect to ω + ω so the result follows from Theorem 2.. The second part of the statement is an obvious consequence of the first one because the operator B := γi satisfies the positive maximum principle with respect to γ. We proceed now to establish a generation result for degenerate secondorder differential operators in the framework of weighted continuous function spaces. To begin with, we introduce some notation we shall deal with throughout the paper. Let J be an arbitrary open interval of R set r := inf J R r 2 := sup J R +. As above, we shall denote by C(J) the space of all real valued continuous functions on J with C b (J) the Banach lattice of all bounded continuous functions endowed with the natural order the uniform norm. We consider the space (2.3) E(J) := f C(J) : there exists f(x) R for every i =, 2 x ri which is a Banach lattice with natural order the uniform norm, the weighted space (2.4) E w (J) := f C(J) : wf E(J), where w is a weight function on J, i.e., w C(J) w(x) > 0 for every x J. Observe that E w (J) is a Banach lattice with respect to the natural order the norm w defined by (2.5) f w := wf, f E w (J). Let us consider α C(J) such that > 0 for every x J, β C(J) γ E(J) set (2.6) γ := sup γ(x). x J Our main aim is to study the second order differential operator (2.7) Lu = αu + βu + γu

5 5 Degenerate differential operators with reflecting barriers 05 defined on the subspace of E w (J) C 2 (J) which is defined as follows. Fix once for all a point x 0 J define the Wronskian ( x ) β(t) (2.8) W (x) := exp α(t) dt, x J. x 0 The domain D N (L) of our operator L is defined by means of the generalized Reflecting Barriers boundary conditions, i.e., (2.9) D N (L) := (u w) (x) u D M (L) : x ri w 2 (x)w (x) = 0 for every i =, 2, where D M (L) denotes the maximal domain for L, i.e., (2.0) D M (L) = u E w (J) C 2 (J) : w(x)(u (x)+ x ri + β(x)u (x)) R for every i =, 2. If w is the constant function, then the boundary conditions defining D N (L) turn into the ordinary Reflecting Barriers boundary conditions x r i u (x) = 0, i =, 2, W (x) which have been studied, e.g., in [9]. Since γ E(J), clearly the operator L maps D N (L) into E w (J). The main generation result for the operator (L, D N (L)) reads as follows. In the sequel we shall denote by J i the interval whose endpoints are x 0 r i, i =, 2. Theorem 2.3. Assume that w C 2 (J) that ( 2w (x) 2 w (x)w(x) ) β(x)w(x)w (x) (2.) x ri w 2 R, (x) set ( 2w (x) 2 w (x)w(x) ) β(x)w(x)w (x) (2.2) ω := sup x J w 2 < +. (x) Suppose that at each endpoint r i one of the following conditions is fulfilled: (i) ( ) αw 2 W L (J i ) (ii) ( ) αw 2 W / L (J i ), w 2 W / L (J i ) (2.3) sup x J i σ(x) x ds α(s)w 2 < +, (s)w (s)

6 06 Francesco Altomare Graziana Musceo 6 where σ(x) J i is defined by σ(x) x w 2 (s)w (s) ds = w(x 0 ) 2. Then the operator (L, D N (L)) defined by (2.7) (2.9) is the generator of a strongly continuous positive semigroup (T (t)) t 0 on E w (J) satisfying (2.4) T (t) e (ω+γ ) t for every t 0, with ω defined by (2.2) γ defined by (2.6). Proof. In order to prove the result, we will use an isometric isomorphism to identify the spaces E w (J) E(J) then we shall study the operator ( L, D N ( L)) on E(J) obtained by similarity from our operator (L, D N (L)). So, let us consider the lattice isometric isomorphism Φ : E w (J) E(J) defined by (2.5) Φ(f)(x) := w(x)f(x) for every f E w (J) x J consider the domain D N ( L) := Φ(D N (L)) = v D M ( L) : v w D N(L), where D M ( L) = Φ(D M (L)) = v E(J) : v w D M(L) = ( v ) = v E(J) C 2 (J) : w(x)l (x) R, i =, 2. x ri w The differential operator on E(J) obtained by similarity through the above isomorphism (2.5) is then L : D N ( L) E(J) defined by ( v ) (2.6) Lv = Φ(L(Φ (v))) = w L. w By an easy computation, we may give an explicit representation of the domains D M ( L) D N ( L) of the operator L in terms of the coefficient α, β γ the weight w. More precisely, since ( v w ( v w ) (x) = v (x)w(x) v(x)w (x) w 2 (x) ) (x) = w(x) v (x) 2 w (x) w 2 (x) v (x) + 2w (x) 2 w (x)w(x) w 3 v(x), (x)

