with growing initial conditions
Gis´ele Ruiz Goldstein, Jerome A. Goldstein ∗, Rosa Maria Mininni, Silvia Romanelli†
September 23, 2018
Abstract
In this paper we solve the problem of the existence and strong continuity of the semigroup associated with the initial value problem for a generalized Cox-Ingersoll-Ross equation for the price of a zero-coupon bond (see [8]), on spaces of continuous functions on [0, ∞) which can grow at infinity. We focus on the Banach spaces
Ys= f ∈ C[0, ∞) : f (x) 1 + xs ∈ C0[0, ∞) , s ≥ 1,
which contain the nonzero constants very common as initial data in the Cauchy prob-lems coming from financial models. In addition, a Feynman-Kac type formula is given. Key words. Cox-Ingersoll-Ross equation, (C0) semigroups, Feynman-Kac type
for-mula, large initial condition.
AMS subject classifications. 35K65, 47D06, 35Q91
1
Introduction
Of concern is the initial value problem studied in [8], ∂u ∂t = ν 2x∂2u ∂x2 + (γ + β x) ∂u ∂x − rx u u(0, x) = f (x), (1)
for x ≥ 0, t ≥ 0. The constant coefficients satisfy ν > 0, γ > 0, β ∈ R and r > 0. This problem generalizes the so-called CIR problem introduced by Cox, Ingersoll and Ross in 1985 ([3]) to price a discount zero-coupon bond, that is a contract promising to pay a certain “face” amount, conventionally taken equal to 1 (currency unit), at a fixed
∗
Department of Mathematical Sciences, University of Memphis, 373 Dunn Hall, Memphis, TN 38152-3240, USA ([email protected], [email protected])
†
Department of Mathematics, University of Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy ([email protected], [email protected])
maturity date T . In the CIR framework, the variable x in (1) denotes the current value of the interest rate assumed to be stochastic, the potential term is −x u (r = 1), the initial condition is f (x) = 1, which corresponds to the face value 1 of the bond at the maturity T , and ν = σ/√2 with σ > 0 being the volatility of the stochastic interest rate (for more details the reader can refer to [8]).
This problem can be compared to the Black-Scholes equation to price European options without dividends ([1]) ∂v ∂t = ν 2x2 ∂2v ∂x2 + µx ∂v ∂x− µv, v(0, x) = g(x) (2)
for x ≥ 0, t ≥ 0, with ν as before and associated constant interest rate µ > 0. Here the most important initial condition is
g(x) = (x − K)+= max{x − K, 0} or g(x) = (K − x)+= max{K − x, 0},
which indicates the payoff of a call (resp. put) European option at a fixed maturity date T , where the strike price K is a given positive constant. Observe that the semigroup governing (2) is easy to construct and study. Indeed, it can factor into the product of the semigroups generated by the commuting operators H1v = ν2x2v00, H2v = µxv0 and
H3u = −µv, while the semigroup governing (1) does not factor into the product of the
four semigroups generated by the operators A1u = ν2xu00, A2u = γu0, A3u = βxu0
and V (x)u = −rx u. Each pair Ai and Aj with i and j different fails to commute. The
noncommuting makes (1) a much harder problem than (2).
As a consequence of the results obtained in [8], we know that the semigroup TCIR
governing (1) is a (C0) semigroup on C0[0, ∞), the space of continuous functions on [0, ∞)
which vanish at ∞ equipped with the sup norm k · k∞. In fall 2017, we learned that the
(C0) property of TCIR on C0[0, ∞) had been established in [4], but there it is expressed
in a very different context using affine processes and Feller semigroups. Our paper [8] contained a completely different proof of the strong continuity and additional results, such as a characterization of the domain of the generator and a new type of Feynman-Kac formula. This is a significant step towards understanding how to represent the solution of the Cauchy problem (1).
It is worth noting that TCIR fails to be of class (C0) on the space C[0, ∞] of all
continuous complex valued functions on [0, ∞) having finite limit at ∞. Thus, the previous results cannot be applied to the special initial function f0(x) ≡ 1. To remedy this, one
must find a space Y such that (1) is governed by a (C0) semigroup on Y and f0 ∈ Y.
The present paper is devoted to solving this problem. We also replace the potential term V (x)u = −rx u in (1) by a more general nonpositive potential term. In Mathematical Finance to make some more assumptions on the derivative’s price dynamics when other issues in the financial markets are considered (e.g. jumps that actually happen because of government fiscal and monetary decisions, changes in investors’ expectations, etc..). We use polynomials, which generalize positive polynomials on (0, ∞).
The spaces that we need are Ys:= C0 [0, ∞), 1 1 + xs
for s > 0. Here, for a weight function 0 < w ∈ C[0, ∞), we define C0([0, ∞), w) = {h ∈ C[0, ∞) : hw ∈ C0[0, ∞)},
and for h ∈ C0([0, ∞), w), the norm khkw is defined by
||h||w = sup x≥0
|h(x)| w(x) < ∞.
