VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA98 10.15559/18-VMSTA98 Research Article On backward Kolmogorov equation related to CIR process MackevičiusVigirdasvigirdas.mackevicius@mif.vu.lt∗ MongirdaitėGabrielė Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius, Lithuania Corresponding author. 2018 63201851113127 20112017 3112018 222018 © 2018 The Author(s). Published by VTeX2018 Open access article under the CC BY license.

We consider the existence of a classical smooth solution to the backward Kolmogorov equation ∂tu(t,x)=Au(t,x),x≥0,t∈[0,T],u(0,x)=f(x),x≥0, where A is the generator of the CIR process, the solution to the stochastic differential equation Xtx=x+∫0tθ(κ−Xsx)ds+σ∫0tXsxdBs,x≥0,t∈[0,T], that is, Af(x)=θ(κ−x)f′(x)+12σ2xf″(x) , x≥0 ( θ,κ,σ>0 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 σ2≤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 (SDE) Xtx=x+∫0tb(Xsx)dt+∫0tσ(Xsx)dBs,X0x=x, and the following parabolic partial differential equation (PDE), called the backward Kolmogorov equation, with initial condition ∂tu(t,x)=Au(t,x),u(0,x)=f(x), where Af=bf′+12σ2f″ 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(Xtx) is a (classical) solution to PDE (1.2). From this by Itô’s formula it follows that the random process Mtx:=u(T−t,Xtx),t∈[0,T], is a martingale with mean EMtx=f(x) satisfying the final condition MTx=f(XTx) . 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.

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 Xtx=x+∫0tθ(κ−Xsx)ds+∫0tσXsxdBs,t∈[0,T], with parameters θ,κ,σ>0 0$]]>, x≥0 , where the diffusion coefficient σ˜(x)=σx 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(Xtx) is a classic solution to the PDE (1.2), where Af(x)=θ(κ−x)f′(x)+12σ2xf″(x),x≥0, 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.

In this paper, in case the coefficients of Eq. (1.3) satisfy the condition σ2≤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 Xtx=(x+Bt)2+Yt , 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(Xtx) after differentiation with respect to x>0 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 Bt by the “opposite” Brownian motion B¯t:=−Bt does not change the distribution of  Xtx .

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 Cpoln(R+) or Cpol∞(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 ‖u‖C2+α([0,T]×[0,∞))≤C‖f‖Cp2+α([0,∞)), in terms of the corresponding Hölder and weighted Hölder space supremum norms.

 Preliminaries ([<xref ref-type="bibr" rid="j_vmsta98_ref_008">8</xref>], Def. 6.1.2.1).

For every δ≥0 and x≥0 , the unique strong solution Y to the equation Yt=x+δt+2∫0tYsdBs,t≥0 is called a squared Bessel process with dimension δ, starting at x (BESQ xδ for short). We further denote it by Ytδ(x) or Yδ(t,x) , and also, Ytδ:=Ytδ(0) .

 (See [<xref ref-type="bibr" rid="j_vmsta98_ref_008">8</xref>], Section 6.1).

Let B=(B1,B2,…,Bn) be a standard n-dimensional Brownian motion, n∈N . Then the process Rt2:=‖z+Bt‖2= ∑i=1n(zi+Bti)2,t≥0, where z=(z1,…,zn)∈Rn , coincides in distribution with Ytn(‖z‖), that is, with a BESQ xn random process starting at x=‖z‖=∑i=1nzi2 . In particular, Ytn(x)=d(x+Bt1)2+ ∑i=2n(Bti)2=d(x+ξt)2+Ytn−1,t≥0, where ξ is a standard normal variable independent of Ytn−1 , and =d means equality in distribution.

 ([<xref ref-type="bibr" rid="j_vmsta98_ref_008">8</xref>], Prop. 6.3.1.1).