7 7 Degenerate differential operators with reflecting barriers 07 then, for every v D N ( L) x J, ( v ) Lv(x) = w(x)l (x) = w ( v ) ( v ) ( v ) = w(x) (x) + β(x)w(x) (x) + γ(x)w(x) (x) = w w w = v (x) + β(x)w(x) 2w (x) v (x)+ w(x) + 2w (x) 2 w (x)w(x) β(x)w(x)w (x) w 2 (x) To simplify the notation we set (2.7) (2.8) (2.9) :=, β(x) := β(x)w(x) 2w (x), w(x) v(x) + γ(x) v(x). γ(x) := [ 2w (x) 2 w (x)w(x) ] β(x)w(x)w (x) w 2 (x) for every x J, so that (2.20) Lv = αv + βv + ( γ + γ) v for every v D N ( L). Note that, since α, β C(J) > 0 for every x J, then α, β C(J) > 0 for every x J as well. Moreover, by (2.), γ E(J) hence (2.2) D M ( L) = v E(J) C 2 (J) : there exists v (x) + β(x)v (x) R for every i =, 2 x r i Moreover, if we consider the new Wronskian ) x β(t) (2.22) W (x) := exp ( α(t) dt, x J, then we get W (x)w (2.23) W 2 (x) (x) = w 2, x J, (x 0 ) hence (2.24) D N ( L) = = x 0 v D M ( L) : x ri v D M ( L) : x ri. v (x) w 2 (x)w (x) = 0 for every i =, 2 v (x) W (x) = 0 for every i =, 2.

8 08 Francesco Altomare Graziana Musceo 8 After this preinaries, from the general theory of strongly continuous semigroup we then know that (L, D N (L)) generates a strongly continuous positive semigroup on E w (J) if only if the same property holds true for ( L, D N ( L)) on E(J). Moreover, the two relevant semigroups are similar between them so they have the same norm (see, e.g., [, p. 43 p. 59]). In order to study the generation property of ( L, D N ( L)) on E(J) we also point out that, denoted by J the two-point compactification of J, then E(J) can be naturally identified with the Banach lattice C( J) through the lattice isometric isomorphism which to every f E(J) associates the unique continuous extension f of f to J. Therefore we can apply Corollary 2.2 by referring to the space E(J) so, since sup ( γ(x) + γ(x)) ω + γ, x J it is enough to show that the incomplete operator Ãv := α v + β v defined on D N (Ã) := D N( L), generates a strongly continuous semigroup of positive contractions on E(J). Note that, from (2.7) (2.23) follows that (2.25) α W = w2 (x 0 ) αw 2 W hence (2.3) is equivalent to require σ(x) ds (2.26) sup x J i x α(s) W (s) < +, where, for every x J i, σ(x) J i it satisfies σ(x) x W (s) ds =. Hypotheses (i) (ii) as well as formulas (2.25) (2.26) imply that all the hypotheses of Theorem 0.2 of [9] are satisfied so that the operator (Ã, DN(Ã)) actually generates a strongly continuous semigroup of positive contractions on E(J) the proof is now complete. Throughout the next sections we shall discuss some applications of Theorem EXAMPLES AND APPLICATIONS In this section we discuss several applications of Theorem 2.3 by presenting different examples of differential operators defined on bounded or unbounded intervals.

