Abstract
Abstract We consider the existence of a classical smooth solution to the backward Kolmogorov equation
Abstract
where A is the generator of the CIR process, the solution to the stochastic differential equation

∂tu(t, x) = Au(t, x), x ≥ 0, t ∈ [0, T ], u(0, x) = f (x),

where A is the generator of the CIR process, the solution to the stochastic differential equation

that is, Af (x) = θ(κ − x)f ′ (x) + 1 2 σ 2 xf ′′ (x), x ≥ 0 (θ, κ, σ > 0). Alfonsi [1] showed that the equation has a smooth solution with partial derivatives of polynomial growth, provided that the initial function f is smooth with derivatives of polynomial growth. His proof was mainly based on the analytical formula for the transition density of the CIR process in the form of a rather complicated function series. In this paper, for a CIR process satisfying the condition σ 2 ≤ 4θκ, we present a direct proof based on the representation of a CIR process in terms of a squared Bessel process and its additivity property.
Introduction
Let us recall the well-known relationship between the one-dimensional stochastic differential equation ( where Af = bf ′ + 1 2 σ 2 f ′′ is the generator of the diffusion defined by SDE (1.1). If the coefficients b, σ : R → R and the initial function f are sufficiently "good," then the function u = u(t, x) := Ef (X x t ) is a (classical) solution to PDE (1.2) . From this by Itô's formula it follows that the random process This fact is essential in rigorous proofs of the convergence rates of weak approximations of SDEs. The higher the convergence rate, the greater smoothness of the coefficients, and the final condition is to be assumed to get a sufficient smoothness of the solution u to (1.2). The question of the existence of smooth classical solutions to the backward Kolmogorov equation is more complicated than it might seem from the first sight. General results typically require smoothness and polynomial growth of several higher-order derivatives of the coefficients; we refer to the book by Kloeden and Platen [9] , Theorem 4.8.6 on p. 153.
Introduction
However, the coefficients of many SDEs used in financial mathematics are not sufficiently good, and therefore the general theory is not applicable. A classic example is the well-known Cox-Ingersoll-Ross (CIR) process [5] , the solution to the SDE
Introduction
with parameters θ, κ, σ > 0, x ≥ 0, where the diffusion coefficientσ(x) = σ √ x has unbounded derivatives. Alfonsi [1, Prop. 4.1] , using the known expression of the transition density of CIR process by a rather complicated function series, gave an ad hoc proof that, indeed, u = u(t, x) := Ef (X x t ) is a classic solution to the PDE (1.2), where
Introduction
is the generator of the CIR process (1.3). Moreover, he proved that if f : R + → R is sufficiently smooth with partial derivatives of polynomial growth, then so is the solution u.
Introduction
In this paper, in case the coefficients of Eq. (1.3) satisfy the condition σ 2 ≤ 4θκ, we give another proof of this result, where we do not use the transition function. We believe that our approach will be applicable to a wider class of "square-root-type" processes for which an explicit form of the transition function is not known (e.g., the well-known square-root stochastic-volatility Heston process [7] ). The main tools are the additivity property of CIR processes and their representation in terms of squared Bessel processes. More precisely, we use, after a smooth time-space transformation, the expression of the solution to Eq. (1.3) in the form X x t = ( √ x+B t ) 2 +Y t , where Y is a squared Bessel process independent from B. The main challenge is the negative powers of x appearing in the expression of u(t, x) = Ef (X x t ) after differentiation with respect to x > 0. To overcome it, we use a "symmetrization" trick (see Step 1 in the proof of Theorem 4) based on the simple fact that replacing B t by the "opposite" Brownian motionB t := −B t does not change the distribution of X x t . Both proofs, Alfonsi's and ours, are "probabilistic." It is interesting whether there are similar results with "nonprobabilistic" proofs in the literature. Equation (1.2) seems to be a very simple equation, with coefficients analytic everywhere and the diffusion nondegenerate everywhere except a single point. However, although there is a vast literature on degenerate parabolic and elliptic equations, we could find only a few related results, which, however, do not include the case of initial functions f from C n pol (R + ) or C ∞ pol (R + ) (see the notation in the Introduction); instead, the boundedness of f and its derivatives is assumed as a rule. For example, general Theorem 1.1 of Feehan and Pop [6] (see also Cerrai [4] ) in our particular (one-dimensional) case gives an a priori estimate of the form
Introduction
, in terms of the corresponding Hölder and weighted Hölder space supremum norms. 