The distribution of CIR process (1.3) can be expressed in terms of a squared Bessel process as follows: Xt(x)=de−θtYδ(σ24θ(eθt−1),x),t≥0, where δ=4θκ/σ2 .

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].

Let (E,A,μ) be a measure space. Let X⊂Rk be an open set, and f:X×E→R be a measurable function. The function f is said to be locally integrable in X if ∫K∫E|f(x,ω)|μ(dω)dx<∞ for all compact sets K⊂X .

(Differentiation under the integral sign; see [<xref ref-type="bibr" rid="j_vmsta98_ref_003">3</xref>], 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 ∂∂xi∫Ef(x,ω)μ(dω)=∫E∂∂xif(x,ω)μ(dω) for almost all x∈X . In particular, if both sides are continuous in X, then we have equality for all x∈X .

As usual, N and R are the sets of natural and real numbers, R+:=[0,∞) , and N‾:=N∪{0} . We denote by Cpoln(R+) the set of n times continuously differentiable functions f:R+→R such that there exist constants Ci≥0 and ki∈N , i=0,1,…,n , such that |f(i)(x)|≤Ci(1+xki),x≥0, for all i=0,1,…,n. Then, following Alfonsi [2], we say that the set of constants { (Ci,ki) , i=0,1,…,n } is good for f. If f∈Cpol∞(R+) , that is, f:R+→R is infinitely differentiable and there exist constants Ci≥0 and ki∈N , i∈N‾ , such that |f(i)(x)|≤Ci(1+xki),x≥0,i∈N‾, then the sequence of constants { (Ci,ki) , 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:

(cf. Alfonsi [<xref ref-type="bibr" rid="j_vmsta98_ref_001">1</xref>], Prop. 4.1).

Let Xt(x)=Xtx be a CIR process with coefficients satisfying the condition σ2≤4θκ and starting at x≥0 . Let f∈Cpolq(R+) for some q≥4 . Then the function u(t,x):=Ef(Xt(x)),x≥0,t∈[0,T], is l times continuously differentiable in x≥0 and l′ times continuously differentiable in t∈[0,T] for l,l′∈N such that 2l+4l′≤q . Moreover, there exist constants C≥0 and k∈N , depending only on a good set {(Ci,ki) , i=0,1,…,q} for f, such that |∂xj∂tiu(t,x)|≤C(1+xk),x≥0,t∈[0,T], for j=0,1,…,l,i=0,1,…,l′ . In particular, u(t,x) is a (classical) solution to the Kolmogorov backward equation (1.2) for (t,x)∈[0,T]×R+ .

As a consequence, if f∈Cpol∞(R+) , then u(t,x) is infinitely differentiable on [0,T]×R+ , and estimate (3.1) holds for all i,j∈N+ with C and k depending on (i,j) and a good sequence {(Ci,ki) , i∈N‾} for f.

We first focus ourselves on the differentiability in x≥0 . By Lemma 2 the process Xt(x) can be reduced, by a space–time transformation, to the BESQ δ process Ytδ(x) with δ=4θκσ2≥1 . Since only bounded smooth functions of t∈[0,T] are involved in (2.3), it suffices to show estimate (3.1) for Ytδ(x) , t∈[0,T˜], instead of Xt(x) , t∈[0,T] , with T˜=1θln(1+4θTσ2) . With an abuse of notation, we further write T instead of T˜ . We proceed by induction on l.

Step 1. Let l=1 . First, suppose that δ=n∈N . By Lemma 1 we have Ytn(x)=d(x+ξt)2+Ytn−1, where ξ∼N(0,1) is independent of Ytn−1 (in the case n=1 , Yt0:=0 ). Denote Yt+(x):=(x+ξt)2,Yt−(x):=(x−ξt)2.