The space Ys corresponds to w(x) =
1
1 + xs. Thus, the weighted space Ys has its norm
depending on s > 0. This will be denoted by k · ks. If s = 0, then C0[0, ∞) is equipped
with the norm
||f ||0=
1 2||f ||∞
for f ∈ C0[0, ∞). In a similar way, if C0(0, ∞) denotes the space of all continuous complex
valued functions on (0, ∞) that vanish at both 0 and ∞, then one can define Xs:= C0
(0, ∞), 1 1 + xs
for s > 0. Notice that Ysis a strictly larger space than Xssince functions h ∈ Ys need not
satisfy the condition that h(x)/(1 + xs) vanishes at the origin.
We remark that the CIR equation is often treated with potential term in (1) missing. Then the generator becomes ν2x d
2
dx2 + (γ + βx)
d
dx, and generators like this have a long history, starting with Feller around 1950. But the treatments of these operators on C[0, ∞], the space of all continuous complex valued functions on [0, ∞) having finite limit at ∞ are often incomplete.
2
Preliminaries
Denote by M(0, ∞) the set of all finite complex Borel measures on (0, ∞). As observed in [6, Section 2], by the Riesz Representation Theorem, the dual space of (C0[0, ∞), || · ||0)
can be identified by M(0, ∞) equipped with the norm ||ψ|| = 2 (Total Variation of ψ).
Similarly we can define the dual space Ys∗ of Ys. Using the pairing, for u ∈ Ys, ψ ∈ Ys∗,
< u, ψ > = Z ∞ 0 u(x) 1 + xs(1 + x s) ψ(dx) = Z ∞ 0 u(x) ψ(dx),
the dual space of Ys can be identified by
Ys∗ = {ϕ ∈ Mloc(0, ∞) : (1 + xs) ϕ(dx) ∈ M(0, ∞)} (3)
for all s > 0 (see [6, Lemmas 2.1–2.2]), with the norm of ϕ ∈ Ys∗ being the total variation of the finite complex measure (1 + xs) ϕ(dx).
Observe that the initial value problem (1) can be written as ut= Bνu + Mru, u(0, x) = f (x), t, x ≥ 0,
with r > 0 and f ∈ C[0, ∞] (resp. f ∈ C0[0, ∞)), where Mris the multiplicative operator,
Mru(x) = −rxu, and the operator Bν is given by
Bνu := ν2x u00+ (γ + βx) u0, (4)
with ν > 0, γ > 0, β ∈ R. The operators Bν and Mr act on C[0, ∞] (resp. C0[0, ∞)). For
any ν > 0, in [8] we considered the operator defined formally by Gν = ν
√ x d
dx; (5)
its square is given by
G2ν = ν2x d 2 dx2 + ν2 2 d dx.
Thus the operator Bν defined by (4) can be represented as a perturbation of G2ν, namely
Bν = G2ν+ (γ − ν2 2 ) d dx+ βx d dx = G2ν+ P1+ P2, (6) where P1= α d dx (with α = γ − ν2 2 ), P2= βx d dx. (7)
According to the results in [8] the operators G2ν, Bν, P1+ P2+ Mr with α > 0, β 6=
0, r > 0 are infinitesimal generators of (C0) semigroups on C0[0, ∞), while the operator
Ar := Bν + Mr, r > 0, generates a once-integrated semigroup on C[0, ∞] which is (C0)
on the space C0[0, ∞). Furthermore, we showed that the semigroups generated by the
operators G2ν and P1+ P2+ Mr have explicit representations on C0[0, ∞).
In this paper our aim is to study the operators G2ν, P1+ P2+ Mr, Ar := Bν+ Mr,
and their associated semigroups on the weighted Banach space Ys, s > 0, in order to solve
the CIR problem (r = 1) mentioned in the Introduction
ut= A1u, u(0, x) ≡ 1, t > 0, x ≥ 0.
Observe that the constant function 1 is in Ysfor all s > 0. Further, we will show that some
generation results work even when the multiplicative operator Mr is replaced by a more
general nonpositive multiplicative operator of type M−P = −P (x)I with P (x) posynomial
(see Definition 1 in the next section). We only need to consider real functions and real Banach spaces. So, from now on, all our function spaces consist of only real functions.
3
Generation results
Recall that if X is a Banach space equipped with norm k · kX, for any x ∈ X, x 6= 0,
we can define the (nonempty) subset I(x) of the dual of X, X∗, by I(x) = {ϕ ∈ X∗ : ||ϕ||X∗ = 1, < x, ϕ >= ||x||X}.
Here < ·, · > is the duality between X and X∗. A linear operator ˆA on X is called quasidissipative if
for any x ∈ D( ˆA) ⊂ X there exists ϕ ∈ I(x) such that
Re < ˆAx, ϕ >≤ ω||x||X (8)
for some constant ω ∈ R depending only on ˆA. Also, ˆA − ω I is called dissipative. Consider the operator Gν defined in (5) and acting on C[0, ∞] with domain
D(Gν) = {u ∈ C[0, ∞] ∩ C1(0, ∞) : Gνu ∈ C[0, ∞]}, (9)
and the operator Gν,sdefined as Gν and acting on Ys(resp. on C0[0, ∞) when s = 0) with
domain
D(Gν,s) = {u ∈ Ys∩ C1(0, ∞) : Gν,su ∈ Ys}. (10)
As in [11, Corollary 4.5], let us consider the family of operators Tν := (Tν(t))t∈Rdefined
by Tν(t) f (x) := f √ x +ν t 2 2! (11) for t ∈ R, x ≥ 0, f ∈ C[0, ∞].
