on the occasion of his 70th birthday
DEGENERATE SECOND ORDER DIFFERENTIAL OPERATORS WITH GENERALIZED REFLECTING
BARRIERS BOUNDARY CONDITIONS
FRANCESCO ALTOMARE and 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=αu00+βu0+γu
generate strongly continuous positive semigroups, in the setting of weighted con- tinuous 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, 35K15, 35J70.
Key words: degenerate second-order differential operator, degenerate evolution equation, weighted continuous function space, strongly continuous positive semigroup, generalized Reflecting Barriers boundary condi- tion.
1. 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
(1.1) ∂u
∂t(x, t)α(x)∂2u
∂x2 +β(x)∂u
∂x(x, t) +γ(x)u(x, t), x∈J, t >0 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 and 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
(1.2) Lu=αu00+βu0+γu,
MATH. REPORTS12(62),2 (2010), 101–117
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 (1.1) and coupled with suitable boundary conditions.
Actually, several authors have been interested in generation results for differential operators of type (1.2). In this respect, we refer, e.g., to [2], [3], [4], [5], [9], [14], [15].
In [7], among other things, we showed a generation result for the operator L defined by (1.2) 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 and we consider other boundary conditions which guarantee further generation results. The starting point is the paper [9] where the authors study the gene- ration property of second-order degenerate differential operators in spaces of bounded continuous functions on a real open interval, under Reflecting Barri- ers boundary conditions.
In the present paper we consider generalized Reflecting Barriers boun- dary conditions which are incorporated into the domain
DN(L) =n
u∈DM(L) : lim
x→ri
(u w)0(x)
w2(x)W(x) = 0 for everyi= 1,2o of the weighted continuous function spaces
Ew(J) :={f ∈C(J) :wf ∈E(J)},
whereJ = (r1, r2) is an open interval,W is the Wronskian function,DM(L) is the maximal domain (see Section 2 for more details),wis a “weight” function on J, i.e., w∈C(J) andw(x)>0 for every x∈J, and
E(J) :=n
f ∈C(J) : there exists lim
x→ri
f(x)∈Rfor everyi= 1,2o . Under suitable assumptions on α, β, γ and w, we then show that the ope- rator (L, DN(L)) generates a strongly continuous (quasicontractive) positive semigroup on Ew(J).
As in [7] our method consists in identifying the spaceEw(J) with E(J) by an isometric isomorphism and 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 = ]0,1[ and on the real interval J = ]0,+∞[, considering in some cases a bounded continuous weight function and in other cases a continuous not bounded weight function. In particular, the last example concerns the
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 characteri- zation of quasicontractive positive semigroup on the Banach latticeC(X) of all real-valued continuous functions defined on a compact space X. This charac- terization 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∈Rif
Au(x0)≤ω u(x0) for every u∈D(A) andx0∈Rsatisfying sup
x∈X
u(x) =u(x0)>0.
For the proof of the next result we refer, e.g., to [1, Theorem 2.2] and [18, Theorem 9.3.3].
Here and in the sequel the symbol I stands for the identity operator.
Theorem 2.1. Consider a linear operator A : D(A) → C(X) defined on a linear subspace D(A) of C(X) and 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
kT(t)k ≤eωt for everyt≥0.
b) (i)There exists λ > ω such that (λI−A)(D(A)) =C(X).
(ii) A verifies the generalized positive maximum principle with res- pect to ω.
As a consequence we have the following generation result for an additive perturbation of (A, D(A)).
Corollary2.2. Let (A, D(A))be the generator of a strongly continuous positive semigroup (T(t))t≥0 on C(X) such that kT(t)k ≤eωt for every t≥0 and for some ω ∈R.
IfB is a bounded linear operator onC(X)satisfying a generalized positive maximum principle with respect to someω1 ∈R, then(A+B, D(A))generates a strongly continuous positive semigroup (S(t))t≥0 onC(X) satisfying
(2.1) kS(t)k ≤e(ω+ω1)t, t≥0.
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 onC(X) satisfying (2.2) kS(t)k ≤e(ω+γ∞)t, t≥0,
where γ∞:= sup
x∈X
γ(x).