Since the distributions of Yt+(x) and Yt−(x) coincide, we have ∂xEf(Ytn(x))=∂xEf(Yt+(x)+Ytn−1)=12[∂xEf(Yt+(x)+Ytn−1)+∂xEf(Yt−(x)+Ytn−1)]=12E[f′(Yt+(x)+Ytn−1)(1+ξtx)(Lemma 3)+f′(Yt−(x)+Ytn−1)(1−ξtx)]=Ef′(Ytn(x))+12txE{ξ[f′(Yt+(x)+Ytn−1)−f′(Yt−(x)+Ytn−1)]}=Ef′(Ytn(x))+12tE(ξg1(x,ξt,Ytn−1))=:P(t,x)+R(t,x),x>0, 0,\end{array}\]]]> where g1(x,a,b):=f′((x+a)2+b)−f′((x−a)2+b)x,x>0,a∈R,b≥0. 0,\hspace{2.5pt}a\in \mathbb{R},\hspace{2.5pt}b\ge 0.\]]]> We now estimate P(t,x) and R(t,x) separately. By the well-known inequality | ∑i=1nai|p≤np−1 ∑i=1n|ai|pfor anyn∈N,p≥1,ai∈R,i=1,2,…,n, we have the following estimates: E(Yt±(x))p=E(x±ξt)2p≤22p−1(xp+E|ξ|2ptp)=22p−1(xp+2pΓ(p+12)πtp)≤C(1+xp),x≥0,t∈[0,T],E(Ytn)p=E( ∑i=1n|Bti|2)p≤np−1 ∑i=1nE|Bti|2p=np2pΓ(p+12)πtp≤C,t∈[0,T], and, as a consequence, E(Ytn(x))p=E(Yt+(x)+Ytn−1)p≤2p−1E((Yt+(x))p+E(Ytn−1)p)≤C(1+xp),x≥0,t∈[0,T]. Now, for P(t,x) , we have |P(t,x)|=E|f′(Ytn(x))|≤C1(1+E(Ytn(x))k1)≤C1(1+C(1+xk1))≤C(1+xk1),x≥0,t∈[0,T], where the constant C depends only on C1 , k1 , T, and n.

At this point, we need the following technical lemma, which we will prove in the Appendix.

For a function f:R+→R , define the function g(x;a,b):=f((x+a)2+b)−f((x−a)2+b)x,x>0,a∈R,b∈R+. 0,\hspace{2.5pt}a\in \mathbb{R},\hspace{2.5pt}b\in \mathbb{R}_{+}.\]]]> If f∈Cpolq(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)∈Cpoll(R+) for all a∈R and b∈R+ . Moreover, there exist constants C≥0 and k∈N , depending only on a good set {(Ci,ki) , i=0,1,…,q} for f, such that |∂xjg(x;a,b)|≤C|a|(1+xk+|a2+b|k),x∈R+,a∈R,b∈R+, for all j=0,1,…,l .

Now consider R(t,x) . Applying Lemma 5 with f′ instead of f (and thus with g1 instead of g), we have |R(t,x)|≤12tE|ξg1(x,ξt,Ytn−1)|≤CtE[ξ2(1+xk2+|(ξt)2+Ytn−1|k2)]≤CtE[ξ2(1+xk2+2k−1((ξt)2k2+(Ytn−1)k2))]≤C(1+xk2),x≥0,t∈[0,T], where the constant C clearly depends only on C2 , k2 , T, and n.

Combining the obtained estimates, we finally get |∂xEf(Xt(x))|≤C(1+xk1)+C(1+xk2)≤C(1+xk),x≥0,t∈[0,T], where k=max{k1,k2} , and the constant C depends only on C1,C2 , k1,k2 , 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.