9 9 Degenerate differential operators with reflecting barriers 09 This examples complement other ones which we discuss in [6] [7] where, however, we considered other boundary conditions only bounded weight functions. 3.. A DIFFERENTIAL OPERATOR ON ], + [ Consider the interval J = ], + [, the weight w(x) := x ( < x) α C(], + [) such that (3.) x + R (3.2) 0 < a b, < x for some a, b R. On the weighted space (3.3) E w (], + [) = f C(], + [) : there exists consider the differential operator x f(x) R x + (3.4) Au := α u defined on (3.5) D N (A) := f(x) R x + u E w (], + [) C 2 (], + [) : Au E w (], + [), x u (x) + u(x) = Since β = γ = 0, then W =, γ = 0 (3.6) ω = 3 4 sup <x x u (x) + 2 u(x) x + x 3/2 = 0. x 2. Choose x 0 = 2; clearly condition (i) of Theorem 2.3 is satisfied for r i =. As regards r 2 = +, clearly w 2 / L ([2, + [) / L ([2, + [) because, α w 2 for x 2, x b x. Finally, in this case, σ(x) = x (x > ) hence σ(x) α(s) w 2 (s) ds σ(x) a s ds = ( ) σ(x) a ln = x a ln + 4/x 2 x x so that (2.3) is satisfied. Summing up, according to Theorem 2.3, we then obtain

10 0 Francesco Altomare Graziana Musceo 0 Corollary 3.. The operator (A, D N (A)) generates a strongly continuous positive semigroup (T (t)) t 0 on E w (], + [) satisfying where ω is defined by (3.6). T (t) e ωt, t 0, 3.2. A DIFFERENTIAL OPERATOR ON ]0, [ EQUIPPED WITH BOUNDED JACOBI WEIGHTS Let consider J = ]0, [, α C (]0, [) such that > 0 for every x ]0, [ the weight function (3.7) w(x) = x p ( x) q, 0 < x <, with 0 < p < 2 0 < q < 2. The weighted function space Ew (J) defined by (2.4) is (3.8) E w (]0, [) = f C(]0, [) : x p ( x) q f(x) R, x on the subspace (3.9) D N (A) := u D M (A) : (uw) (x) w 2 (x)w (x) = x = x [x( x)u (x) + (p (p + q)x)u(x)] x p+ ( x) q+ = 0 define the operator A : D N (A) E w (]0, [) by (3.0) Au(x) := (α u ) (x) = u (x) + α (x)u (x) for every u E w (]0, [) x ]0, [. For this operator we have the following generation result. Corollary 3.2. Assume that (3.) x 2 ( x) x 2 R x α (x) x( x) R is satisfied. Then the operator (A, D N (A)) defined by (3.9) (3.0) is the generator of a strongly continuous positive semigroup (T (t)) t 0 on E w (]0, [). Moreover, set (3.2) α 0 := sup x ]0,[ We have x 2 ( x) 2 α 0 := sup x ]0,[ (3.3) T (t) e ωt α (x) x( x).

11 Degenerate differential operators with reflecting barriers for every t 0, with ω α 0 maxq (q + ), p (p + ) + α 0 maxp, q. Proof. Observe that γ 0 consequently, by (2.6), γ 0. Differentiating the weight w we get (3.4) w (x) = p x p ( x) q q x p ( x) q (3.5) w (x) = p(p )x p 2 ( x) q 2pqx p ( x) q + q(q )x p ( x) q 2. By a direct computation, which we omit for brevity, replacing (3.4) (3.5) in formula (2.), we obtain (3.6) ( 2w (x) 2 w (x)w(x) ) β(x)w(x)w (x) w 2 (x) = (p + q) (p + q + )x2 2p(p + q + )x + p(p + ) x 2 ( x) 2 x 0 = + α (x) Therefore, condition (2.) is satisfied since (3.) holds. Now fix a point x 0 ]0, [ consider the Wronskian ( x α ) (t) (3.7) W (x) = exp α(t) dt = α(x 0), 0 < x <, (3.8) where K := α(x 0 ). Then w 2 (x)w (x) L (]0, x 0 [) w 2 (x)w (x) = K x 2p ( x) 2q, Kx 2p = + ( x) 2q x (p + q)x p. x( x) w 2 (x)w (x) L (]x 0, [). So, hypothesis (i) of Theorem 2.3 is fulfilled so the operator (A, D N (A)) is the generator of a strongly continuous semigroup (T (t)) t 0 on E w (J). Now, on account of (3.2) (3.6), ( 2w (x) 2 w (x)w(x) ) β(x)w(x)w (x) w 2 α 0 p (p + ) + α 0 p (x) ( 2w (x) 2 w (x)w(x) ) β(x)w(x)w (x) x w 2 (x) α 0 q(q + ) + α 0 q.