Proof. SinceB is bounded, it is well-known that (A+B, D(A)) generates a strongly continuous semigroup onC(X) (see, e.g., [11, Theorem 1.3, p. 158]).
Therefore, for sufficiently large λ >0, we get (λI−(A+B))(D(A)) =C(X).
On the other hand, A+B satisfies the generalized positive maximum principle with respect to ω+ω1 and so the result follows from Theorem 2.1.
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 second- order differential operators in the framework of weighted continuous function spaces. To begin with, we introduce some notation we shall deal with through- out the paper.
LetJ be an arbitrary open interval ofRand set r1 := infJ ∈R∪ {−∞}
and r2 := supJ ∈R∪ {+∞}. As above, we shall denote byC(J) the space of all real valued continuous functions on J and with Cb(J) the Banach lattice of all bounded continuous functions endowed with the natural order and the uniform norm k · k∞. We consider the space
(2.3) E(J) :=n
f ∈C(J) : there exists lim
x→rif(x)∈Rfor everyi= 1,2o which is a Banach lattice with natural order and the uniform norm, and the weighted space
(2.4) Ew(J) :={f ∈C(J) :wf ∈E(J)},
where w is a “weight” function on J, i.e., w ∈C(J) and w(x) >0 for every x ∈ J. Observe that Ew(J) is a Banach lattice with respect to the natural order and the norm k · kw defined by
(2.5) kfkw :=kwfk∞, f ∈Ew(J).
Let us consider α∈ C(J) such that α(x) >0 for every x∈J,β ∈C(J) and γ ∈E(J) and set
(2.6) γ∞:= sup
x∈J
γ(x).
Our main aim is to study the second order differential operator
(2.7) Lu=αu00+βu0+γu
defined on the subspace of Ew(J)∩C2(J) which is defined as follows. Fix once and for all a point x0 ∈J and define the Wronskian
(2.8) W(x) := exp
− Z x
x0
β(t) α(t)dt
, x∈J.
The domain DN(L) of our operator L is defined by means of the generalized Reflecting Barriers boundary conditions, i.e.,
(2.9) DN(L) :=
n
u∈DM(L) : lim
x→ri
(u w)0(x)
w2(x)W(x) = 0 for everyi= 1,2 o
, where DM(L) denotes the maximal domain forL, i.e.,
DM(L) = n
u∈Ew(J)∩C2(J) : lim
x→riw(x)(α(x)u00(x)+
(2.10)
+β(x)u0(x))∈Rfor everyi= 1,2o .
Ifw is the constant function1, then the boundary conditions defining DN(L) turn into the ordinary Reflecting Barriers boundary conditions
x→rlimi u0(x)
W(x) = 0, i= 1,2, which have been studied, e.g., in [9].
Since γ ∈ E(J), clearly the operator L maps DN(L) into Ew(J). The main generation result for the operator (L, DN(L)) reads as follows. In the sequel we shall denote byJithe interval whose endpoints arex0andri,i= 1,2.
Theorem 2.3. Assume that w∈C2(J) and that
(2.11) lim
x→ri
α(x) 2w0(x)2−w00(x)w(x)
−β(x)w(x)w0(x)
w2(x) ∈R,
and set
(2.12) ω := sup
x∈J
α(x) 2w0(x)2−w00(x)w(x)
−β(x)w(x)w0(x)
w2(x) <+∞.
Suppose that at each endpoint ri one of the following conditions is fulfilled:
(i) αw12W
∈L1(Ji) (ii) αw12W
∈/ L1(Ji), w2W /∈L1(Ji) and
(2.13) sup
x∈Ji
Z σ(x)
x
ds α(s)w2(s)W(s)
<+∞,
where σ(x) ∈ Ji is defined by |Rσ(x)
x w2(s)W(s) ds| =w(x0)2. Then the ope- rator (L, DN(L))defined by (2.7)and (2.9)is the generator of a strongly con- tinuous positive semigroup (T(t))t≥0 onEw(J) satisfying
(2.14) kT(t)k ≤e(ω+γ∞)t
for every t≥0, withω defined by(2.12) and γ∞ defined by(2.6).