Let n<δ<n+1 , n∈N . According to [8, Prop. 6.2.1.1], Ytδ(x) has the same distribution as the affine sum of two independent BESQ processes, namely, Ytδ(x)=dλY˜tn(x)+λ2Yˆtn+1(x), where Y˜tn(x) and Yˆtn+1(x) are two independent BESQ processes of dimensions n and n+1 , respectively, starting at x, and λ1=n+1−δ∈(0,1) , λ2=1−λ1=δ−n∈(0,1) (so that δ=λ1n+λ2(n+1) ). Using the estimates just obtained for δ∈N , we have ∂xEf(Ytδ(x))=∂xEf(λ1Y˜tn(x)+λ2Yˆtn+1(x))=12[∂xEf(λ1(Y˜t+(x)+Y˜tn−1)+λ2(Yˆt+(x)+Yˆtn))+∂xEf(λ1(Y˜t−(x)+Y˜tn−1)+λ2(Yˆt−(x)+Yˆtn))], where Y˜t+(x):=(x+ξ˜t)2,Y˜t−(x):=(x−ξ˜t)2,Yˆt+(x):=(x+ξˆt)2,Yˆt−(x):=(x−ξˆt)2,Y˜tn−1:= ∑i=1n−1(B˜ti)2,Yˆtn:= ∑i=1n(Bˆti)2, with independent standard normal variables ξ˜ and ξˆ and standard Brownian motions B˜i and Bˆi . Using again the fact that the distributions of Y˜t±(x) and Yˆt±(x) coincide and proceeding as in (3.3), we have ∂xEf(Ytδ(x))=12[∂xEf(λ1((x+ξ˜t)2+Y˜tn−1)+λ2((x+ξˆt)2+Yˆtn))+∂xEf(λ1((x−ξ˜t)2+Y˜tn−1)+λ2((x−ξˆt)2+Yˆtn))]=12[Ef′(λ1((x+ξ˜t)2+Y˜tn−1)+λ2((x+ξˆt)2+Yˆtn))(1+(λ1ξ˜+λ2ξˆ)tx)+Ef′(λ1((x−ξ˜t)2+Y˜tn−1)+λ2((x−ξˆt)2+Yˆtn))(1−(λ1ξ˜+λ2ξˆ)tx)]=Ef′(Ytδ(x))+t2E[(λ1ξ˜+λ2ξˆ)×g1(x,(λ1ξ˜+λ2ξˆ)t,λ1λ2(ξ˜−ξˆ)2t+λ1Y˜tn−1+λ2Yˆtn)]=:P1(t,x)+R1(t,x).

Combination of estimates (3.4) and (3.5) leads to the estimate |P1(t,x)|=|Ef′(Ytδ(x))|≤C1E(1+|Ytδ(x)|k1)≤C1E(1+4k1−1|λ1k1((Y˜t+(x))k1+(Y˜tn−1)k1)+λ2k1((Yˆt+(x))k1+(Yˆtn)k1)|)≤C(1+xk1),x≥0,t∈[0,T], where the constant C depends only on C1 , k1 , T, and n. By Lemma 5, similarly to estimate (3.8), we have |R1(t,x)|≤CE[(λ1ξ˜+λ2ξˆ)2(1+xk+|(λ1ξ˜+λ2ξˆ)2t+λ1λ2(ξ˜−ξˆ)2t+λ1Y˜tn−1+λ2Yˆtn|k)]≤C(1+xk),x≥0,t∈[0,T], where the constant C depends only on C2 , k2 , T, and n. Combining the last two estimates, we get |∂xEf(Xt(x))|≤C(1+xk),x≥0,t∈[0,T], where k=max{k1,k2} , and the constant C depends only on C1,C2 , k1,k2 , and T.