12 2 Francesco Altomare Graziana Musceo 2 Therefore, by (2.4), T (t) e ωt for every t 0, with ω α 0 maxq (q + ), p (p + ) + α 0 maxp, q A DIFFERENTIAL OPERATOR ON ]0, [ EQUIPPED WITH UNBOUNDED JACOBI WEIGHTS Consider J = ]0, [, α C(]0, [) such that > 0 for every x ]0, [ the weight function (3.9) w(x) = x p ( x) q, 0 < x <, where p > 0 q > 0. Consider the weighted space (3.20) E w f(x) (]0, [) = f C(]0, [) : x p ( x) q R x the operator A : D N (A) E w (]0, [) defined by (3.2) Au(x) := u (x) for every u D N (A) x ]0, [, where (3.22) D N (A) := u D M (]0, [) : x (uw) (x) w 2 (x)w (x) = = x x p ( x) q u (x) + [q x p ( x) q p x p ( x) q ]u(x) = 0 This operator generates a strongly continuous positive semigroup as stated in the following result. Corollary 3.3. Assume that 2 < min(p, q) that exist 2 r < 2p+ 2 s < 2q + such that x R \ 0 r x ( x) R \ 0. s Then the operator (A, D N (A)) defined by (3.2) (3.22) is the generator of a strongly continuous positive semigroup (T (t)) t 0 on E w (]0, [). Moreover, set (3.23) Λ := sup x ]0,[ x 2 ( x) 2 we have T (t) e w t for every t 0, with ω Λ maxp (p ), q (q )..

13 3 Degenerate differential operators with reflecting barriers 3 Proof. First note that it follows from the assumptions that = 0 x x x 2 ( x) 2 R. Moreover, since β = γ = 0, we have γ = 0 W =. Differentiating the weight function, for 0 < x <, we get w (x) = w (x) = (q + p)x p x p+ ( x) q+ q (q + ) x p ( x) q+2 2p q p (p + ) x p+ + ( x) q+ x p+2 ( x) q. Replacing these expression in formula (2.), we obtain ( 2w (x) 2 w (x)w(x) ) w 2 (x) = (q + p)(q + p )x2 2p(p + q )x + p(p ) x 2 ( x) 2 hence condition (2.) of Theorem 2.3 is satisfied. Now we proceed to check condition (i) of Theorem 2.3, because of the symmetry of the assumptions on 0, it is sufficient to consider only the endpoint r = 0. Consider the function ϕ(x) := w 2 (x)w (x) = x2p ( x) 2q, 0 < x <. If 2p r, then ϕ(x) = 0; if 2p < r, ϕ(x) = + x 0 xr 2p ϕ(x) + = 0 hence, in both cases, ϕ L (]0, /2]). Therefore, according to Theorem 2.3, the operator (A, D N (A)) is the generator of a strongly continuous positive semigroup (T (t)) t 0 on E w (]0, [). Now, taking (3.23) into account, hence, by (2.4), ( 2w (x) 2 w (x)w(x) ) w 2 (x) = Λ p (p ) ( 2w (x) 2 w (x)w(x) ) x w 2 Λ q (q ) (x) T (t) e ω t for every t 0, with ω Λ maxp (p ), q (q ).

14 4 Francesco Altomare Graziana Musceo ANOTHER LOOK TO THE BLACK-SCHOLES EQUATION We close this section with an application to a model arising in Mathematical Finance; it concerns the widely famous Black-Scholes equation in the theory of the option pricing, which may be written as (3.24) c σ2 (x, t) = t 2 x2 2 c c (x, t) + rx (x, t) rc(x, t), x > 0, t > 0, x2 x where c = c(x, t) is the no-arbitrage price of an option, x is the price of the underlying asset t the time to expiry. In the simple form (3.24), the positive parameters σ r, denoting the volatility the riskless interest rate respectively, are assumed to be constant over the time. This mathematical model is generally coupled with some initial boundary conditions such as (3.25) c(x, 0) = Φ 0 (x), x > 0, c(0, t) = 0, t > 0, where Φ 0 is a function of x which depends on the model under consideration. For a more detailed discussion about equation (3.24) for the meaning of the parameters σ r the above boundary conditions, we refer, e.g., to [8], [0], [2], [3], [6], [9]. The differential operator associated with the Black-Scholes equation is the complete operator (3.26) Lu(x) = σ2 2 x2 u (x) + rxu (x) ru(x) for every x J := ]0, + [. Note that in this case, according to (2.7), = σ2 2 x2, β(x) = rx γ(x) = r, x ]0, + [. Therefore, for x 0 =, the Wronskian turns into W (x) = where k := 2r. x k σ 2 We are interested in studying the generation property of the operator (3.26) in some weighted continuous function spaces. Actually, in such settings some results have been already obtained in [2] [7], by associating with the operator L maximal boundary conditions or Wentzell boundary conditions by considering bounded weight functions. In the present context we shall consider unbounded polynomial weight (3.27) w m (x) := + x m, x > 0, m > 0,

