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SDE are bounded C∞ functions with bounded derivatives of any order and that, on the other hand, the Hormand¨er condition holds, then the solution of the equation is a smooth functional in Malliavin’s sense, and it is non- degenerate at any fixed positive time. Then the general criterion given by Malliavin [15] allows one to say that the law of such a random variable is absolutely continuous with respect to the Lebesgue measure, and its density is a smooth function (see [16] for a general presentation of this topic). The aim of this paper is to relax the aforementioned conditions on the coefficients: roughly speaking, we assume that the coefficients are smooth only on an open domain D and have bounded partial derivatives therein. Moreover, we assume that the nondegeneracy condition on the diffusion coefficient holds true on D only. Under these assumptions, we prove that the law of a strong solution to the equation admits a smooth density on D (Theorem 2.1). Furthermore, when D is the complementary of a compact ball and the coefficients satisfy some additional assumptions on D (mainly dealing with their growth and the one of derivatives), we give upper bounds for the density for large values of the state variable (Theorem 2.2). We will occasionally refer to these aymptotic estimates of the density as “tail estimates” or estimates on the density’s “tails.” Local results have already been obtained by Kusuoka and Stroock in [13], Section 4. Here the authors work under local regularity and nondegeneracy hypotheses too, but the bounds they provide on the density are mostly signif- icant on the diagonal (i.e., close to starting point) and in the small time limit, while they are not appropriate for tail estimates. Moreover, the constants appearing in the estimates are not explicit (cf. (4.7)–(4.9) in Theorem 4.5 and the corresponding estimates in Corollary 4.10, [13]). In the present pa- per, we provide upper bounds that are suitable for tail estimates, and we find out the explicit dependence of the bounding constants with respect to the coefficients of the SDE and their derivatives. Our bounds turn out to be applicable to the case of diffusions with tails stronger than gaussian. This is the case for square-root diffusions, which are our major example of interest (see Section 3). Also, our approach is substantially different from the one in [13]. In particular, we rely on a Fourier transform argument, employing a technique to estimate the Fourier transform of the process inspired from the work of Fournier in [8] and of Bally in [2] and relying on specifically- designed Malliavin calculus techniques. We estimate the density pt(y) of the diffusion at a point y D performing an integration by parts that involves the contribution of the∈ Brownian noise only on an arbitrarily small time interval [t δ, t]. This allows us to gain a free parameter δ that we can even- tually optimize,− and the appropriate choice of δ proves to be a key point in our argument. We do not study here the regularity with respect to initial condition (which may be the subject of future work). SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 3
Our study is motivated by applications to Finance, in particular by the study of models for financial securities which rely on SDEs with non-Lipschitz coefficients. As it is well known, a celebrated process with square-root diffu- sion coefficient was proposed by Cox, Ingersoll and Ross in [5] as a model for short interest rates and was later employed by Heston in [10] to model the stochastic volatilities of assets. The stochastic-αβρ or SABR model in [9] is based on the following mixing of local and stochastic volatility dynamics: β 1 dXt = σtXt dWt , dσ = νσ dB , σ = α, t t t 0 1 2 2 1 2 where 0 <β< 1, Bt = ρWt + 1 ρ Wt , (W ,W ) is a standard Brownian motion, and ρ [ 1, 1] is the correlation− parameter. In this paper, we apply p our results to one-dimensional∈ − SDEs of the form t t (1.1) X = x + (a(X ) b(X )X ) ds + γ(X )Xα dW , t s − s s s s t Z0 Z0 ∞ where α [1/2, 1) and a, b and γ are Cb functions. When the coefficients a, b, σ are constant,∈ the solutions to this class of equations include the classical CIR process (α = 1/2) and a subclass of the CEV local volatility diffusions (when a =0 and b = r). As pointed out by Bossy and Diop in [4], SDEs with square-root terms− and coefficients depending on the level of the state variable arise as well in the modeling of turbulent flows in fluid mechanics. It is well known that for a CIR process, the density of Xt is known explic- itly. The main contribution of our results lies in the fact that they apply to the more general framework of SDEs whose coefficients are functions of the state variable, thus when explicit computations are no longer possible. Theorem 2.2 directly applies to (1.1) and allows one to show that Xt ad- mits a smooth density on (0, + ): under some additional conditions on the coefficients (mainly dealing with∞ their asymptotic behavior at and zero), we give exponential-type upper bounds for the density at infinity∞ (Proposi- tion 3.3) and study the explosive behavior of the density at zero (Proposi- tion 3.4). The explicit expression of the density for the classical CIR process shows that our estimates are in the good range. The paper is organized follows: in Section 2 we present our main results on SDEs with locally smooth coefficients (Section 2.1), and we collect all the technical elements we need to give their proofs. In particular, in Sec- tion 2.2 we recall the basic tools of Malliavin calculus on the Wiener space, which will be used in Section 2.3 to obtain some explicit estimates of the L2-norms of the weights involved in the integration by parts formula. This is done following some standard techniques of estimation of Sobolev norms and inverse moments of the determinant of the Malliavin matrix (as in [16] and [6], Section 4), but in our computations we explicitly pop out the depen- dence with respect to the coefficients of the SDE and their derivatives. This 4 S. DE MARCO further allows us to obtain the explicit asymptotic estimates on the density. Section 2.4 is devoted to the proof of the theorems stated in Section 2.1. We employ the Fourier transform argument and the optimized integration by parts we have discussed above. Finally, in Section 3 we apply our results to the solutions of (1.1).