Step 2. Let l=2 . From Step 1 we have ∂xEf(Ytn(x))=Ef′(Ytn(x))+12tE(ξg1(x,ξt,Ytn−1)). Therefore, ∂x2Ef(Ytn(x))=∂xEf′(Ytn(x))+12tE(ξ∂xg1(x,ξt,Ytn−1))=:P2(t,x)+R2(t,x). From estimate (3.9) with f replaced by f′ we obtain |P2(t,x)|≤C(1+xk3),x≥0,t∈[0,T], where the constant C depends only on C1 , C3 , k1 , k3 , T, and n. For R2(t,x) , applying Lemma 5 once more to  g1 instead of g, we get |R2(t,x)|≤12tE|ξ∂xg1(x,ξt,Ytn−1)|≤CtE(ξ2(1+xk+|ξt|k+(Ytn−1)k))≤C(1+xk),x≥0,t∈[0,T], where the constants C and k∈N depend only on {(Ci,ki) , i=1,2,3,4} , T, and n. Combining the obtained estimates, we finally get |∂x2Ef(Xt(x))|≤C(1+xk),x≥0,t∈[0,T], where the constants C and k∈N depend only on {(Ci,ki) , i=1,2,3,4 }, T, and n.

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 ∂xmEf(Ytn(x))=∂xm−1Ef′(Ytn(x))+12t∂xm−1E(ξg1(x,ξt,Ytn−1))=:Pm(t,x)+Rm(t,x). Then, similarly to estimates (3.6) and (3.10), we have |Pm(t,x)|≤C(1+xkm), where the constant C depends only on {(Ci,ki) , i=1,3,…,2m−1} , T, and n.

For Rm(t,x) , applying Lemma 5 to g1 instead of g, we get |Rm(t,x)|≤12tE|ξ∂xm−1g1(x,ξt,Ytn−1)|≤C(1+xk), where the constants C and k∈N depend only on {(Ci,ki) , i=1,…,2m} , T, and n. Combining the obtained estimates, we get |∂xmEf(Xt(x))|≤C(1+xk),x>0,t∈[0,T], 0,\hspace{2.5pt}t\in [0,T],\end{array}\]]]> where the constants C and k∈N depend only on {(Ci,ki) , i=1,…,2m} , T, and n. Thus, Theorem 4 is proved for all l∈N .

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 l′≥1 such that 4l′+2l≤q , ∂xl∂tl′u(t,x)=∂xl(θ(κ−x)∂x∂tl′−1u(t,x)+σ22x∂x2∂tl′−1u(t,x))=σ22x∂xl+2∂tl′−1u(t,x)+(lσ22+θ(κ−x))∂xl+1∂tl′−1u(t,x)−lθ∂xl∂tl′−1u(t,x).  □

 Appendix: Proof of Lemma <xref rid="j_vmsta98_stat_009">5</xref>

First, let n=5 ( l=2 ), that is, f∈C5(R+) . Then, denoting A:=a2+b , for i=0,…,4 , we have gi(x;a,b)=f(i)(A+x+2ax)−f(i)(A+x−2ax)x=1xf(i)(A+x+2axs)|s=−1s=1=1x∫−11f(i+1)(A+x+2axs)2axds=2a∫−11f(i+1)(A+x+2axs)ds,x>0. 0.\end{array}\]]]> From this it follows that limx↓0gi(x;a,b)=2a∫−11f(i+1)(A)ds=4af(i+1)(A). In particular, every function gi , i=0,…,4 , is continuously extendable to the whole half-line R+=[0,∞) by defining gi(0;a,b):=4af(i+1)(A) .

Let, moreover, f∈Cpol5(R+) with the estimates |f(i)(x)|≤Ci(1+xki),x≥0,i=0,1,…,5, for some constants Ci>0 0$]]> and ki∈N , i=0,1,…,5 .

Then we have the estimate |gi(x;a,b)|≤2|a|∫−11|f(i+1)(A+x+2axs)|ds≤4|a|Ci+1(1+(A+x+2|a|x)ki+1)≤4|a|Ci+1(1+(A+a2+2x)ki+1)≤C|a|(1+Aki+1+xki+1),x≥0, where C depends on Ci+1 and ki+1 only.