Lemma 1. Fix ν > 0. The family Tν defined in (11) is a (C0) group of isometries on
C[0, ∞] having as generator Gν with domain D(Gν) given by (9).
Proof. For any t ∈ R and x ≥ 0 let us define ξt(x) :=
√ x +νt
2 2
and observe that the Cauchy problem ut= ν
√
xux, u(0, x) = f (x), t ∈ R, x ≥ 0
is solved by u(t, x) = f (ξt(x)). In addition, it can be easily seen that (Tν(t))t∈Ris a group
of bounded linear continuous operators on C[0, ∞] which consists of isometries. Now, let us prove that the group Tν is strongly continuous. Preliminarily, we remark that
lim
and
lim
t→0ξt(x) = x (12)
uniformly on compact sets of [0, ∞). Let us fix f ∈ C[0, ∞]. We have to prove that lim
t→0kTν(t)f − f k∞= limt→0kf (ξt) − f k∞= 0. (13)
Observe that f (x) converges as x approaches to +∞ and denote by L ∈ R its limit. It follows that
lim
x→∞f (x) = L = limx→∞f (ξt(x)).
Hence, fixing any ε > 0, there exists M > 0 such that sup
x>M
|f (ξt(x)) − L| <
ε 2 uniformly for t ∈ R and
sup x>M |f (x) − L| < ε 2. Therefore, sup x>M |f (ξt(x)) − f (x)| < ε (14)
uniformly for t ∈ R. On the other hand, from the uniform continuity of f and (12), we deduce that there exists δ > 0 such that, for any t ∈ R,
0 < |t| < δ implies sup
0≤x≤M
|f (ξt(x)) − f (x)| < ε. (15)
From (14) and (15) we can deduce that, for any t ∈ R, 0 < |t| < δ implies sup
x∈R
|f (ξt(x)) − f (x)| = kTν(t)f − f k∞< ε,
and the assertion (13) follows. Let us consider the resolvent equation
λu − Gνu = h (16)
for λ ∈ R, λ 6= 0, and h ∈ C[0, ∞]. The general solution is parametrized by u(0) ∈ R, but only one solution is bounded and in C[0, ∞]. Suppose λ > 0 and h ∈ C[0, ∞] satisfies
h(x) = a ∈ R
for any x > b. As a, b vary in R the functions h describe a dense subspace of C[0, ∞]. Observe that (16) implies
λu(x) ν√x − u
0
(x) = h(x)
ν√x, x > 0, and u(0) = c is to be determined. Thus,
d dx(e −2λ √ x ν u(x)) = e− 2λ√x ν u0(x) − λu(x) ν√x = −e−2λ √ x ν h(x) ν√x (17)
for any x > 0. Integrating (17) gives u(x) = e2λ √ x ν u(0) − e 2λ√x ν Z x 0 e−2λ √ y ν h(y) ν√y dy. For x > b we obtain that
u(x) = e2λ √ x ν u(0) − Z b 0 e−2λ √ y ν h(y) ν√y dy − Z x b e−2λ √ y ν a ν√y dy . Since Z x b e− 2λ√y ν a ν√y dy = −a λ e−2λ √ x ν − e− 2λ√b ν := C(a, b, ν, λ, x), this yields that
lim x→∞C(a, b, ν, λ, x) = a λe −2λ√b ν := C(a, b, ν, λ). Because e2λ √ x
ν → ∞ as x → ∞, the only choice of u(0) that makes u ∈ C[0, ∞] is
u(0) = Z b 0 e−2λ √ y ν h(y) ν√y dy + C(a, b, ν, λ). (18) A similar calculation works for λ < 0. Thus, Gν = Gν generates a (C0) group on C[0, ∞]
and for λu − Gνu = h with h(x) = a for x > b, u(0) is given by (18).
Now, let us focus on the family of operators Tν acting on the space Ys.
Lemma 2. Fix ν > 0. The family Tν = (Tν(t))t∈R, defined in (11) is a (C0)
quasicon-tractive group on Ys, for any s ≥ 1/2.
Proof. Fix ν > 0. We will prove that, for any s ≥ 1/2 : i) The operators ±Gν,s are quasidissipative on Ys.
ii) The operators ±Gν,s satisfy the range condition on Ys.
iii) The group Tν is strongly continuous on the space Ys and there exists ω ∈ R+ such
that
||Tν(t)f ||s≤ eω |t|||f ||s,
for any t ∈ R, f ∈ Ys.
Proof of i). Step 1. For ν > 0 and s ≥ 12 we consider the operator Gν,s, see (??),
(10). In order to prove the assertion, let 0 6= f ∈ D(Gν,s) and x0 ≥ 0 be such that for
w(x) := 1
1 + xs, s > 0, the real function |f (x)| w(x) is maximized at x0, i.e.,
|f (x0)| w(x0) = ||f ||s.