2. Smoothness and tail estimates of densities for SDEs with locally smooth coefficients.
2.1. Main results. In what follows, b and σj are measurable functions m m m from R into R , j = 1,...,d. For y0 R and R> 0, we denote by BR(y0) ∈ m [resp., BR(y0)] the open (resp., closed) ball BR(y0)= y R : y y0 < R m { ∈ | − | } [resp., BR(y0)= y R : y y0 R ], where stands for the Euclidean { ∈ | − |≤ } |∞ · | norm. We follow the usual notation denoting Cb (A) the class of infinitely differentiable functions on the open set A Rm which are bounded together with their partial derivatives of any order.⊆ For a multi-index α 1,...,m k, k ∈ { } k 1, ∂ denotes the partial derivative ∂ . α ∂x ···∂x ≥ α1 αk Let 0 < R 1 and y Rm be given. We consider the SDE ≤ 0 ∈ t d t i i i i j Xt = x + b (Xs) ds + σj(Xs) dWs , Z0 j=1 Z0 (2.1) X t [0, T ], i = 1,...,m, ∈ for a finite T > 0 and x Rm, and assume that the following hold: ∈ ∞ Rm (H1) (local smoothness) b, σj Cb (B5R(y0); ); (H2) (local ellipticity) σσ∗(y)∈ c I for every y B (y ), for some 0 < ≥ y0,R m ∈ 3R 0 cy0,R < 1; (H3) existence of strong solutions holds for the couple (b, σ).
Let then (Xt; t [0, T ]) denote a strong solution of (2.1). Our first main result follows. ∈
Theorem 2.1. Assume (H1), (H2) and (H3). Then for any initial con- m dition x R and any 0
The proofs of these results will be given in Section 2.4. 6 S. DE MARCO
Notation. Through the rest of the paper, , will denote the Euclidean scalar product in Rm, while the notation willh· be·i used both for the absolute value of real numbers and for the Euclidean|·| norm in Rm. Furthermore, when Θ= θ1,...,θν is a family of parameters, unless differently specified by CΘ, we denote a constant depending on the θi’s but not on any of the other existing variables. All constants of such a type may vary from line to line, but always depend only on the θi’s. For functions of one variable, the kth dkf (k) derivative will be denoted by dxk or f . We will follow the convention of summation over repeated indexes, wherever present.
2.2. Elements of Malliavin calculus. We recall hereafter some elements of Malliavin calculus on the Wiener space, following [16]. 1 d Let W = (Wt ,...,Wt ; t 0) be a d-dimensional Brownian motion defined on the canonical space (Ω, ≥, P). For fixed T > 0, let be the Hilbert space 2 d F Hd T j j = L ([0, T ]; R ). For any h we set W (h)= h (s) dWs , and H ∈H j=1 0 consider the family L2(Ω, , P) of smooth random variables defined by S⊂ F P R = F : F = f(W (h ),...,W (h )); h ,...,h ; f C∞ (Rn); n 1 , S { 1 n 1 n ∈H ∈ pol ≥ } ∞ ∞ where Cpol denotes the class of C functions which have polynomial growth together with their derivatives of any order. The Malliavin derivative of F is the d-dimensional stochastic process DF = (D1F,...,DdF ; r [0, T ])∈ defined S by r r ∈ n j ∂f j DrF = (W (h1),...,W (hn))hi (r), j = 1,...,d. ∂xi Xi=1 For any positive integer k, the kth order derivative of F is obtained by iterat- k ing the derivative operator: for any multi-index α = (α1,...,αk) 1,...,d k α1,...,αk α1 αk ∈ { } and (r1, . . ., rk) [0, T ] , we set Dr1,...,rk F := Dr1 Drk F . Given p 1 and positive integer∈k, for every F we define the seminorm· · · ≥ ∈ S k 1/p E p E (h) p F k,p = [ F ]+ [ D F ⊗h ] , k k | | k kH ! Xh=1 where 1/2 (k) α1,...,αk 2 D F k = D F dr dr , H⊗ r1,...,rk 1 k k k [0,T ]k | | · · · |αX|=k Z k and the sum is taken over all the multi-indexes α = (α1,...,αk) 1,...,d . We denote with Dk,p the completion of with respect to the∈ { seminorms} D∞ Dk,p S k,p, and we set = p≥1 k≥1 . We may occasionally refer to F k,p k·kas the stochastic Sobolev norm of F . k k T T SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 7
In a similar way, for any separable Hilbert space V we can define the analogous spaces Dk,p(V ) and D∞(V ) of V -valued random variables with the corresponding k,p,V semi-norms (the smooth functionals being now of the form F = nk · kF v , where F and v V ). In particular, for any j=1 j j j ∈ S j ∈ Rd-valued process (u ; s t) such that u Dk,p for all s [0,t] and P s ≤ s ∈ ∈ k (h) u + D u h+1 < , P-a.s., k kH k kH⊗ ∞ Xh=1 we have k 1/p E p E (h) p u k,p,H = [ u H]+ [ D u ⊗h+1 ] . k k k k k kH ! Xh=1 Finally, we denote by δ the adjoint operator of D. One of the main applications of Malliavin calculus consists of showing that the law of a nondegenerate random vector F = (F 1,...,F m) (D∞)m admits an infinitely differentiable density. The property of nondegeneracy,∈ understood in the sense of the Malliavin covariance matrix, is introduced in the following:
Definition 2.1. A random vector F = (F 1,...,F m) (D∞)m, m 1, ∈ ≥ is said to be nondegenerate if its Malliavin covariance matrix σF , defined by (σ ) = DF i, DF j , i, j = 1,...,m, F i,j h iH is invertible a.s. and moreover E[det(σ )−p] < F ∞ for all p 1. ≥ The key tool to prove smoothness of the density for a nondegenerate random vector is the following integration by parts formula (cf. [16]).