Introduction
is called a squared Bessel process with dimension δ, starting at x (BESQ δ x for short). We further denote it by Y δ t (x) or Y δ (t, x), and also, Y
Introduction
, that is, with a BESQ n x random process starting at 
Introduction
3)
Introduction
We will frequently use differentiation under the integral sign (in particular, under the expectation sign). Without special mentioning, this will be clearly justified by Lemma 3, which seems to be a folklore theorem; we refer to technical report [3] .
Introduction
k be an open set, and f : X × E → R be a measurable function. The function f is said to be locally
Introduction
Lemma 3 (Differentiation under the integral sign; see [3] , Thm. 4.1). Let (E, A, µ), X, and let f be as in Definition 2. Suppose that f has partial derivatives ∂f ∂xi (x, ω) for all (x, ω) ∈ X × E and that both f and ∂f ∂xi are locally integrable in X. Then
Introduction
for almost all x ∈ X. In particular, if both sides are continuous in X, then we have equality for all x ∈ X.
Introduction
Notation. As usual, N and R are the sets of natural and real numbers, R + := [0, ∞), and N := N ∪ {0}. We denote by C n pol (R + ) the set of n times continuously differentiable functions f : R + → R such that there exist constants C i ≥ 0 and k i ∈ N, i = 0, 1, . . . , n, such that
Introduction
for all i = 0, 1, . . . , n. Then, following Alfonsi [2] , we say that the set of constants
Introduction
, that is, f : R + → R is infinitely differentiable and there exist constants C i ≥ 0 and k i ∈ N, i ∈ N, such that
Introduction
then the sequence of constants {(C i , k i ), i ∈ N} is said to be good for f . Finally, by C ≥ 0 and k ∈ N we will denote constants that depend only on the good set of a function f and may very from line to line.
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
Our main result is a direct proof of the following:
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
t be a CIR process with coefficients satisfying the condition σ 2 ≤ 4θκ and starting at
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
, and estimate (3.1) holds for all i, j ∈ N + with C and k depending on (i, j) and a good sequence
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
Proof. We first focus ourselves on the differentiability in x ≥ 0. By Lemma 2 the process X t (x) can be reduced, by a space-time transformation, to the BESQ δ process Y 
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
With an abuse of notation, we further write T instead ofT . We proceed by induction on l.
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
Step 1. Let l = 1. First, suppose that δ = n ∈ N. By Lemma 1 we have 
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
Since the distributions of
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
where
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
We now estimate P (t, x) and R(t, x) separately. By the well-known inequality
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
we have the following estimates:
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
and, as a consequence,
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
Now, for P (t, x), we have 6) where the constant C depends only on C 1 , k 1 , T , and n. At this point, we need the following technical lemma, which we will prove in the Appendix.
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
Lemma 5. For a function f : R + → R, define the function
Existence and properties of a solution to backward Kolmogorov equation related to CIR process
If f ∈ C q pol (R + ) for some q = 2l + 1 ∈ N (l ∈ N), then the function g is extendable to a continuous function on R + × R × R + such that g(·; a, b) ∈ C l pol (R + ) for all a ∈ R and b ∈ R + . Moreover, there exist constants C ≥ 0 and k ∈ N, depending only on a good set {(C i , k i ), i = 0, 1, . . . , q} for f , such that
7)
for all j = 0, 1, . . . , l.
Now consider R(t, x)
. Applying Lemma 5 with f ′ instead of f (and thus with g 1 instead of g), we have
Now consider R(t, x)
where the constant C clearly depends only on C 2 , k 2 , T , and n.
Now consider R(t, x)
Combining the obtained estimates, we finally get
Now consider R(t, x)
where k = max{k 1 , k 2 }, and the constant C depends only on C 1 , C 2 , k 1 , k 2 , T , and n. Now consider the general case where δ ≥ 1, δ / ∈ N. Note that we consider the general case only for l = 1 because the reasoning for higher-order derivatives is the same.
Now consider R(t, x)
Let n < δ < n + 1, n ∈ N. According to [8, Prop. 83.161.120.155], Y δ t (x) has the same distribution as the affine sum of two independent BESQ processes, namely,
Now consider R(t, x)
where Y n t (x) and Y n+1 t (x) are two independent BESQ processes of dimensions n and n + 1, respectively, starting at x, and λ 1 = n + 1 − δ ∈ (0, 1), λ 2 = 1 − λ 1 = δ − n ∈ (0, 1) (so that δ = λ 1 n + λ 2 (n + 1)). Using the estimates just obtained for δ ∈ N, we have
Now consider R(t, x)
with independent standard normal variablesξ andξ and standard Brownian motions B i and B i . Using again the fact that the distributions ofỸ and proceeding as in (3.3) , we have
Now consider R(t, x)
Combination of estimates (3.4) and (3.5) leads to the estimate
Now consider R(t, x)
where the constant C depends only on C 1 , k 1 , T , and n. By Lemma 5, similarly to estimate (3.8), we have
Now consider R(t, x)
where the constant C depends only on C 2 , k 2 , T , and n. Combining the last two estimates, we get 9) where k = max{k 1 , k 2 }, and the constant C depends only on C 1 , C 2 , k 1 , k 2 , and T .