Proof. In order to prove the result, we will use an isometric isomorphism to identify the spaces Ew(J) and E(J) and then we shall study the operator (L, De N(L)) one E(J) obtained by similarity from our operator (L, DN(L)). So, let us consider the lattice isometric isomorphism Φ : Ew(J)→E(J) defined by
(2.15) Φ(f)(x) :=w(x)f(x)
for every f ∈Ew(J) andx∈J and consider the domain DN(L) := Φ(De N(L)) =n
v∈DM(L) :e v
w ∈DN(L)o , where
DM(L) = Φ(De M(L)) = n
v∈E(J) : v
w ∈DM(L) o
=
=n
v∈E(J)∩C2(J) : lim
x→riw(x)Lv w
(x)∈R, i= 1,2o .
The differential operator on E(J) obtained by similarity through the above isomorphism (2.15) is then Le:DN(eL)→E(J) defined by
(2.16) Lve = Φ(L(Φ−1(v))) =w Lv w
.
By an easy computation, we may give an explicit representation of the domains DM(L) ande DN(L) and of the operatore Le in terms of the coefficientα,β and γ and the weight w. More precisely, since
v w
0
(x) = v0(x)w(x)−v(x)w0(x) w2(x)
and v
w 00
(x) = 1
w(x)v00(x)−2w0(x)
w2(x)v0(x) +2w0(x)2−w00(x)w(x) w3(x) v(x),
then, for every v∈DN(L) ande x∈J, Lv(x) =e w(x)L
v w
(x) =
=α(x)w(x) v
w 00
(x) +β(x)w(x) v
w 0
(x) +γ(x)w(x) v
w
(x) =
=α(x)v00(x) +β(x)w(x)−2α(x)w0(x)
w(x) v0(x)+
+2α(x)w0(x)2−α(x)w00(x)w(x)−β(x)w(x)w0(x)
w2(x) v(x) +γ(x)v(x).
To simplify the notation we set
α(x) :=e α(x), (2.17)
β(x) :=e β(x)w(x)−2α(x)w0(x)
w(x) ,
(2.18)
eγ(x) := α(x)
2w0(x)2−w00(x)w(x)
−β(x)w(x)w0(x) w2(x)
(2.19)
for every x∈J, so that
(2.20) Lve =αve 00+βve 0+ (eγ+γ)v
for every v ∈ DN(L). Note that, sincee α, β ∈ C(J) and α(x) > 0 for every x ∈J, then α,e βe∈C(J) and α(x)e >0 for every x ∈J as well. Moreover, by (2.11), eγ ∈E(J) and hence
DM(L) =e n
v ∈E(J)∩C2(J) : there exists (2.21)
x→rlimiα(x)ve 00(x) +β(x)ve 0(x)∈R for everyi= 1,2 o
. Moreover, if we consider the new Wronskian
(2.22) Wf(x) := exp − Z x
x0
β(t)e α(t)e dt
!
, x∈J, then we get
(2.23) Wf(x) = W(x)w2(x)
w2(x0) , x∈J, hence
DN(L) =e n
v∈DM(L) : lime
x→ri
v0(x)
w2(x)W(x) = 0 for every i= 1,2 o (2.24)
= n
v∈DM(L) : lime
x→ri
v0(x)
Wf(x) = 0 for every i= 1,2 o
.
After this preliminaries, from the general theory of strongly continuous semi- group we then know that (L, DN(L)) generates a strongly continuous positive semigroup onEw(J) if and only if the same property holds true for (L, De N(L))e onE(J). Moreover, the two relevant semigroups are similar between them and so they have the same norm (see, e.g., [11, p. 43 and p. 59]).
In order to study the generation property of (eL, DN(L)) one E(J) we also point out that, denoted by Jethe two-point compactification of J, then E(J) can be naturally identified with the Banach lattice C(Je) through the lattice isometric isomorphism which to every f ∈E(J) associates the unique continuous extension feof f toJ.e
Therefore we can apply Corollary 2.2 by referring to the spaceE(J) and so, since
sup
x∈J
(eγ(x) +γ(x))≤ω+γ∞, it is enough to show that the incomplete operator
Ave :=α ve 00+β ve 0
defined on DN(A) :=e DN(L), generates a strongly continuous semigroup ofe positive contractions on E(J).