Proposition 2.1. Let F = (F 1,...,F m) (D∞)m, m 1, be a nonde- D∞ ∈ ∞ Rm ≥ generate random vector. Let G and φ Cpol( ). Then for any k 1 ∈ ∈ k ≥ and any multi-index α = (α1,...,αk) 1,...,m there exists a random variable H (F, G) D∞ such that ∈ { } α ∈ (2.8) E[∂αφ(F )G]= E[φ(F )Hα(F, G)], where the Hα(F, G) are recursively defined by
Hα(F, G)= H(αk)(F,H(α1,...,αk−1)(F, G)), m −1 j H(i)(F, G)= δ(G(σF )i,jDF ). Xj=1 8 S. DE MARCO
2.3. Explicit bounds on integration by parts formula for diffusion pro- cesses. The notation of this section is somehow cumbersome, as we try to keep our bounds as general and as accurate as possible. The framework will nevertheless considerably simplify in Section 2.4, when we will give the proofs of the results stated in Section 2.1. Throughout this section, X = (Xt; t 0) will denote the unique strong solution of the SDE ≥ t d t i i i i j (2.9) Xt = x + B (Xs) ds + Aj(Xs) dWs , t 0, i = 1,...,m, 0 0 ≥ Z Xj=1 Z Rm i i ∞ Rm where x and B , Aj b ( ) for all i = 1,...,m and j = 1,...,d. We assume that∈ the diffusion matrix∈C A satisfies the following ellipticity condition at starting point x: (E) A(x)A(x)∗ c I , for some c > 0, where ∗ stays for matrix trans- ≥ ∗ m ∗ · position. Without loss of generality, we will suppose c∗ < 1. We recall that the first-variation process of X is the matrix-valued process i ∂Xt (Yt)i,j = , i, j = 1,...,m, ∂xj which satisfies the following equation, written in matrix form: t d t l dYt = Im + ∂B(Xs)Ys ds + ∂Al(Xs)Ys dWs, 0 0 Z Xl=1 Z where ∂B and ∂Al are, respectively, the m m matrices of components i i × (∂B)i,j = ∂jB and (∂Al)i,j = ∂jAl. By means of Itˆo’s formula, one shows −1 that Yt is invertible and that the inverse Zt := Yt satisfies the equation t d Z = I Z ∂B(X ) (∂A (X ))2 ds t m − s s − l s 0 ( l=1 ) (2.10) Z X d t l Zs ∂Al(Xs) dWs. − 0 Xl=1 Z Additional notation. For k 0, we define ≥m i B k =1+ sup ∂αB (x) , | | x∈Rm | | i=1 0≤|α|≤k (2.11) X X i A k =1+ sup ∂αAj(x) , | | x∈Rm | | Xi,j 0≤|Xα|≤k where α is the length of the multi-index α. Then, for p 1 and t 0 we set | | ≥ ≥ p/2 1/2 p t (t |B|1+|A|1) (2.12) ep(t) := e SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 9 and
p/2 1/2 2 p Z t (t (|B|1+|A|1)+|A|1) (2.13) ep (t) := e . The constants in (2.12) and (2.13) naturally arise when estimating the mo- ments of the random variables Xt, Yt and Zt. Indeed, the results given in the following proposition can be easily obtained from (2.9) and (2.10) applying Burkholder’s inequality and Gronwall’s lemma.
Proposition 2.2. For every p> 1 there exists a positive constant Cp,m depending on p and m but not on the bounds on B and A and their deriva- tives such that, for every 0 s t T , ≤ ≤ ≤ E i i p p/2 1/2 p (2.14) (i) sup Xr Xs Cp,m(t s) ((t s) B 0 + A 0) , s≤r≤t | − | ≤ − − | | | | h i E p Z Cp,m (2.15) (ii) sup [ (Zs)i,j ] Cp,mep (t) s≤t | | ≤ for all i, j = 1,...,m.
For any t> 0, the iterated Malliavin derivative of Xt is the solution of a linear SDE. The coefficients of this equation are bounded, and hence it is once again a straightforward application of Gronwall’s lemma to show that α1,...,αk the random variables Dr1,...,rk Xt have moments of any order which are finite and uniformly bounded in r1, . . . , rk. This is indeed the content of [16], The- orem 2.2.2. The following lemma highlights the explicit constants appearing in the estimates of the Lp-norms of the iterated derivative, expressing them in terms of the bounds (2.11) on A and B.