Now let us concentrate ourselves on the derivatives of g=g0 with respect to x. We have g0′(x;a,b)=2a∫−11f″(A+x+2axs)(1+asx)ds=2a∫−11f″(A+x+2axs)ds+2a2x∫−11f″(A+x+2axs)sds=g1(x;a,b)+2a2x∫−11(f″(A+x)+∫0sf‴(A+x+2axu)2axdu)sds=g1(x;a,b)+2a2f″(A+x)x∫−11sds+4a3∫−11∫0sf‴(A+x+2axu)dusds=g1(x;a,b)+4a3∫−11∫0sf‴(A+x+2axu)dusds,x>0. 0.\end{array}\]]]>

(Note that the term at the negative power of x, that is, at 1/x , vanishes since ∫−11sds=0 .) From this it follows that there exists the limit limx↓0g0′(x;a,b)=limx↓0g1(x;a,b)+4a3∫−11∫0sf‴(A)dusds=4af″(A)+8a33f‴(A). In particular, the function g=g0 is continuously differentiable at x=0 and thus belongs to C1(R+) since g0′(0;a,b)=limx↓0g0′(x;a,b) by the Lagrange theorem.

If, moreover, f∈Cpol5(R+) satisfies estimates (2.4) for i≤5 , then we have the corresponding estimate for g0′ : |g0′(x;a,b)|≤|g1(x;a,b)|+4|a|3∫−11∫−ss|f‴(A+x+2axu)|du|s|ds≤C2|a|(1+Ak2+xk2)+4|a|3∫01∫−ssC3(1+(A+x+2|a|xu)k3)duds≤C2|a|(1+Ak2+xk2)+4|a|3C3(1+Ak3+xk3)≤C|a|(1+Ak+xk),x≥0, where C and k depend on C2,3 , k2,3 , and A=a2+b only.

Thus, we have proved that g=g0∈Cpol1(R+) , provided that f∈Cpol5(R+). (In fact, for estimate (A.3), it suffices that f∈Cpol3(R+). ) More precisely, if |f(i)(x)|≤Ci(1+xki),x≥0,i=1,2,3, then |g0(j)(x;a,b)|≤C(1+Ak+xk),x≥0,j=0,1, where the constants C>0 0$]]> and k∈N depend only on Ci and ki , i=1,2,3 , and, in particular, on a good set of the function f∈Cpol5(R+) .

Now, let us proceed to the second derivative of g0 . From Eq. (A.2) we have g0″(x;a,b)=g1′(x;a,b)+4a3∫−11∫0sf(4)(A+x+2axu)(1+aux)dusds=(2a∫−11f(2)(A+x+2axs)ds)′+4a3∫−11∫0sf(4)(A+x+2axu)dusds+4a4x∫−11∫0sf(4)(A+x+2axu)udusds=2a∫−11f(3)(A+x+2axs)(1+asx)ds+4a3∫−11∫0sf(4)(A+x+2axu)dusds+4a4x∫−11∫0s[f(4)(A+x)+∫0uf(5)(A+x+2axv)2axdv]udusds=2a∫−11f(3)(A+x+2axs)ds+2a2x∫−11[f(3)(A+x)+∫0sf(4)(A+x+2axu)2axdu]sds+4a3∫−11∫0sf(4)(A+x+2axu)dusds+4a4xf(4)(A)∫−11∫0sudusds+8a5∫−11∫0s∫0uf(5)(A+x+2axv)dvudusds=2a∫−11f(3)(A+x+2axs)ds+4a3∫−11∫0sf(4)(A+x+2axu)dusds+4a3∫−11∫0sf(4)(A+x+2axu)dusds+8a5∫−11∫0s∫0uf(5)(A+x+2axv)dvudusds,x>0. 0.\end{array}\]]]> (Note that, again, the term at the negative power of x, that is, at 1/x , vanishes since ∫−11∫0sudusds=0 .) In particular, again by the Lagrange theorem, g0 is twice continuously differentiable on the whole half-line R+ since there exists the finite limit limx↓0g0″(x;a,b)=limx↓0g1′(x;a,b)+4a3f(4)(A)∫−11∫0sdusds+8a5f(5)(A)∫−11∫0s∫0udvudusds=4af(3)(A)+16a3f(4)(A)3+16a5f(5)(A)15. If, moreover, f∈Cpol5(R+) satisfies estimates (A.1), then we have the corresponding estimate for  g0″ : |g0″(x,a,b)|≤2|a|∫−11|f(3)(A+x+2axs)|ds+8|a|3∫−11∫−ss|f(4)(A+x+2axu)|du|s|ds+8|a|5∫−11∫−ss∫−uu|f(5)(A+x+2axv)|dv|u|du|s|ds≤2|a|∫−11C3(1+(A+x+2|a|xs)k3)ds+8|a|3∫01∫−ssC4(1+(A+x+2|a|xu)k4)duds+8|a|5∫01∫−ss∫−uuC5(1+(A+x+2|a|xv)k5)dvduds≤C|a|(1+Ak+xk),x≥0, where the constants C>0 0$]]> and k∈N depend only on Ci and ki , i=3 ,4,5, and, in particular, on a good set of the function f∈Cpol5(R+) .