Note that, from (2.17) and (2.23) follows that
(2.25) 1
αfeW = w2(x0) αw2W and hence (2.13) is equivalent to require
(2.26) sup
x∈Ji
Z σ(x)
x
ds α(s)e Wf(s)
<+∞, where, for every x∈Ji,σ(x)∈Ji and it satisfies
Rσ(x)
x Wf(s) ds = 1.
Hypotheses (i) and (ii) as well as formulas (2.25) and (2.26) imply that all the hypotheses of Theorem 0.2 of [9] are satisfied so that the operator (A, De N(A)) actually generates a strongly continuous semigroup of positivee contractions on E(J) and the proof is now complete.
Throughout the next sections we shall discuss some applications of Theo- rem 2.3.
3. EXAMPLES AND APPLICATIONS
In this section we discuss several applications of Theorem 2.3 by pre- senting different examples of differential operators defined on bounded or un- bounded intervals.
This examples complement other ones which we discuss in [6] and [7]
where, however, we considered other boundary conditions and only bounded weight functions.
3.1. A DIFFERENTIAL OPERATOR ON ]1,+∞[
Consider the intervalJ = ]1,+∞[, the weight w(x) := √
x (1 < x) and α ∈C(]1,+∞[) such that
(3.1) lim
x→1+α(x)∈R and
(3.2) 0< a≤α(x)≤b, 1< x for some a, b∈R. On the weighted space
Ew(]1,+∞[) =n
f ∈C(]1,+∞[) : there exists lim
x→1+f(x)∈R (3.3)
and lim
x→+∞
√x f(x)∈R o
consider the differential operator
(3.4) Au:=α u00
defined on
DN(A) :=
n
u∈Ew(]1,+∞[)∩C2(]1,+∞[) :Au∈Ew(]1,+∞[), (3.5)
x→1lim+u0(x) +1
2u(x) = 0 and lim
x→+∞
x u0(x) +12u(x) x3/2 = 0
o . Since β =γ = 0, then W =1,γ∞= 0 and
(3.6) ω= 3
4 sup
1<x
α(x) x2 .
Choose x0= 2; clearly condition (i) of Theorem 2.3 is satisfied for ri = 1. As regards r2 = +∞, clearly w2 ∈/ L1([2,+∞[) and α w12 ∈/ L1([2,+∞[) because, forx≥2, α(x)1 x ≥ b x1 . Finally, in this case,σ(x) =√
x2+ 4 (x >1) and hence
Z σ(x) x
1
α(s)w2(s)ds
≤ 1 a
Z σ(x) x
1
sds= 1 aln
σ(x) x
= 1 a lnp
1 + 4/x2 so that (2.13) is satisfied.
Summing up, according to Theorem 2.3, we then obtain
Corollary 3.1. The operator (A, DN(A)) generates a strongly conti- nuous positive semigroup (T(t))t≥0 on Ew(]1,+∞[) satisfying
kT(t)k ≤eωt, t≥0, where ω is defined by (3.6).
3.2. A DIFFERENTIAL OPERATOR ON ]0,1[
EQUIPPED WITH BOUNDED JACOBI WEIGHTS
Let consider J = ]0,1[, α ∈ C1(]0,1[) such that α(x) > 0 for every x∈]0,1[ and the weight function
(3.7) w(x) =xp(1−x)q, 0< x <1,
with 0 < p < 12 and 0< q < 12. The weighted function space Ew(J) defined by (2.4) is
(3.8) Ew(]0,1[) = n
f ∈C(]0,1[) : lim
x→0+
x→1−
xp(1−x)qf(x)∈R o
, and on the subspace
DN(A) :=n
u∈DM(A) : lim
x→0+
x→1−
(uw)0(x) w2(x)W(x) = (3.9)
= lim
x→0+
x→1−
α(x)[x(1−x)u0(x) + (p−(p+q)x)u(x)]
xp+1(1−x)q+1 = 0o define the operator A:DN(A)→Ew(]0,1[) by
(3.10) Au(x) := (α u0)0(x) =α(x)u00(x) +α0(x)u0(x)
for every u∈Ew(]0,1[) andx∈]0,1[. For this operator we have the following generation result.