Lemma 2.1. For every k 1 and every p> 1 there exist a positive inte- ≥ ger γk,p and a positive constant Ck,p depending on k,p but not on the bounds on B and A and their derivatives such that, for any t> 0,
E j1,...,jk i p sup [ Dr1,...,rk Xt ] r1,...,rk≤t | | (2.16) 2 C A kp (t1/2 B + A )(k+1) pe (t)γk,p , ≤ k,p| |k−1 | |k | |k p for all i = 1,...,m and (j ,...,j ) 1,...,d k. 1 k ∈ { } The proof of this result is based on some standard but rather cumbersome computations; hence we leave it for Appendix A.1. We rather give hereafter the proof of some estimates which follow easily from Lemma 2.1 and will be useful in the following sections. 10 S. DE MARCO
Corollary 1. For any k 1 and p> 1, there exists a positive con- ≥ stant Ck,p depending only on k and p such that, for any t> 0, E (k) i p 1/p k/2 k 1/2 (k+1)2 (i) [ D X k ] Ck,pt A (t B k + A k) k t kH⊗ ≤ | |k−1 | | | | (2.17) e (t)γk,p ; × p k 1/2 (ii) φ(Xt) k,p Ck,p φ k(1 + (t t ) ) (2.18) k k ≤ | | ∨ 2 A k (t1/2 B + A )(k+2) e (t)kγk,p , × | |k−1 | |k | |k p where (i) holds for i = 1,...,m and (ii) for any φ C∞(Rm). ∈
Proof. (i) Employing the definition of H⊗k and Lemma 2.1, a sim- ple computation holds. k · k E[ D(k)Xi p ]1/p k t kH⊗k 1/p k(p/2−1) E α1,...,αk i p Ck,p t sup Dr1,...,rk Xt dr1 drk ≤ [0,t]k |α|=k | | · · · Z h i 2 C tk/2 A k (t1/2 B + A )(k+1) e (t)γk,p/p, ≤ k,p | |k−1 | |k | |k p hence we get bound (2.17). (ii) We start from the definition of and write k · kk,p k 1/p E p E (h) p φ(Xt) k,p = [ φ(Xt) ]+ [ D φ(Xt) h ] k k | | k kH⊗ h=1 ! (2.19) X k E (h) p 1/p φ 0 + [ D φ(Xt) h ] . ≤ k k k kH⊗ Xh=1 Using the notation introduced in the proof Lemma 2.1, we have ν D(h)φ(X )= ∂ ∂ φ(X ) D(card(Il))Xkl , t k1 · · · kν t t I1,...,IνX={1,...,h} Yl=1 where, with a slight abuse of notation, we have now written D(h) for the generic derivative of order h. Repeatedly applying H¨older’s inequality for Sobolev norms and using bound (2.17), we get E (h) p [ D φ(Xt) h ] k kH⊗ ν p E (hl) kl ch,p ∂k1 ∂kν φ(Xt) D Xt ≤ " · · · ⊗h # h1,...,hXν =1,...,h Yl=1 H h1+···+hν =h
SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 11
ν l l p E (hl) i 2 p 1/2 ch,p φ h sup [ D Xt hl ] ≤ k k i=1,...,m k kH⊗ h1,...,hXν=1,...,h Yl=1 h1+···+hν =h 2 c φ p th/2 A h (t1/2 B + A )(h+2) pe (t)hγh,p . ≤ h,pk kh{ | |h−1 | |h | |h } p By means of this bound, from (2.19) we get the desired estimate when setting C max c : h k . k,p ≥ { h,p ≤ } We need a last preliminary result on the inverse moments of the determi- nant of the Malliavin covariance matrix of Xt. This result is again achieved with some standard arguments, but, as in Lemma 2.1, the next lemma finds out the explicit constants appearing in the estimate of the Lp-norms of −1 det(σXt ) . Lemma 2.2. For every p> 1 and t> 0, (2.20) E[ det σ −p]1/p C eZ (t)Cp,m,d K (t, c ), | Xt | ≤ p,m,d 4(mp+1) m ∗ where 4 m 1 K (t, c )=1+ + 1 + (t1/2 B A 3 + A 2)2(m+1) m ∗ tc 2(m+1) k k0k k2 k k1 ∗ c∗ for some positive constant Cp,m,d depending on p,m and d but not on the bounds on B and A and their derivatives. The proof is once again postponed to Appendix A.1. We now come to the main result of this section. We give an estimate of 2 the L -norm of the random variables Hα involved in the integration by parts formula (2.8), when F = Xt. The proof follows the arguments of [6], proof of Lemma 4.11, but is given in the general setting of an integration by parts of order k N, and moreover it takes advantage of the explicit bounds which have been∈ obtained in Corollary 1 and Lemma 2.2. We give this result employing some slightly more compact notation, defining P (t)= t1/2 B + A , k | |k | |k P A(t)= A P (t). k | |k k+1