Without any loss of generality, we may assume f is real and f (x0) > 0. We choose notation
in the usual way and write < f, δx0 >= f (x0), where δx0 is the Dirac measure. From (3)
it follows that
Assume now x0 > 0 and define g(x) := f (x) w(x) for any x > 0. Then
g0(x0) = 0 if and only if f0(x0) = −
f (x0)
w(x0)
w0(x0) = ||f ||ss xs−10 ,
and so we deduce that
< Gν,sf, ϕ >= ν √ x0f0(x0) w(x0) = ||f ||sνs xs−1/20 1 + xs 0 ≤ νs ||f ||s, because x s−1/2 0
1 + xs0 ∈ (0, 1] for all s ≥ 1/2 and x0 > 0. Indeed, if 0 < x0 < 1, then the assertion is trivial. If x0 ≥ 1, then taking into account that 1 + xs0 ≥ xs0, we have that
xs−1/20 1 + xs 0 ≤ x s−1/2 0 xs 0 = √1 x0 ≤ 1.
Then condition (8) is satisfied with ω = ωs := ν s and s ≥ 1/2. In case x0 = 0, then f
has a positive maximum at 0. Thus, d+
dxf (x)|x=0≤ 0 whence
< Gν,sf, ϕ >≤ 0.
Step 2. We consider the operator −Gν,s. From the above arguments it follows that
< −Gν,sf, ϕ >= −||f ||sνs
xs−1/20 1 + xs 0
≤ 0
for all x0 > 0 and s ≥ 1/2. If in the above inequality we let x0→ 0, then
< −Gν,sf, ϕ >≤ 0.
Then the operator −Gν,s is dissipative, and hence quasi-dissipative on Ys for all s ≥ 1/2.
Thus the proof of assertion i) is complete.
Proof of ii). Fix s > 0. Let h ∈ Ys and observe that 1+xh s ∈ C0[0, ∞). Since Gν
generates Tν on C[0, ∞] and Tν leaves invariant C0[0, ∞) (as in [8, Proposition 3]) we
deduce that for λ sufficiently large, we can find v ∈ C0[0, ∞) ∩ C1(0, ∞) with Gνv ∈
C0[0, ∞) such that
λv − Gνv =
h 1 + xs
and the assertion follows. Similar arguments work for −Gν,s.
Thus for any s > 0,
Ds := {˜h : ˜h(x) =
h(x)
1 + xs, h ∈ Ys}
is in the range of λ − Gν,s for large real λ, and Ds is dense in Ys. It follows that both
±Gν,s are m-quasidissipative. This works for all s ≥ 1/2. And we may choose ω = νs in all cases.
Lemma 2, [8, Lemma 1], and Romanov’s theorem (see [10]) imply the following result. Theorem 1. For any ν > 0, the closure of the operator G2ν,s = ν2x d
2 dx2 + ν2 2 d dx with domain D(G2 ν,s) = {u ∈ D(Gν,s) : Gν,su ∈ D(Gν,s)},
generates a positive (C0) quasicontractive semigroup Wν = (Wν(t))t≥0 on Ys, s ≥ 1/2,
having the explicit representation Wν(t)f (x) = et G 2 νf (x) = Z +∞ −∞ p(t, y) f h√ x +ν y 2 i2 dy, t > 0, x ≥ 0,
for f ∈ Ys, where p is the probability density function of the normal distribution with mean
zero and variance 2t,
p(t, y) = √1 4π te
−y2 4 t.
As our next step, according to (6), we consider the operator Qsacting on Ys such that
Qsu(x) := (α + β x) u0(x) for x ≥ 0 . Here α > 0 and β 6= 0.
Theorem 2. Assume ν > 0, α > 0 and Pi,s, i = 1, 2 defined as Pi in (7), acting on Ys.
Then the closure of the operator
Bν,s= G2ν,s+ P1,s+ P2,s
= G2ν,s+ Qs
with domain D(Bν,s) = D(G2ν,s) generates a positive quasicontractive (C0) semigroup on
Ys, for all s ≥ 1.
To prove this theorem we need the following lemmas.
Lemma 3. The closure of the operator P1,s defined as P1,su(x) := α u0(x), for x ≥ 0 and
α > 0, with domain
D(P1.s) = {u ∈ Ys∩ C1(0, ∞) : u0 ∈ Ys},
generates a positive quasicontractive (C0) semigroup on Ys for all s ≥ 1, given by
etP1,sf (x) = f (x + α t), t, x ≥ 0.
Proof. Let s ≥ 1. We first prove that for any u ∈ D(P1,s),
there exists ϕ ∈ I(u) such that < P1,su, ϕ >≤ωes||u||s (quasidissipativity)
By using the same arguments as in the proof of Lemma 1 i), consider 0 6= f ∈ D(P1,s),
with x0 ≥ 0 a maximizing value yielding ||f ||s, and ϕ = δx0w(x0) ∈ I(f ), where w(x) =
1
1 + xs and f (x0) > 0. We deduce that, if x0 > 0,
< P1,sf, ϕ >= α f0(x0) w(x0) = ||f ||sαs xs−10 1 + xs 0 ≤ αs ||f ||s, because x s−1 0 1 + xs0 ∈ (0, 1] for all s ≥ 1.
