A generalized Cox-Ingersoll-Ross equation with growing initial conditions

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 \begin{document}$ [0, \infty) $\end{document} which can grow at infinity. We focus on the Banach spaces \begin{document}$ Y_{s} = \left\{f\in C[0,\infty): \dfrac{f(x)}{1+x^{s}}\in C_0[0,\infty)\right\},\qquad s\ge 1, $\end{document} which contain the nonzero constants very common as initial data in the Cauchy problems coming from financial models. In addition, a Feynman-Kac type formula is given.


1.
Introduction. Of concern is the initial value problem studied in [8], 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 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]).
for x ≥ 0, t ≥ 0, with ν as before and associated constant interest rate µ > 0. Here the most important initial condition is 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 H 1 v = ν 2 x 2 v , H 2 v = µxv and H 3 u = −µv, while the semigroup governing (1) does not factor into the product of the four semigroups generated by the operators A 1 u = ν 2 xu , A 2 u = γu , A 3 u = βxu and V (x)u = −rx u. Each pair A i and A j 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 T CIR governing (1) is a (C 0 ) semigroup on C 0 [0, ∞), the space of continuous functions on [0, ∞) which vanish at ∞ equipped with the sup norm · ∞ . In fall 2017, we learned that the (C 0 ) property of T CIR on C 0 [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 T CIR fails to be of class (C 0 ) 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 f 0 (x) ≡ 1. To remedy this, one must find a space Y such that (1) is governed by a (C 0 ) semigroup on Y and f 0 ∈ 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 that means to run some scenarios for the derivative's price dynamics when other issues in financial markets are considered (e.g. jumps that actually happen because of government fiscal and monetary decisions, changes in investors' expectations, etc.). We use posynomials, which generalize positive polynomials on (0, ∞).
The spaces that we need are Here, for a weight function 0 < w ∈ C[0, ∞), we define and for h ∈ C 0 ([0, ∞), w), the norm h w is defined by The space Y s corresponds to w(x) = 1 1 + x s . Thus, the weighted space Y s has its norm depending on s > 0. This will be denoted by · s . If s = 0, then C 0 [0, ∞) is equipped with the norm In a similar way, if C 0 (0, ∞) denotes the space of all continuous complex valued functions on (0, ∞) that vanish at both 0 and ∞, then one can define Notice that Y s is a strictly larger space than X s since functions h ∈ Y s need not satisfy the condition that h(x)/(1 + x s ) vanishes at the origin.
We remark that the CIR equation is often treated with potential term in (1) missing. Then the generator becomes ν 2 x d 2 dx 2 + (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 (C 0 [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 Y * s of Y s . Using the pairing, for u ∈ Y s , ψ ∈ Y * s , the dual space of Y s can be identified by for all s > 0 (see [6, Lemmas 2.1-2.2]), with the norm of ϕ ∈ Y * s being the total variation of the finite complex measure (1 + x s ) ϕ(dx).
Observe that the initial value problem (1) can be written as where M r is the multiplication operator, M r u = −rxu, and the operator B ν is given by with ν > 0, γ > 0, β ∈ R. The operators B ν and M r act on C[0, ∞] (resp. C 0 [0, ∞)).
For any ν > 0, in [8] we considered the operator defined formally by its square is given by Thus the operator B ν defined by (4) can be represented as a perturbation of G 2 ν , namely According to the results in [8] the operators G 2 ν , B ν , P 1 + P 2 + M r with α > 0, β = 0, r > 0 are infinitesimal generators of (C 0 ) semigroups on C 0 [0, ∞), while the operator B ν + M r , r > 0, generates a once-integrated semigroup on C[0, ∞] which is (C 0 ) on the space C 0 [0, ∞). Furthermore, we showed that the semigroups generated by the operators G 2 ν and P 1 + P 2 + M r have explicit representations on C 0 [0, ∞).