Theorem 2.3. For every k 1 there exists a positive constant Ck = C such that, for any multi-index≥ α 1,...,m k, any G D∞ and t> 0, k,m,d ∈ { } ∈ H (X , G) k α t k0,2 −k/2 k(k−1)/2 m k(k+3)/2 (2.21) C G k+1 (t t )(t K (t, c )) ≤ kk kk,2 ∨ m ∗ A φk Ck Z Z Ck (P ) (e (t) e k+2 (t)) (e (t) e (t)) , × k 8 ∨ 2 32m+4 ∨ 2k+4m+4 where Km(t, c∗) has been defined in Lemma 2.2, and 2 φk = 3m(k + 4) . 12 S. DE MARCO
Remark 2.1. Estimate (2.21) is rather involved. For our purposes, the most important elements are the dependence with respect to time of the −k/2 k(k−1)/2 A factor t t and the coefficient Pk containing the bounds on ∨ m the derivatives of the coefficients. We remark that the factor t Km(t, c∗) is bounded for t close to zero. Moreover, when t< 1, the factor t−k/2 tk(k−1)/2 reduces to t−k/2. ∨
Proof of Theorem 2.3. We write σt = σXt for simplicity of nota- tion. We first use the continuity of δ (see [6], Proposition 4.5) and H¨older’s inequalities for Sobolev norms to obtain H (X , G) k (α1,...,αk) t k0,2
= H(αk)(Xt,H(α1,...,αk−1)(Xt, G)) 0,2 (2.22) k k m −1 j = δ(H(α1,...,αk−1)(Xt, G)(σt )αk,jDXt )
j=1 0,2 X m
C H (X , G) (σ−1) DXj . ≤ mk (α1,...,αk−1) t k1,4 k t αk,jk1,8k t k1,8,H Xj=1 To estimate the last factor we can directly use the definition of k,p,H and apply Corollary 1. The major part of the efforts in the rest of thek · k proof will −1 be targeted on the estimation of (σt )i,j k,p. We claim that for any k 1, p>k 1 andk for all i, j = 1,...,m, ≥ −1 −1 k/2−1 m 1+k (σ )i,j k,p ck,p(t t )(t Km(t, c∗)) (2.23) k t k ≤ ∨ A φ′ +2(k+4)2 c Z c P (t) k e (t) k,p e (t) k,p , × k p 4(mp+1) where φ′ = 2(k + 1)(m 1), k − and ck,p is a positive constant depending also on m, d but not on t and on the bounds on B and A and their derivatives. Iterating process (2.22) and repeatedly using estimates (2.23) and (2.17), one easily obtains the desired estimate H (X , G) k (α1,...,αk) t k0,2 m k C G k+1 (t K (t, c )) ≤ k,m,dk kk,2 m ∗ k (t−1 th/2−1)(t th)1/2(tmK (t, c ))h × ∨ ∨ m ∗ hY=1 A k(φ′ +2(k+4)2+(k+1)2) P (t) k × k k ch,m,d Z ch,m,d e h+2 (t) e (t) × 2 4(2h+2m+1) hY=1 SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 13
m k(k+3)/2 C G k+1 (t K (t, c )) ≤ k,m,dk kk,2 m ∗ (t−k/2 tk(k−1)/2)P A(t)φk × ∨ k Ck,m,d Z Z Ck,m,d (e (t) e k+2 (t)) (e (t) e (t)) . × 8 ∨ 2 32m+4 ∨ 2k+4m+4 Proof of (2.23). We follow [6], proof of Lemma 4.11. We start from the definition of and write k · kk,p k 1/p −1 E −1 p E (h) −1 p (2.24) (σt )i,j k,p = [ (σt )i,j ]+ [ D (σt )i,j ⊗h ] . k k | | k kH ! Xh=1 For the first term, we simply use Cramer’s formula for matrix inversion, (σ−1) = (det σ )−1σ(i,j), | t i,j| t t (i,j) where σt denotes the (i, j) minor of σt. We then apply H¨older’s inequality and bounds (2.17) and (2.20) and get
E[ (σ−1) p] c(1) E[det(σ )−2p]E[ σ(i,j) −2p] 1/2 | t i,j| ≤ p,m{ t | t | } 1/2 (1) E −2p E i 4(m−1)p cp,m [det(σt) ] sup DXt H ≤ i k k n h io (2.25) c(1) t−p(tmK (t, c ))p ≤ p,m,d m ∗ A (t1/2 B + A )4 2(m−1)p × {| |0 | |1 | |1 } c(1) c(1) e (t) p,m,d eZ (t) p,m,d , × p 4(mp+1) where Km(t, c∗) is the constant defined in Lemma 2.2. To estimate the second term, as done in [6], proof of Lemma 4.11, we iterate the chain rule for D m D(σ−1) = (σ−1) D(σ ) (σ−1) . t i,j − t i,a t a,b t b,j a,bX=1 We take advantage of the notation introduced in the proof of Lemma 2.1 and for (β ,...,β ) 1,...,m k, k 1, we write 1 k ∈ { } ≥ Dβ1,...,βk (σ−1) | r1,...,rk t i,j| m (σ−1) (σ−1) ≤ | t i,a1 t b1,a2 · · · I1∪···∪IXν ={1,...,k} a1,...,aXν =1 b1,...,bν =1 (2.26) (σ−1) (σ−1) × t bν−1,aν t bν ,j| β(I1) β(Iν ) D (σt)a ,b D (σt)a ,b . × | r(I1) 1 1 · · · r(Iν ) ν ν | 14 S. DE MARCO