Suppose |f (x)| w(x) is maximized at x0 = 0. Without loss of generality we suppose
f (0) is positive. Then < P1,sf, ϕ >= α f0(0) ≤ 0 since α > 0 and f0(0) ≤ 0. But (f w)0(0) = f0(0) w(0) + f (0) w0(0) = f0(0) + f (0) w0(0) ≤ 0, because w0(x) = − s x s−1 (1 + xs)2,
w0(0) = −1 if s = 1 and w0(0) = 0 if s > 1. It follows that < P1,sf, ϕ >≤ α ||f ||s
if s ≥ 1 in case x0 = 0. Then the quasidissipativity is satisfied withωes:= αs and s ≥ 1. Concerning the range condition, we observe that P1,0 generates the (C0) contraction
semigroup (TP1,0(t)f (x) = f (x + αt), for α > 0, x ≥ 0, t ≥ 0, on the space C0[0, ∞). Thus,
by the Hille-Yosida theorem, the range of λI − P1,0 contains C0[0, ∞) for each λ > 0.
This yields that on the weighted Ys space, for large enough λ > 0, the range of λI − P1,s
contains C0[0, ∞) which is dense in Ys. Hence the closure of P1,s on Ys is essentially
m-quasidissipative.
Lemma 4. For any s ≥ 1, β 6= 0, the closure of the operator P2,s defined as P2,su(x) :=
β x u0(x), x ≥ 0, with domain
D(P2,s) = {u ∈ Ys∩ C1(0, ∞) : xu0 ∈ Ys}
generates a positive quasicontractive (C0) semigroup on Ys given by
etP2,sf (x) = f (x eβ t), t, x ≥ 0.
Proof. As was the case for Lemma 3, it is enough to verify the quasidissipativity and the range condition. Observe that for any u ∈ D(P2,s), there exists ϕ ∈ I(u) such that
< P2,su, ϕ >≤ ωs∗||u||s for some constant ω∗s ∈ R depending only on P2,s and s.
Hence, by using the same arguments as in the proof of Lemma 3, quasidissipativity holds for ωs∗ := |β| s and s ≥ 1.
The range condition follows by using analogous arguments as in the proof of Lemma 3. Indeed, for any real β 6= 0, the realization of P2,0 on C0[0, ∞) generates a (C0) group of
isometries given by TP2,0f (x) = f (xe
βt) for x ≥ 0, t ∈ R. By the Hille-Yosida theorem, P 2,0
and −P2,0 are both m-dissipative, thus the equation u − βxu0 = h has a unique solution
in the space C0[0, ∞) for any initial condition u(0) = a and any h ∈ Y0. In particular, for
any large enough λ > 0, C0[0, ∞) is in the range of λI − P2,0 and C0[0, ∞) is dense in all
our weighted Ys spaces. Hence the closure of P2,s is m-quasidissipative.
Let us define the domain of the operator Qs = P1,s+ P2,s on the space Ys to be
D(Qs) = D(P1,s) ∩ D(P2,s) = {u ∈ Ys∩ C1(0, ∞) : u0, xu0∈ Ys},
for any s > 0.
Lemma 5. For any s ≥ 1, α > 0, β ∈ R, the closure of the operator Qs generates a (C0)
quasicontractive semigroup Vαβ = Vαβ(t) t≥0 on Ys given by Vαβ(t)f (x) = et Qsf (x) = f et βx +α β (e t β− 1) , t ≥ 0, x ≥ 0, f ∈ Ys. (19)
Proof. Here assume β 6= 0. If β = 0, we can replace et ββ−1 by t, its limit as β → 0. In [8, Lemma 3] we observed that the solution of the Cauchy problem
vt= (α + β x) vx = Q0v, t, x ≥ 0
v(0, x) = f (x), x ≥ 0 where Q0 acts on C0[0, ∞), is given by
v(t, x) = f et βx + α β (e t β− 1) .
Both P1,s and P2,sare quasidissipative on Ys for s ≥ 1. It follows that Qs= P1,s+ P2,s
has quasidissipative closure on Ys for s ≥ 1. The constant ωs fo Qs is the sum ofωes and ωs∗ corresponding to P1,s and P2,s respectively.
From [8, Lemma 3] it follows that, for any large enough λ > 0, the range of λI − Q0
is dense in C0[0, ∞). This implies the density of the range of λI − Qs in Ys for all s > 0.
Thus the closure Qs is m-quasidissipative for s ≥ 1 and we are done.
Proof of Theorem 2. The quasidissipativity of Bν,s is an immediate consequence of
The-orem 1 and Lemma 5. The range condition follows from the range condition proved in [8] for C0[0, ∞) since the range of λI − Bν,s contains C0[0, ∞) which is dense in Ys for all
s > 0.