In this paper our aim is to study the operators G 2 ν , P 1 + P 2 + M r , A r := B ν + M r , and their associated semigroups on the weighted Banach space Y s , s > 0, in order to solve the CIR problem (r = 1) mentioned in the Introduction Observe that the constant function 1 is in Y s for all s > 0. Further, we will show that some generation results work even when the multiplication operator M r is replaced by a more general nonpositive multiplication operator of type M −P = −P (x)I with P (x) posynomial (see Definition 3.8 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 · X , for any x ∈ X, x = 0, we can define the (nonempty) subset I(x) of the dual of X, X * , by Here < ·, · > is the duality between X and X * . A linear operatorÂ on X is called quasidissipative if for some constant ω ∈ R depending only onÂ. Also,Â − ω I is called dissipative.
Consider the operator G ν defined in (5) and acting on C[0, ∞] with domain and the operator G ν,s defined as G ν and acting on Y s with domain If As in [11,Corollary 4.5], let us consider the family of operators T ν := (T ν (t)) t∈R defined by given by (7).
Proof. For any t ∈ R and x ≥ 0 let us define and observe that the Cauchy problem ). In addition, it can be easily seen that (T ν (t)) t∈R is a group of bounded linear continuous operators on C[0, ∞] which consists of isometries. Now, let us prove that the group T ν is strongly continuous. Preliminarily, we remark that uniformly on compact sets of [0, ∞). Let us fix f ∈ C[0, ∞]. We have to prove that Observe that f (x) converges as x approaches to +∞ and denote by L ∈ R its limit. It follows that lim Hence, fixing any ε > 0, there exists M > 0 such that uniformly for t ∈ R. On the other hand, from the uniform continuity of f and (10), we deduce that there exists δ > 0 such that, for any t ∈ R, From (12) and (13) we can deduce that, for any t ∈ R, and the assertion (11) follows. Let us consider the resolvent equation for λ ∈ R, λ = 0, and h ∈ C[0, ∞]. The general solution is parametrized by u(0) ∈ R, but only one solution is bounded and in Observe that (14) implies for any x > 0. Integrating (15) gives A similar calculation works for λ < 0. Thus, Now, let us focus on the family of operators T ν acting on the space Y s .
Proof. Fix ν > 0. We will prove that, for any s ≥ 1/2 : i) The operators ±G ν,s are quasidissipative on Y s .
ii) The operators ±G ν,s satisfy the range condition on Y s .
iii) The group T ν is strongly continuous on the space Y s and there exists ω ∈ R + such that ||T ν (t)f || s ≤ e ω |t| ||f || s , Proof of i).
Step 1. For ν > 0 and s ≥ 1 2 we consider the operator G ν,s , see (5), (8). In order to prove the assertion, let 0 = f ∈ D(G ν,s ) and x 0 ≥ 0 be such that for w(x) := 1 1 + x s , s > 0, the real function |f (x)| w(x) is maximized at x 0 , i.e., |f Without any loss of generality, we may assume f is real and f (x 0 ) > 0. We choose notation in the usual way and write < f, δ x0 >= f (x 0 ), where δ x0 is the Dirac measure. From (3) it follows that Assume now x 0 > 0 and define g(x) := f (x) w(x) for any x > 0. Then , and so we deduce that Then condition (6) is satisfied with ω = ω s := ν s and s ≥ 1/2. In case x 0 = 0, then f has a positive maximum at 0. Thus, Step 2. We consider the operator −G ν,s . From the above arguments it follows that for all x 0 > 0 and s ≥ 1/2. If in the above inequality we let x 0 → 0, then Then the operator −G ν,s is dissipative, and hence quasi-dissipative on Y s for all s ≥ 1/2. Thus the proof of assertion i) is complete.
h 1 + x s and the assertion follows. Similar arguments work for −G ν,s .