We repeatedly apply H¨older’s inequality for Sobolev norms to (2.26) and get E (k) −1 p [ D (σ )i,j k ] k t kH⊗ (2) E −1 ν+1 ck,p,m sup [ (σt )a,b ≤ a,a1,...,aν =1,...,m k k1,...,kν =1,...,k X n b,b1,...,bν =1,...,m k1+···+kν =k D(k1)(σ ) × t a1,b1 · · · (kν ) p (2.27) D (σt)a ,b k ] × ν ν kH⊗ o c(2) ≤ k,p,m E −1 (ν+1)p sup sup [ (σt )a,b ] × k1,...,kν =1,...,k(a,b=1,...,m | | k1+···+kν =k ν l l E (kl) 2 p 1/2 sup [ D (σt)a ,b k ] , l l H⊗ l × al,bl=1,...,m k k ) Yl=1 where, as in the proof of Corollary 1, we have written D(kl) for the generic (kl) derivative of order kl. To estimate D (σt)al,bl we use bound (2.17) and get E (k) p [ D (σt)i,j k ] k kH⊗ k p k t E D(h)D Xi D(k−h)D Xj ≤ h s t · s t " 0 ⊗k # h=0 Z H X k (2.28) c(3) E[ D(h)DXi 2p ]1/2 ≤ k,p k t kH⊗h+1 Xh=0 E[ D(k−h)DXj 2p ]1/2 × k t kH⊗k−h+1 (3) (k+2)p 2 c t(k/2+1)p A (t1/2 B + A )2(k+2) pe (t)2γk+1,p , ≤ k,p | |k | |k+1 | |k+1 p where we have once again applied H¨older’s inequality for Sobolev norms in the second step. Using (2.28) together with (2.27), bound (2.25) and (2.24) and observing m that t Km(t) is greater than one for all the values of t, we finally obtain (σ−1) c (t−1 tk/2−1)(tmK (t, c ))1+k k t i,jkk,p ≤ k,p ∨ m ∗ ′ φ +k(k+2) 1/2 φ′ +2(k+4)2 A k (t B + A ) k × | |k | |k+1 | |k+1 e (t)ck,p eZ (t)ck,p , × p 4(mp+1) for a positive constant ck,p depending also on m, d. Estimate (2.23) follows. SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 15
2.4. Proofs of Theorems 2.1 and 2.2. We now come to the proof of the results stated in Section 2.1. We recall that an Rm-valued random vector X m is said to admit a density on an open set A R if X A possesses a density, being the law of X. It is equivalent to∈ say thatL | LX (2.29) E[f(X)] = f(x)p(x) dx R Z holds for all f C (R) such that supp(f) A, for some positive p L1(A). ∈ b ⊂ ∈ We refer to the setting of Section 2.1. We recall that X = (Xt; t [0, T ]) denotes a strong solution of ∈ t d t i i i i j Xt = x + b (Xs) ds + σj(Xs) dWs , Z0 j=1 Z0 (2.30) X t [0, T ], i = 1,...,m, ∈ where b and σ satisfy the assumptions (H1)–(H3). For k 1 and f Ck(Rm), we denote ≥ ∈
(2.31) f k,BR(y0) =1+ sup ∂αf(x) , | | x∈BR(y0) | | |αX|≤k where the sum is taken over all the multi-index α 1,...,m k. Let us define the following “local” version of the constants appearing∈ { in the} estimates of the previous section: P (t,y )= t1/2 b + σ , k 0 | |k,B5R(y0) | |k,B5R(y0) P σ(t,y )= σ P (t,y ), k 0 | |k,B5R(y0) k+1 0 Z 1/2 2 P (t,y0)= t ( b + σ )+ σ , 1 | |1,B5R(y0) | |1,B5R(y0) | |1,B5R(y0) C 1/2 3 2 2(m+1) P (t,y0) = (t b σ + σ ) , m | |0,B5R(y0)| |2,B5R(y0) | |1,B5R(y0) m m 4 C Cm(t,y0)= t + 2(m+1) (1+ Pm (t,y0)), cy0 p/2 p ep(t,y0) = exp(t P1(t,y0) ), Z p/2 Z p ep (t,y0) = exp(t P1 (t,y0) ). In order to prove Theorem 2.1, we simplify this rather heavy notation in- troducing a constant that contains the factors appearing in estimate (2.21) in Theorem 2.3 (recall the constant φk defined there) 2 mk(mk+3)/2 σ φmk+(mk+2) Θk(t,y0, γ)= Cm(t,y0) Pmk(t,y0) γ (e (t,y ) e mk+2 (t,y )) × 8 0 ∨ 2 0 (eZ (t,y ) eZ (t,y ))γ. × 32m+4 0 ∨ 2mk+4m+4 0 16 S. DE MARCO
As addressed in Section 2.1, the following theorem is a more detailed version of Theorem 2.1. In particular, it provides the explicit expression of the constant Λk appearing in estimates (2.2) and (2.3).