Using a terminology due to Dick Duffin (see [5]), we introduce the definition of a posynomial.
Definition 1. We call a function f : [0, ∞) → R a posynomial if for any x ≥ 0, f (x) = N X j=1 rjxkj ≥ 0,
where N ≥ 1, each rj ≥ 0 and 0 ≤ k1 < k2< ... < kN.
Let us fix a weight function 0 < w ∈ C[0, ∞) and denote by C0,w the Banach space
C0([0, ∞), w) already introduced in Section 1. Then the following result holds. Its easy
proof is omitted.
Lemma 6. Let P be a posynomial and let M−P be the multiplication operator
M−Pf (x) = −P (x)f (x), x ≥ 0
with domain
D(M−P) = {f ∈ C0,w : M−Pf ∈ C0,w}.
Then M−P generates a (C0) contraction semigroup on C0,w given by
etM−Pf (x) = e−tP (x)f (x), x ≥ 0
for any f ∈ C0,w.
Remark 1. The previous Lemma works in the particular case of the posynomial P (x) = s + r xk, x ≥ 0, with s ≥ 0, r > 0 and 0 < k < ∞.
The next theorem generalizes the result proved in [8, Theorem 3] in the case of the more general multiplication operator M−P.
Theorem 3. Let us assume γ > 0 and consider the operator Bν,0= G2ν,0+Q0 with domain
D0(Bν,0) = {u ∈ D(G2ν,0) : Gν,0(0) = 0 = G2ν,0(0)}.
Then the operator AP,0:= Bν,0+ M−P with domain
D(AP,0) = {u ∈ D0(Bν,0) : M−Pu ∈ Y0} (20)
is essentially m-dissipative (and densely defined) on C0[0, ∞).
Proof: Let h ∈ C0[0, ∞), λ > 0. We want solve the equation
λu − AP,0u = h. (21)
Without any loss of generality, it is enough to do this for h ≥ 0. For any m ∈ N, consider the sequence
Wm(x) =
(
−P (x), if x ∈ [0, m], −P (m), if x ∈ [m, ∞).
Note that, by the properties of P ,
0 ≥ Wm(x) ≥ Wm+1(x) ≥ −P (x) (22)
for all x ≥ 0, m ∈ N, and Wm(x) → −P (x) as m → ∞, for all x ≥ 0. Thus the
multiplication operator MWm is a bounded perturbation of Bν,0 in C0[0, ∞). It follows
that Am,0 := Bν,0 + MWm with domain D(Am,0) = D(Bν,0) generates a positive (C0)
contraction semigroup on C0[0, ∞). Therefore for any m ∈ N there exists a unique solution
um∈ D(Bν,0) of
λu − Am,0u = h. (23)
Also, (22) implies that 0 ≤ um+1(x) ≤ um(x) for all x ≥ 0, m ∈ N. Since h ≥ 0 and
(λI − Am,0)−1 is positive, for λ > 0, kumk ≤ ku1k < ∞, so by (22) supmkAm,0umk < ∞.
Rewrite (23) as λum− (ν2x u00m+ (γ + βx) u0m+ Wmum) = h or x u00m+ (γ + βx) u0m+ (Wm− λI)um = −h (24) by replacing each c ∈ {γ, β, Wm, h, λ} by ˜c = c
ν2 and then erasing each tilde. Multiply
(24) by xγ−1eβx to get d dx xγeβxu0m(x) = xγ−1eβx((λ − Wm)um(x) − h(x)). (25)
Let > 0 be given. Integrating (25) over [, x] with x ≤ 1/ gives |xγeβxu0m(x) − γeβu0m()| ≤ K
for some K > 0 and all m ∈ N and x ∈ [, 1/].
Since umsatisfies (24) on [, 1/], the operator at the left hand side of (24) is uniformly
elliptic on [, 1/], and supm||um|| < ∞, we have that
d dx x
γeβxu0
m(x) , xγeβxu0m(x) and
u0m(x) are all uniformly bounded on [, 1/], for all m ∈ N, as is u00mby subtraction (by (24)).
By the Arzela-Ascoli theorem and Cantor diagonalization, there exists a subsequence {vn}
of {um} such that
vn→ u in Cloc2 (0, ∞).
To prove that the pointwise solution u of (21) belongs to D(AP,0), it remains to control
u near x = ∞ and near to x = 0.
Note that 0 ≤ u(x) ≤ u1(x) → 0 as x → ∞, because u1 ∈ C0[0, ∞). Further,
u, u0, u00∈ C(0, ∞), thus u ∈ C2(0, ∞) ∩ C(0, ∞] and u(∞) = 0. This takes care of u near
x = ∞.