Thus for any s > 0, is in the range of λ − G ν,s for large real λ, and D s is dense in Y s . It follows that both ±G ν,s are m-quasidissipative. This works for all s ≥ 1/2. And we may choose ω = νs in all cases. Part iii) now follows from the Hille-Yosida Theorem.
for f ∈ Y s , where p is the probability density function of the normal distribution with mean zero and variance 2t, Before stating the next theorem we need the following lemmas. Proof. Let s ≥ 1. We first prove that for any u ∈ D(P 1,s ), there exists ϕ ∈ I(u) such that < P 1,s u, ϕ >≤ ω s ||u|| s (quasidissipativity) for some constant ω s ∈ R depending only on P 1,s and s. By using the same arguments as in the proof of Lemma 3.2 i), consider 0 = f ∈ D(P 1,s ), with x 0 ≥ 0 a maximizing value yielding ||f || s , and ϕ = δ x0 w(x 0 ) ∈ I(f ), where w(x) = 1 1 + x s and f (x 0 ) > 0. We deduce that, if x 0 > 0, Concerning the range condition, we observe that the realization P 1,0 of P 1,s on C 0 [0, ∞), generates the (C 0 ) contraction semigroup T P1,0 (t)f (x) = f (x + αt), for α > 0, x ≥ 0, t ≥ 0, on the space C 0 [0, ∞). Thus, by the Hille-Yosida theorem, the range of λI − P 1,0 contains C 0 [0, ∞) for each λ > 0. This yields that on the weighted Y s space, for large enough λ > 0, the range of λI − P 1,s contains C 0 [0, ∞) which is dense in Y s . Hence P 1,s on Y s is essentially m-quasidissipative.
Lemma 3.5. For any s ≥ 1, β = 0, the closure of the operator P 2,s defined as Proof. As was the case for Lemma 3.4, it is enough to verify the quasidissipativity and the range condition. Observe that for any u ∈ D(P 2,s ), there exists ϕ ∈ I(u) such that < P 2,s u, ϕ >≤ ω * s ||u|| s for some constant ω * s ∈ R depending only on P 2,s and s.
Hence, by using the same arguments as in the proof of Lemma 3.4, quasidissipativity holds for ω * s := |β| s and s ≥ 1. The range condition follows by using analogous arguments as in the proof of Lemma 3.4. Indeed, for any real β = 0, the realization P 2,0 of P 2,s on C 0 [0, ∞) generates a (C 0 ) group of isometries given by T P2,0 f (x) = f (xe βt ) for x ≥ 0, t ∈ R. By the Hille-Yosida theorem, P 2,0 and −P 2,0 are both m-dissipative, thus the equation u − βxu = h has a unique solution in the space C 0 [0, ∞) for any initial condition u(0) = a and any h ∈ Y 0 . In particular, for any large enough λ > 0, C 0 [0, ∞) is in the range of λI − P 2,0 and C 0 [0, ∞) is dense in all our weighted Y s spaces. Hence the closure of P 2,s is m-quasidissipative.
Let us define the domain of the operator Q s = P 1,s + P 2,s on the space Y s to be for any s > 0. Lemma 3.6. For any s ≥ 1, α > 0, β ∈ R, the closure of the operator Q s generates a (C 0 ) quasicontractive semigroup V αβ = V αβ (t) t≥0 on Y s given by Proof. Here assume β = 0. If β = 0, we can replace e t β −1 β by t, its limit as β → 0. In [8,Lemma 3] we observed that the solution of the Cauchy problem where Q 0 := P 1,0 + P 2,0 acts on C 0 [0, ∞) (see the definition of P 1,0 , respectively P 2,0 , in the proof of Lemma 3.4, respectively of Lemma 3.5 is given by Both P 1,s and P 2,s are quasidissipative on Y s for s ≥ 1. It follows that Q s = P 1,s + P 2,s has quasidissipative closure on Y s for s ≥ 1. The constant ω s for Q s is the sum of ω s and ω * s corresponding to P 1,s and P 2,s respectively. From [8,Lemma 3] it follows that, for any large enough λ > 0, the range of λI − Q 0 is dense in C 0 [0, ∞). This implies the density of the range of λI − Q s in Y s for all s > 0. Thus the closure Q s is m-quasidissipative for s ≥ 1 and we are done.