Now, for l>2 2$]]>, we can proceed similarly. For f∈Cpol2l+1(R+) , denote F0,q=F0,q(x,a,b):=∫−11f(q)(A+x+2axs)ds,Fp,q=Fp,q(x,a,b):=∫−11∫0s1⋯∫0spf(q)(A+x+2axsp+1)dsp+1…s2ds2s1ds1,p=1,…,l,q=l+1,…,2l+1. Then, in addition to the first two derivatives g0′(x,a,b)=a(2F0,2+4a2F1,3)andg0″(x,a,b)=a(2F0,3+8a2F1,4+8a4F2,5), we get: g0‴(x,a,b)=a(2F0,4+12a2F1,5+24a4F2,6+16a6F3,7),…………………………g0(l)(x,a,b)=a ∑j=0lcj,la2jFj,l+j+1(x;a,b), where cj,l , 0≤j≤l , are some constants. Note that, as before, in the right-hand side of Eq. (A.5), there are no negative powers of x, so that g0 is l times continuously differentiable on the whole half-line R+ , provided that f∈Cpol2l+1(R+) . Moreover, as before, from (A.5) we get the following estimates for g0(r) : |g0(r)(x,a,b)|≤C|a|(1+Ak+xk),x≥0,r=0,1,…,l, where the constants C>0 0$]]> and k∈N depend only on Ci and ki , i=0,…,2l+1 , that is, only on a good set of the function f∈Cpol2l+1(R+) .  □

 Acknowledgement

We thank the anonymous reviewer for comments that enabled us to improve the presentation of the paper.

 References A. Alfonsi. On the discretization schemes for the CIR (and Bessel squared) processes. Monte Carlo Methods Appl, pages 355–384, 2005. MR2186814 A. Alfonsi. High order discretization schemes for the CIR process: Application to affine term structure and Heston models. Mathematics of Computation, 79(269):306–237, 2010. MR2552224 S. Cheng. Differentiation under the integral sign with weak derivatives. Technical report, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.525.2529&rep=rep1&type=pdf, 2010. S. Cerrai, Second Order PDE’s in Finite and Infinite Dimension: A Probabilistic Approach, Springer, 2001. MR1840644 J.C. Cox, J.E. Ingersoll, and S.A. Ross. A theory of the term structure of interest rates. Econometrica, 53:385–407, 1985. MR0785475 P.M.N. Feehan and C.A. Pop, A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients, J. Differential Equations, 254:4401–4445, 2013. MR3040945 S. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6:327–343, 1993. M. Jeanblanc, M. Yor, and M. Chesney. Mathematical Methods for Financial Markets. Springer, 2009. MR2568861 P.E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992 (2nd corrected printing 1995). MR1214374