Corollary 3.2. Assume that
(3.11) lim
x→0+
x→1−
α(x)
x2(1−x)2 ∈R and lim
x→0+
x→1−
α0(x) x(1−x) ∈R
is satisfied. Then the operator (A, DN(A)) defined by (3.9) and (3.10) is the generator of a strongly continuous positive semigroup (T(t))t≥0 on Ew(]0,1[).
Moreover, set
(3.12) α0 := sup
x∈]0,1[
α(x)
x2(1−x)2 and α00:= sup
x∈]0,1[
α0(x) x(1−x). We have
(3.13) kT(t)k ≤eωt
for every t≥0, withω ≤α0max{q(q+ 1), p(p+ 1)}+α00max{p, q}.
Proof. Observe thatγ ≡0 and consequently, by (2.6),γ∞≡0. Differen- tiating the weight w we get
(3.14) w0(x) =p xp−1(1−x)q−q xp(1−x)q−1 and
w00(x) =p(p−1)xp−2(1−x)q−2pqxp−1(1−x)q−1+q(q−1)xp(1−x)q−2. (3.15)
By a direct computation, which we omit for brevity, and replacing (3.14) and (3.15) in formula (2.11), we obtain
α(x) 2w0(x)2−w00(x)w(x)
−β(x)w(x)w0(x)
w2(x) =
(3.16)
=α(x)(p+q) (p+q+ 1)x2−2p(p+q+ 1)x+p(p+ 1)
x2(1−x)2 +α0(x)(p+q)x−p x(1−x) . Therefore, condition (2.11) is satisfied since (3.11) holds.
Now fix a point x0 ∈]0,1[ and consider the Wronskian (3.17) W(x) = exp
− Z x
x0
α0(t) α(t) dt
= α(x0)
α(x) , 0< x <1, and
(3.18) 1
α(x)w2(x)W(x) = 1
K x2p(1−x)2q, where K:=α(x0). Then
lim
x→0+
x→1−
1
Kx2p(1−x)2q = +∞
and
1
α(x)w2(x)W(x) ∈L1(]0, x0[) and 1
α(x)w2(x)W(x) ∈L1(]x0,1[).
So, hypothesis (i) of Theorem 2.3 is fulfilled and so the operator (A, DN(A)) is the generator of a strongly continuous semigroup (T(t))t≥0 on Ew(J).
Now, on account of (3.12) and (3.16),
x→0lim+
α(x) 2w0(x)2−w00(x)w(x)
−β(x)w(x)w0(x)
w2(x) ≤α0p(p+ 1) +α00 p and
x→1lim−
α(x) 2w0(x)2−w00(x)w(x)
−β(x)w(x)w0(x)
w2(x) ≤α0q(q+ 1) +α00q.
Therefore, by (2.14),
kT(t)k ≤eωt
for every t≥0, with ω≤α0max{q(q+ 1), p(p+ 1)}+α00max{p, q}.
3.3. A DIFFERENTIAL OPERATOR ON ]0,1[
EQUIPPED WITH UNBOUNDED JACOBI WEIGHTS
Consider J = ]0,1[, α ∈ C(]0,1[) such thatα(x)> 0 for every x ∈]0,1[
and the weight function
(3.19) w(x) = 1
xp(1−x)q, 0< x <1, where p >0 andq >0. Consider the weighted space (3.20) Ew(]0,1[) =
n
f ∈C(]0,1[) : lim
x→0+
x→1−
f(x)
xp(1−x)q ∈R o
and the operatorA:DN(A)→Ew(]0,1[) defined by
(3.21) Au(x) :=α(x)u00(x)
for every u∈DN(A) and x∈]0,1[, where DN(A) :=n
u∈DM(]0,1[) : lim
x→0+
x→1−
(uw)0(x) w2(x)W(x) = (3.22)
= lim
x→0+
x→1−
xp(1−x)qu0(x) + [q xp(1−x)q−1−p xp−1(1−x)q]u(x) = 0o . This operator generates a strongly continuous positive semigroup as stated in the following result.
Corollary3.3. Assume that 12 <min(p, q)and that exist2≤r <2p+1 and 2≤s <2q+ 1 such that lim
x→0+ α(x)
xr ∈R\ {0} and lim
x→1− α(x)
(1−x)s ∈R\ {0}.