Concerning the case of x near to 0, let us recall a result of [2]. Let U (x, t) for x, t ≥ 0 satisfy Ut= 1 2Uxx+ c1 xUx, U (x, 0) = f (x),
with Neumann boundary condition at the origin, Ux(0, t) = 0 (= f0(0)). Then Z(x, t) := U √ x,t 2 satisfies Zt= xZxx+ c2Zx, Z(x, 0) = g(x) = f ( √ x), where c2 = c1 + 1
2 > 0, which is valid when c1 > −1/2. The boundary condition for Z resulting from the boundary condition for U (which is independent of c2> 0) is
√
xZx(x, t) → 0 (26)
as x → 0+ for all t ≥ 0. Z satisfies Zt = (A + B) Z in C0[0, ∞) if we take c2 =
γ ν2 > 0
and β = 0. But for the operator Gν,0 = ν
√
x D = ν√x d
dx with boundary condition Gν,0u(x) → 0 as x → 0+, i.e. (26), we get
G2ν,0u(x) = ν2√xD(√x u0(x)) = ν2(x u00(x) + 1 2u
0(x)) = ν2(x D2+1
2D)u(x). And for u ∈ D(G2ν,0), Gν,0u(x) and G2ν,0u(x) vanish when x → 0+. In [11, Theorem 3.4]
and in the results proved in [8] we showed that Gν,0 is the infinitesimal operator of a non
(C0), but a once integrated positive contraction semigroup on C0[0, ∞). The Wentzell
boundary condition for G2ν,0 is Gν,0u(0) = G2ν,0u(0) = 0, and G2ν,0 is m-dissipative and
densely defined on C0[0, ∞). Now we must strengthen this to strong continuity.
Recall that α = γ − ν2/2. We now give a new direct proof that the Wentzell boundary condition for Bν,0 on C0[0, ∞), namely Bν,0u(0) = 0, is independent of γ > 0 and β ∈ R.
Thus the Wentzell boundary condition is independent of ν, γ and β, provided that γ > 0. The next Lemma is an extension of the Kallman-Rota inequality [12].
Lemma 7. Let L generate a strongly measurable contraction semigroup {esL: s ≥ 0} on a Banach space Z. Then for all f ∈ D(L2),
||L f ||2 ≤ 4||L2f || ||f ||.
Proof: {esL : s ≥ 0} is strongly continuous for t > 0, but it need not be of class (C0).
The usual Kallman-Rota inequality assumes that {esL : s ≥ 0} is of class (C0), but the
conclusion holds in the more general case when L need not be densely defined. Let f ∈ D(L2). Then by Taylor’s formula,
etLf = f + t Lf + Z t 0 (t − s)esLL2f ds, whence Lf = e tLf − f t + 1 t Z t 0 (t − s)esLL2f ds, and ||Lf || ≤ 2 t||f || + 1 t Z t 0 (t − s)||L2f || ds = 2 t||f || + t 2||L 2f ||.
Minimizing over t > 0 gives, if L2f 6= 0, t = 2 ||f ||1/2||L2f ||−1/2, and so
||Lf || ≤ 2||L2f ||1/2||f ||1/2≤ δ||L2f || +1
δ||f || (27) for every δ > 0. This is also valid if L2f = 0, which implies Lf = 0. Then Lemma now
follows. 2
First we take β = 0. Thus L in Lemma 7 is a dissipative Kato perturbation of L2 (and so is −L if L is the infinitesimal operator of a group of isometries). Therefore, L2+ k L is m-dissipative on D(L2) for all k ≥ 0 (and for all k ∈ R in the isometric group case).
Thus the Wentzell boundary condition for G2ν,0+ k Gν,0 is independent of k and has
the form Gν,0u(0) = G2ν,0u(0) = 0 for u ∈ D( bBν,0), where bBν,0 = ν2x D2+ γ D, and this
holds for all γ > 0 since it holds for all relevant k (see (27)).
We take now β 6= 0. Note that for x > 0 small, bBν,0u(x) = Bν,0u(x) −
√
x(β√xu0(x)). Since bBν,0u(x) → 0 and
√
x(β√xu0(x)) → 0 as x → 0+, it follows that Bν,0u(0) = 0.
Thus, u, AP,0u ∈ C0[0, ∞), Bν,0u(0) = 0 and so u ∈ D(AP,0).
This now completes our proof of strong continuity of the our problem on C0[0, ∞),
including the characterization of D(Bν,0) which is independent of γ and β, provided γ > 0.
Theorem 3 is now fully proved.
Hence the claim follows. 2
Remark 2. At x = ∞, the multiplicative operator M−P is unbounded, so it preserves the
boundary condition u(∞) = 0 whenever u ∈ D(AP,0), but D(AP,0) will be smaller than
D(Bν,0) because u(x) → 0 as x → ∞ does not imply P (x) u(x) → 0 as x → ∞.
Theorem 4. Assume that ν > 0, γ > 0, β ∈ R, P is a posynomial. Then the closure of the operator AP,s with domain
D(AP,s) = {u ∈ D(G2ν,s) : Gν,s(0) = 0 = G2ν,s(0), M−Pu ∈ Ys}
generates a positive (C0) quasicontractive semigroup on Ys for all s ≥ 1.
Proof. The quasidissipative assertion follows from the previous results. The range condi-tion is a consequence of the range condicondi-tion proved in the previous theorem for C0[0, ∞)
since the range of λI − Bν,s contains C0[0, ∞) which is dense in Ys for all s > 0.