Theorem 2.4. Assume (H1), (H2) and (H3). Then, for any initial con- m dition x R and any 0 m ihξ,yi Proposition 2.3. Let µ be a probability law on R , and µ(ξ)= R e µ(dy) its characteristic function. If µ is integrable, then µ is absolutely con-× R tinuous w.r.t. the Lebesgue measure, and b 1 b −ihξ,yi (2.35) p(y)= m e µ(ξ) dξ (2π) Rm Z is a continuous version of its density. If moreoverb (2.36) ξ k µ(ξ) dξ < Rm | | | | ∞ Z N ∞ holds for any k , then p is of classb C and for any multi-index α = (α ,...,α ) 1,...,m∈ k, 1 k ∈ { } k k αj −ihξ,yi ∂αp(y) = ( i) ξ e µ(ξ) dξ. − R ! Z jY=1 b SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 17 Proof of Theorem 2.4. Step 1 (“localized” characteristic function). Fix a t in (0, T ]. Let φ C∞(Rm) be such that 1 φ 1 R ∈ b BR(0) ≤ R ≤ B2R(0) and φ 2kR−k. We first observe that if m = E[φ (X y )] is zero, | R|k ≤ 0 R t − 0 then it just follows that p 0 is a density for Xt on BR(y0). Otherwise, we consider the law on R≡m such that Lt,y0 1 E (2.37) f(y) t,y0 (dy)= [f(Xt)φR(Xt y0)], Rm L m − Z 0 Rm ′ for all f Cb( ). If t,y0 possesses a density, say pt,y0 , it follows that ∈ ′ L pt,y0 (y) := m0pt,y0 is a density for Xt on BR(y0). Indeed, for any f Cb such that supp(f) B (y ), (2.37) implies ∈ ⊂ R 0 ′ f(y)pt,y0 (y) dy = f(y)m0pt,y0 (y) dy Rm Rm Z Z = m0 f(y) t,y0 (dy) Rm L Z = E[f(Xt)]. If the characteristic function of Lt,y0 1 ihξ,yi E ihξ,Xti pt,y0 (ξ)= e t,y0 (dy)= [e φR(Xt y0)] Rm L m − Z 0 is integrable, then by Proposition 2.3 admits a density. Hence, we focus b Lt,y0 on the integrability of pt,y0 ; in particular, we show that condition (2.36) of Proposition 2.3 holds true for all k N. ∈ Moreover, the inversionb formula (2.35) yields the representation for pt,y0 m p (y):= m p′ (y)= 0 e−ihξ,yip (ξ) dξ t,y0 0 t,y0 m t,y0 (2π) Rm (2.38) Z 1 −ihξ,yiE ihξ,Xti b = m e [e φR(Xt y0)] dξ. (2π) Rm − Z Step 2 (localization). We define the coefficients i i b (y)= b (ψ(y y0)), (2.39) − σi (y)= σi (ψ(y y )), j j − 0 where ψ C∞(Rm; Rm) (a truncation function) is defined by ∈ y, if y 4R, y | |≤ ψ(y)= 5 , if y 5R, ( y | |≥ | | m and ψ(y) B5R(0) for all y R . ψ can be defined in such a way that, for all i =∈ 1,...,m, ψi ∈1 and ψi 2k−2R−(k−1) for all k 2. As k k1 ≤ k kk ≤ ≥ 18 S. DE MARCO ∞ a consequence of (H1), the b and σ defined in this way are Cb -extensions of b B4R(y0) and σ B4R(y0). Furthermore, there exist constants c = ck,m such that| | i −(k−1) i b k ck,mR b k,B5R(y0), (2.40) | | ≤ | | σi c R−(k−1) σi | j|k ≤ k,m | j|k,B5R(y0) and by (H2), for any y B (y ) the matrix σ(y) is elliptic ∈ 3R 0 (2.41) σ σ∗(y) c I , y B (y ). ≥ y0,R m ∈ 3R 0 m For y R we denote by X(y) = (Xs(y); 0 s t) the unique strong solu- tion of∈ the equation ≤ ≤ s d s i i i i j Xs(y)= y + b (Xu(y)) du + σj(Xu(y)) dWu , Z0 j=1 Z0 (2.42) X 0 s t, i = 1,...,m. ≤ ≤ Let now 0 <δ νt,δ = inf s t δ : Xs B3R(y0) , (2.43) { ≥ − ∈ } τ = inf s ν : X / B (y ) t,δ { ≥ t,δ s ∈ 4R 0 } and inf ∅ = . Suppose that φ (X y ) > 0, so that X B (y ) and { } ∞ R t − 0 t ∈ 2R 0 ν We now claim that for all q> 0 the following estimate holds: P φR(Xt y0) > 0, sup Xs(Xν ) Xν R − 0≤s≤δ| − |≥ (2.44) −q q/2 q cq,mR δ P0(δ, y0) P inf Xs y0 3R , ≤ t−δ≤s≤t| − |≤ for some positive constant cq,m. Estimate (2.44) will be proved later on. On the other hand, E[eihξ,Xtiφ (X y )1 ] R t − 0 {φR(Xt−y0)>0,t−δ=ν k times k times such that α = km. Hence, recalling| {z } that|∂ {zeihξ,x}i = iξkeihξ,xi, | | xk E[eihξ,Xδ(y)iφ (X (y) y )] | R δ − 0 | 1 (2.46) E[∂ eihξ,Xδ (y)iφ (X (y) y )] ≤ m ξi k | α R δ − 0 | i=1| | 1 Q E[ H (X (y), φ (X (y) y )) ], ≤ m ξi k | α δ R δ − 0 | i=1 | | for any y B3R(y0). Q ∈ We need to separately estimate φ(Xδ(y) y0) |α|,2|α|+1 . By Corollary 1, this is given by k − k −mk 1/2 mk φ(Xδ(y) y0) mk+1 ck,mR (1 + δ ) σ k − kmk,2 ≤ | |mk−1,B5R(y0) 2 (mk+2) ck,m P (y , δ) e mk+1 (δ) × mk 0 2 2 −mk σ (mk+2) ck,m