Then the operator(A, DN(A))defined by(3.21)and(3.22)is the genera- tor of a strongly continuous positive semigroup (T(t))t≥0 onEw(]0,1[). More- over, set
(3.23) Λ := sup
x∈]0,1[
α(x) x2(1−x)2 we have
kT(t)k ≤ew t
for every t≥0, withω ≤Λ max{p(p−1), q(q−1)}.
Proof. First note that it follows from the assumptions that lim
x→0+
x→1−
α(x) = 0 and lim
x→0+
x→1−
α(x)
x2(1−x)2 ∈R. Moreover, since β =γ = 0, we haveγ∞= 0 andW =1.
Differentiating the weight function, for 0< x <1, we get w0(x) = (q+p)x−p
xp+1(1−x)q+1 and
w00(x) = q(q+ 1)
xp(1−x)q+2 − 2p q
xp+1(1−x)q+1 + p(p+ 1) xp+2(1−x)q. Replacing these expression in formula (2.11), we obtain
α(x) 2w0(x)2−w00(x)w(x)
w2(x) =
=α(x)(q+p)(q+p−1)x2−2p(p+q−1)x+p(p−1) x2(1−x)2
and hence condition (2.11) of Theorem 2.3 is satisfied.
Now we proceed to check condition (i) of Theorem 2.3 and, because of the symmetry of the assumptions on 0 and 1, it is sufficient to consider only the endpoint r1 = 0. Consider the function
ϕ(x) := 1
α(x)w2(x)W(x) = x2p(1−x)2q
α(x) , 0< x <1.
If 2p≤r, then lim
x→0+ϕ(x) = 0; if 2p < r, lim
x→0+ϕ(x) = +∞and lim
x→0+xr−2pϕ(x)
= 0 and hence, in both cases, ϕ∈L1(]0,1/2]).
Therefore, according to Theorem 2.3, the operator (A, DN(A)) is the generator of a strongly continuous positive semigroup (T(t))t≥0 onEw(]0,1[).
Now, taking (3.23) into account,
x→0lim+
α(x) 2w0(x)2−w00(x)w(x)
w2(x) ≤Λp(p−1) and
x→1lim−
α(x) 2w0(x)2−w00(x)w(x)
w2(x) ≤Λq(q−1) and hence, by (2.14),
kT(t)k ≤eω t
for every t≥0, with ω≤Λ max{p(p−1), q(q−1)}.
3.4. ANOTHER LOOK TO THE BLACK-SCHOLES EQUATION
We close this section with an application to a model arising in Mathe- matical Finance; it concerns the widely famous Black-Scholes equation in the theory of the option pricing, which may be written as
(3.24) ∂c
∂t(x, t) = σ2
2 x2 ∂2c
∂x2(x, t) +rx ∂c
∂x(x, t)−rc(x, t), x >0, t >0, where c= c(x, t) is the no-arbitrage price of an option, x is the price of the underlying asset and tthe time to expiry.
In the simple form (3.24), the positive parametersσ andr, denoting the volatility and 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
c(x,0) = Φ0(x), x >0, c(0, t) = 0, t >0, (3.25)
where Φ0 is a function ofx which depends on the model under consideration.
For a more detailed discussion about equation (3.24) and for the meaning of the parameters σ and r and the above boundary conditions, we refer, e.g., to [8], [10], [12], [13], [16], [19].
The differential operator associated with the Black-Scholes equation is the complete operator
(3.26) Lu(x) = σ2
2 x2u00(x) +rxu0(x)−ru(x) for every x∈J := ]0,+∞[.
Note that in this case, according to (2.7), α(x) = σ22 x2,β(x) = rx and γ(x) = −r, x ∈]0,+∞[. Therefore, for x0 = 1, the Wronskian turns into W(x) = x1k wherek:= 2rσ2.
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] and [7], by associating with the operator L maximal boundary conditions or Wentzell boundary conditions and by considering bounded weight functions.
In the present context we shall consider unbounded polynomial weight (3.27) wm(x) := 1 +xm, x >0, m >0,
and generalized Reflecting Barrier conditions which are described as follows.