Now consider R(t, x)
Step 2. Let l = 2. From Step 1 we have
Now consider R(t, x)
From estimate (3.9) with f replaced by f ′ we obtain
Now consider R(t, x)
where the constant C depends only on C 1 , C 3 , k 1 , k 3 , T , and n. For R 2 (t, x), applying Lemma 5 once more to g 1 instead of g, we get
Now consider R(t, x)
where the constants C and k ∈ N depend only on {(C i , k i ), i = 1, 2, 3, 4}, T , and n.
Now consider R(t, x)
Combining the obtained estimates, we finally get
Now consider R(t, x)
where the constants C and k ∈ N depend only on {(C i , k i ), i = 1, 2, 3, 4}, T , and n.
Now consider R(t, x)
Step 3. Now we may continue by induction on l. Suppose that estimate (3.1) is valid for l = m − 1. Let us show that it is still valid for l = m. The arguments are similar to those in the case m = 2 (Step 2). We have
Now consider R(t, x)
Then, similarly to estimates (3.6) and (3.10), we have
Now consider R(t, x)
where the constant C depends only on {(C i , k i ), i = 1, 3, . . . , 2m − 1}, T , and n. For R m (t, x), applying Lemma 5 to g 1 instead of g, we get
Now consider R(t, x)
where the constants C and k ∈ N depend only on {(C i , k i ), i = 1, . . . , 2m}, T , and n. Combining the obtained estimates, we get
Now consider R(t, x)
where the constants C and k ∈ N depend only on {(C i , k i ), i = 1, . . . , 2m}, T , and n. Thus, Theorem 4 is proved for all l ∈ N.
Now consider R(t, x)
Step 4. As in Alfonsi [1, p. 28], inequality (3.1) for the derivatives with respect to t and mixed derivatives follows automatically by an induction on l ′ using that, for
Now consider R(t, x)
A Appendix: Proof of Lemma 5 Proof. First, let n = 5 (l = 2), that is, f ∈ C 5 (R + ). Then, denoting A := a 2 + b, for i = 0, . . . , 4, we have
Now consider R(t, x)
From this it follows that
Now consider R(t, x)
In particular, every function g i , i = 0, . . . , 4, is continuously extendable to the whole half-line
Now consider R(t, x)
pol (R + ) with the estimates
Now consider R(t, x)
for some constants C i > 0 and k i ∈ N, i = 0, 1, . . . , 5.
Now consider R(t, x)
Then we have the estimate
Now consider R(t, x)
where C depends on C i+1 and k i+1 only. Now let us concentrate ourselves on the derivatives of g = g 0 with respect to x. We have In particular, the function g = g 0 is continuously differentiable at x = 0 and thus belongs to C 1 (R + ) since g 
Now consider R(t, x)
where C and k depend on C 2,3 , k 2,3 , and A = a 2 + b only.
Now consider R(t, x)
Thus, we have proved that
Now consider R(t, x)
(Note that, again, the term at the negative power of x, that is, at 1/ √ x, vanishes since 1 −1 s 0 u du s ds = 0.) In particular, again by the Lagrange theorem, g 0 is twice continuously differentiable on the whole half-line R + since there exists the finite limit
Now consider R(t, x)
If, moreover, f ∈ C where the constants C > 0 and k ∈ N depend only on C i and k i , i = 3,4,5, and, in particular, on a good set of the function f ∈ C 5 pol (R + ). Now, for l > 2, we can proceed similarly. For f ∈ C 2l+1 pol (R + ), denote where c j,l , 0 ≤ j ≤ l, are some constants. Note that, as before, in the right-hand side of Eq. (A.5), there are no negative powers of x, so that g 0 is l times continuously differentiable on the whole half-line R + , provided that f ∈ C 2l+1 pol (R + ). Moreover, as before, from (A.5) we get the following estimates for g where the constants C > 0 and k ∈ N depend only on C i and k i , i = 0, . . . , 2l + 1, that is, only on a good set of the function f ∈ C 2l+1 pol (R + ).

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