4
A Feynman-Kac type formula
As already mentioned in the Introduction, in [8] we proved a new type of Feynman-Kac formula for the generalized CIR problem (1) on C0[0, ∞). Consider the posynomial
Pr(x) = rx, x ≥ 0, with r > 0. Denote Thus the problem (1) can be written as
du
where
C1= G2ν,0, C2 = Q0+ M−Pr,
and M−Pr reduces to the multiplication operator Mr defined in Section 2. The Trotter
product formula implies that the semigroup generated by C = C1+ C2 is given by
et Cf = lim n→∞ ent C1e t nC2 n f. (28)
Using our explicit formulas for et C1 and et C2, we found a very complicated explicit
for-mula for (ent C1e t
nC2)nf for n = 2k, for any positive integer k. Using this formula in
(28) gives our Feynman-Kac formula. In the Feynman case of the Schr¨odinger equation, zn= (e
t nC1e
t
nC2)nf was regarded as a Riemann sum for the Feynman path integral. The
latter integral does not exist in the usual measure theoretic context, but it is very useful nonetheless. For the heat equation with a potential, Kac showed directly that zn
con-verges to a Wiener integral solution of the heat equation. Our zn looks nothing like an
approximation to an integral. But still, zn converges to the desired solution. We showed
this in [8] for C0[0, ∞) and here the analogous proof establishes it for Ys, s > 1. This is
our Theorem 5.
Before stating Theorem 5, we note the following result.
Lemma 8. For any s > 1, α > 0, β ∈ R, r > 0, the closure of the operator Qs+ M−Pr
with domain
D(Qs+ M−Pr) = {u ∈ D(Qs) : M−Pru ∈ Ys}
generates a (C0) semigroup (Uαβ(t))t≥0 on Ys satisfying
Uαβ(t) f (x) = exp −r β (et β− 1) (α β + x) − α t f et βx +α β (e t β− 1) (29) for t ≥ 0, x ≥ 0.
Proof. As in Lemma 5 assume β 6= 0. If β = 0, we can replace et ββ−1 by t, its limit as β → 0. The assertion follows from [8, Theorem 4] where we showed that the family (Uαβ(t))t≥0 of linear operators satysfying (29) is a (C0) semigroup on C0[0, ∞) obtained
by the approximating formula
Uαβ(t) f = lim m→∞ emt Q0e t mM−Pr , for all t ≥ 0 and f ∈ C0[0, ∞).
Theorem 5. Assume that ν > 0, γ > 0, β ∈ R, r > 0, Pr(x) = rx. Then the semigroup
TAP,s(t) t≥0 on Ys with s > 1 is given by TAP,s(t)f (x) = lim n→∞ Wν(nt) Uαβ(nt) n f (30) = lim n→∞ Z +∞ −∞ . . . | {z } n times Z +∞ −∞ L(t, n, ν, {yj}1≤j≤n, x, f ) · n Y j=1 p(t n, yj) dy1· · · dyn. (31)
with n = 2k for k ∈ N, and for L given by L(t, n, ν, {yj}1≤j≤n, x, f ) = eZ( t n,S(y1,x))+Z( t n,S(y2,R( t n,S(yn(1,x)))))
· eZ(nt,S(y3,R(nt,S(y2,R(nt,S(y1,x))))))+···+Z(nt,S(yn,R(nt,S(yn−1,R(nt,··· ,R(nt,S(y1,x)))))))
· f (R(t n, S(yn, R( t n, S(yn−1, R( t n, . . . , R( t n, S(y1, x)))))))), where R(t, S(y, x)) = etβx +α β(e tβ− 1) + etβν y (√x +ν y 4 ), Z(t, S(y, x)) = −r β hα β (e t β− 1) + (et β − 1) x − α ti− rh(etβ− 1) β ν y ( √ x +ν y 4 ) i . and t, x ≥ 0. The convergence is uniform for x ∈ [0, ∞) and for t in bounded intervals of [0, ∞).
Proof. See the proof of [8, Theorem 5].
Remark 3. Theorem 5 implies that the unique solution of the CIR problem (1) in the special case r = 1 and f (x) = 1, as mentioned in the Introduction, is given by
u(t, x) = lim n→∞ Z +∞ −∞ . . . | {z } n times Z +∞ −∞ ˜ L(t, n, ν, {yj}1≤j≤n, x) · n Y j=1 p(t n, yj) dy1· · · dyn, (32) where ˜ L(t, n, ν, {yj}1≤j≤n, x) = eZ( t n,S(y1,x))+Z( t n,S(y2,R( t n,S(yn(1,x))))) · eZ(nt,S(y3,R( t n,S(y2,R( t n,S(y1,x))))))+···+Z( t n,S(yn,R( t n,S(yn−1,R( t n,··· ,R( t n,S(y1,x))))))). Acknowledgements
We thank Gerald Teschl for his helpful comments. R.M. Mininni and S. Romanelli are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Appli-cazioni (GNAMPA), Istituto Nazionale di Alta Matematica (INdAM). The authors have been partially supported by the GNAMPA research project 2017 “Problemi ellittici e parabolici ed applicazioni all’Economia e alla Finanza”.
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