Theorem 3.7. Assume ν > 0 and α > 0. Then the closure of the operator Proof. The quasidissipativity of B ν,s is an immediate consequence of Theorem 3.3 and Lemma 3.6. The range condition follows from the range condition proved in [8] Using a terminology due to Dick Duffin (see [5]), we introduce the definition of a posynomial.
Let us fix a weight function 0 < w ∈ C[0, ∞) and denote by C 0,w the Banach space C 0 ([0, ∞), w) already introduced in Section 1. Then the following result holds. Its easy proof is omitted. Lemma 3.9. Let P be a posynomial and let M −P be the multiplication operator Then M −P generates a (C 0 ) contraction semigroup on C 0,w given by Remark 1. The previous Lemma works in the particular case of the posynomial P (x) = s + r x k , 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 . Proof. Let h ∈ C 0 [0, ∞), λ > 0. We want solve the equation Without any loss of generality, it is enough to do this for h ≥ 0. For any m ∈ N, consider the sequence Note that, by the properties of P , Let > 0 be given. Integrating To prove that the pointwise solution u of (17) belongs to D(A P,0 ), it remains to control u near x = ∞ and near x = 0.
The next Lemma is an extension of the Kallman-Rota inequality [12].
Proof. The semigroup {e sL : s ≥ 0} is strongly continuous for t > 0, but it need not be of class (C 0 ). The usual Kallman-Rota inequality assumes that {e sL : s ≥ 0} is of class (C 0 ), but the conclusion holds in the more general case when L need not be densely defined.
Let f ∈ D(L 2 ). Then by Taylor's formula, Minimizing over t > 0 gives, if L 2 f = 0, t = 2 ||f || 1/2 ||L 2 f || −1/2 , and so for every δ > 0. This is also valid if L 2 f = 0, which implies Lf = 0. Then Lemma now follows. First we take β = 0. Thus L in Lemma 3.11 is a dissipative Kato perturbation of L 2 (and so is −L if L is the infinitesimal operator of a group of isometries). Therefore, L 2 + k L is m-dissipative on D(L 2 ) for all k ≥ 0 (and for all k ∈ R in the isometric group case).
We take now β = 0. Observe that for x > 0 small, This now completes our proof of strong continuity on C 0 [0, ∞), including the characterization of D(B ν,0 ) which is independent of γ and β, provided γ > 0. Theorem 3.10 is now fully proved.
Hence the claim follows.
Remark 2. At x = ∞, the multiplication operator M −P is unbounded, so it preserves the boundary condition u(∞) = 0 whenever u ∈ D(A P,0 ), but D(A P,0 ) will be smaller than D(B ν,0 ) because u(x) → 0 as x → ∞ does not imply P (x) u(x) → 0 as x → ∞. Proof. The quasidissipative assertion follows from the previous results. The range condition is a consequence of the range condition proved in the previous theorem for C 0 [0, ∞) since the range of λI − B ν,s contains C 0 [0, ∞) which is dense in Y s 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 C 0 [0, ∞). Consider the posynomial P r (x) = rx, x ≥ 0, with r > 0. Thus the problem (1) can be written as du dt = C 1 u + C 2 u, u(0) = f where C 1 = G 2 ν,0 , C 2 = Q 0 + M −Pr , and M −Pr reduces to the multiplication operator M r defined in Section 2. The Trotter product formula implies that the semigroup generated by C = C 1 + C 2 is given by Using our explicit formulas for e t C1 and e t C2 , we found a very complicated explicit formula for (e t n C1 e t n C2 ) n f for n = 2 k , for any positive integer k. Using this formula in (24) gives our Feynman-Kac formula. In the Feynman case of the Schrödinger equation, z n = (e t n C1 e t n C2 ) n f 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 z n converges to a Wiener integral solution of the heat equation. Our z n looks nothing like an approximation to an integral. But still, z n converges to the desired solution. We showed this in [8] for C 0 [0, ∞) and here the analogous proof establishes it for Y s , s ≥ 1. This is our Theorem 4.2.
Before stating Theorem 4.2, we note the following result.