The relevant weighted space associated with wm is Cwm(]0,+∞[) = (3.28)
= n
f ∈C(]0,+∞[) : there exists lim
x→0+
x→+∞
(1 +xm)f(x)∈R o
=
=n
f ∈C(]0,+∞[) : lim
x→0+f(x)∈Rand lim
x→+∞(1 +xm)f(x)∈R o
. If u∈DM(L) then, for every x >0,
(u wm)0(x)
w2m(x)W(x) = xk(1 +xm)u0(x) +m xm+k−1u(x) (1 +xm)2
and hence, if 1< k <2m+ 1, DN(L) =
n
u∈DM(L) : lim
x→0+
x→+∞
(u wm)0(x) w2m(x)W(x) = 0
o
= (3.29)
=n
u∈DM(L) : lim
x→0+xku0(x) = 0 and lim
x→+∞
xk
1 +xmu0(x) = 0o . The following generation result holds true.
Corollary 3.4. Assume that 1 < k < 2m + 1. Then the operator (L, DN(L)) defined by (3.26) and (3.29) is the generator of a strongly con- tinuous positive semigroup (T(t))t≥0 onCwm(]0,+∞[) satisfying
(3.30) kT(t)k ≤eωmt
for every t ≥0, where ωm ≤(m+ 1) σ2
2 m−r
. Therefore, if m < k then
t→+∞lim kT(t)k= 0.
Proof. We will verify that L satisfies the hypotheses of Theorem 2.3.
Differentiating the weight wm = 1 +xm we get
w0m(x) =mxm−1 and wm00(x) =m(m−1)xm−2 and hence
lim
x→0+
x→+∞
α(x) 2w0m(x)2−w00m(x)wm(x)
−β(x)wm(x)w0m(x)
wm2(x) =
(3.31)
= lim
x→0+
x→+∞
σ2
2 m(m+ 1)−rm
x2m−
σ2
2 m(m−1) +rm xm
(1 +xm)2 ≤
≤ σ2
2 m(m+ 1)−rm <+∞
and so condition (2.11) is fulfilled. Now, observe that 1
α(x)w2m(x)W(x) = 1
C x2−k(1 +xm)2, where C:= σ22, and hence α w12
mW ∈L1(]0,1]) as well as α w12
mW ∈L1([1,+∞[).
Thus 0 and +∞satisfies hypothesis (i) in Theorem 2.3 and hence the operator (L, DN(L)) generates a strongly continuous positive semigroup (T(t))t≥0 on Cwm(]0,+∞[). Finally, observe that, by (3.31), ω≤ σ22 m(m+ 1)−rmwhere ω is defined by (2.12); moreover, γ∞:= sup
x∈J
γ(x) =−r, therefore by (2.14) it follows that
(3.32) kT(t)k ≤e(ω−r)t≤eωmt for every t≥0.
As a consequence, the initial boundary value problem associated with the Black-Scholes equation and with initial datum Φ0∈DN(L)
∂c
∂t(x, t) = σ2
2 x2 ∂2c
∂x2(x, t) +rx ∂c
∂x(x, t)−rc(x, t), x >0, t >0,
x→0lim+xk ∂c
∂x(x, t) = 0, t≥0,
x→+∞lim xk 1 +xm
∂c
∂x(x, t) = 0, t≥0,
c(x,0) = Φ0(x), x >0,
x→0lim+c(x, t)∈Rand lim
x→+∞(1 +xm)c(x, t)∈R, t≥0, (3.6)
has a unique solution c : ]0,+∞[×[0,+∞[→ R given by c(x, t) = T(t)Φ0(x), x >0,t >0. Moreover,c(·, t)∈DM(L) for everyt≥0, and
|c(x, t)| ≤ eωmt
1 +xmkΦ0kwm, x >0, t≥0.
Finally, if m < k, lim
t→+∞c(x, t) = 0 uniformly with respect to x >0.
Acknowledgement. This work has been partially supported by the Research Project
“Real Analysis and Functional Analytic Methods for Differential Problems and Ap- proximation Problems”, University of Bari, 2009.
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Received 15 October 2009 Dipartimento di Matematica
Universit`a degli Studi di Bari Via E. Orabona, 4
70125 Bari, Italia