15 5 Degenerate differential operators with reflecting barriers 5 generalized Reflecting Barrier conditions which are described as follows. The relevant weighted space associated with w m is (3.28) C wm (]0, + [) = = f C(]0, + [) : there exists ( + x m )f(x) R = = x + f C(]0, + [) : f(x) R x + xm ) f(x) R. If u D M (L) then, for every x > 0, (u w m ) (x) wm(x) 2 W (x) = xk ( + x m ) u (x) + m x m+k u(x) ( + x m ) 2 hence, if < k < 2m +, (3.29) D N (L) = u D M (L) : x + = u D M (L) : x 0 xk u (x) = 0 + The following generation result holds true. (u w m ) (x) wm(x)w 2 (x) = 0 = x + x k + x m u (x) = 0. Corollary 3.4. Assume that < k < 2m +. Then the operator (L, D N (L)) defined by (3.26) (3.29) is the generator of a strongly continuous positive semigroup (T (t)) t 0 on C wm (]0, + [) satisfying (3.30) T (t) e ωm t for every t 0, where ω m (m + ) T (t) = 0. t + ( σ 2 2 m r ). Therefore, if m < k then Proof. We will verify that L satisfies the hypotheses of Theorem 2.3. Differentiating the weight w m = + x m we get hence (3.3) w m(x) = mx m w m(x) = m(m )x m 2 x + = x + ( 2w m(x) 2 w m(x)w m (x) ) β(x)w m (x)w m(x) wm(x) 2 = ( ) ( ) σ 2 2 m(m + ) rm x 2m σ 2 2 m(m ) + rm x m σ2 2 ( + x m ) 2 m(m + ) rm < +

16 6 Francesco Altomare Graziana Musceo 6 so condition (2.) is fulfilled. Now, observe that w 2 m(x)w (x) = C x 2 k ( + x m ) 2, where C := σ2 2, hence α wm 2 W L (]0, ]) as well as α wm 2 W L ([,+ [). Thus 0 + satisfies hypothesis (i) in Theorem 2.3 hence the operator (L, D N (L)) generates a strongly continuous positive semigroup (T (t)) t 0 on C wm (]0, + [). Finally, observe that, by (3.3), ω σ2 2 m(m + ) rm where ω is defined by (2.2); moreover, γ := sup γ(x) = r, therefore by (2.4) it follows that x J (3.32) T (t) e (ω r) t e ωm t for every t 0. As a consequence, the initial boundary value problem associated with the Black-Scholes equation with initial datum Φ 0 D N (L) (3.6) c σ2 (x, t) = t 2 x2 2 c c (x, t) + rx (x, t) rc(x, t), x > 0, t > 0, x2 x c xk x + (x, t) = 0, t 0, x x k c + x m (x, t) = 0, t 0, x c(x, 0) = Φ 0 (x), x > 0, c(x, t) R ( + x + xm ) c(x, t) R, t 0, has a unique solution c : ]0, + [ [0, + [ R given by c(x, t) = T (t)φ 0 (x), x > 0, t > 0. Moreover, c(, t) D M (L) for every t 0, Finally, if m < k, c(x, t) eωm t + x m Φ 0 wm, x > 0, t 0. c(x, t) = 0 uniformly with respect to x > 0. t + Acknowledgement. This work has been partially supported by the Research Project Real Analysis Functional Analytic Methods for Differential Problems Approximation Problems, University of Bari, 2009.

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