c R P (y , δ) e mk+1 (δ) ≤ k,m mk 0 2 for some positive constant ck,m. Then, from (2.44), (2.45), (2.46) and The- orem 2.3 it follows that (2.47) m p (ξ) C P (δ,t,y )I (ξ,δ,y ) 0| t,y0 |≤ k,q R 0 k,q 0 b 20 S. DE MARCO for some constant Ck,q depending also on m and d, with PR(δ,t,y0)= P inf Xs y0 3R t−δ≤s≤t| − |≤ and R−mk I (ξ,δ,y )= R−qδq/2P (δ, y )q + δ−mk/2Θ (δ, y ,C ). k,q 0 0 0 m ξi k k 0 k,q i=1 | | Estimate (2.47) holds simultaneously for any ξ Rm, 0 <δ PR(δ(ξ),t,y0) P inf Xs y0 3R = Pt(y0) ≤ t/2∨(t−1)≤s≤t| − |≤ and I (ξ, δ(ξ),y ) R−q( ξ −qa/2 (t 1)q/2)P (t 1,y )q k,q 0 ≤ k k ∧ ∧ 0 ∧ 0 (2.48) + R−mk( ξ −k(1−ma/2) ξ −k(t 1)−mk/2) k k ∨ k k ∧ Θ (t 1,y ,C ), × k ∧ 0 k,q since δ P0(δ, y0) and δ Θk(δ, y0,Ck,q) are increasing; hence P0(δ(ξ),y0) P (t 1→,y ) and the same→ holds for Θ . ≤ 0 ∧ 0 k We consider the leading terms determining the decay of Ik,q(ξ, δ(ξ),y0) with respect to ξ and impose qa ma (2.49) = k 1 . 2 − 2 qa ma k Setting a = 1/m, (2.49) yields q = mk, hence 2 = k(1 2 )= 2 . Therefore, we get the bound − −mk −k/2 mk/2 mk Ik,q∗ (ξ, δ(ξ),y0) R ( ξ (t 1) )P0(t 1,y0) k ≤ k k ∧ ∧ ∧ (2.50) + R−mk( ξ −k/2 (t 1)−mk/2 ξ −k) k k ∨ ∧ k k Θk(t 1,y0,Ck,q∗ ) × ∧ k with q∗ = mk. Estimate (2.50) holds for any k 1 and ξ = 0, and then it k ≥ 6 proves that the function pt,y0 (y) defined in (2.38) is in fact well defined and infinitely differentiable with respect to y. SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 21 Let us come to estimate (2.33). We take (2.38) and cut off the integration m i over a region I of finite Lebesgue measure on which ξ = i=1 ξ remains smaller than a given constant. That is, we write k k | | Q 1 −ihξ,yiE ihξ,Xti pt,y0 (y)= m e [e φR(Xt y0)] dξ (2π) Rm − Z 1 E[φ (X y )] dξ ≤ (2π)m R t − 0 ZI −ihξ,yi ihξ,Xti + e E[e φR(Xt y0)] dξ c − ZI 1 P( X y < 2R)λ (I) ≤ (2π)m | t − 0| m + C ∗ P (y ) I ∗ (ξ, δ(ξ),y ) dξ , k,qk t 0 k,qk 0 c ZI m where λm denotes the Lebesgue measure on R . As we have seen, the last term is such that I ∗ (ξ, δ(ξ),y ) dξ k,qk 0 c ZI −mk mk −k/2 R P0(t 1,y0) ξ dξ ≤ ∧ c | | ZI −mk −mk/2 −k + R Θ (t 1,y ,C ∗) (t 1) ξ dξ k 0 k,qk ∧ ∧ c −m | | ZI ∩{ξ : |ξ|<(t∧1) } + ξ −k/2 dξ . c −m | | ZI ∩{ξ : |ξ|≥(t∧1) } Now, since −k/2 −k/2 (1) ξ dξ ξ dξ = ck < c −m | | ≤ c | | ∞ ZI ∩{ξ : |ξ|≥(t∧1) } ZI and −mk/2 −k −mk/2 (2) (t 1) ξ dξ (t 1) ck < ∧ c −m | | ≤ ∧ ∞ ZI ∩{ξ : |ξ|<(t∧1) } hold for any k 3, we then take k = 3 and get the estimate ≥ p (y) C∗P (y )[1+R−3m(P (t 1,y )3m +(t 1)−3m/2Θ (t 1,y ,C ))] t,y0 ≤ t 0 0 ∧ 0 ∧ 3 ∧ 0 m,d ∗ for every y BR(y0), for a positive constant C , estimate (2.33) then follows. For estimate∈ (2.34) on the derivatives we proceed in the same way, observing l −k/2 l αj that for α 1,...,m , ξ j=1 ξ is integrable at as soon as k 2l + 3. ∈ { } | | × | | ∞ ≥ Q 22 S. DE MARCO Proof of (2.44). We remark that φR(Xt y0) t δ ν t t δ ν t, X B (y ) , hence { − } ⊆ { − ≤ ≤ }⊆ { − ≤ ≤ ν ∈ 3R 0 } P φR(XT y0) > 0, sup Xs(Xν ) Xν R − 0≤s≤δ| − |≥ P t δ ν t, Xν B3R(y0), sup Xs(Xν ) Xν R ≤ − ≤ ≤ ∈ 0≤s≤δ| − |≥ −q q R P(t δ ν t) sup E sup Xs(y) y . ≤ − ≤ ≤ y∈B (y ) 0≤s≤δ| − | 3R 0 h i Using boundedness of coefficients of (2.42), it is easy to show that q q/2 q E sup Xs(y) y cq,mδ P0(δ, y0) 0≤s≤δ| − | ≤ h i for some positive constant cq,m. As addressed in Section 2.1, the constants appearing in the definition of ′ Λk (2.32) can be considerably simplified under assumptions (H1 ) and (H4), resulting in some polynomial-type bounds. The following result corresponds to Theorem 2.2; in the presents statement, we explicitly give the expression ′ of the exponent qk(q) appearing in bound (2.6). Theorem 2.5. Assume (H1′) and (H3). m (a) For any initial condition x R and any 0 (c) If moreover sup0≤s≤t Xs has finite moments of all orders, then for every p> 0 and every k 1| there| exist positive constants C such that ≥ k,p,T 1 −p pt(y) C3,p,T 1+ y , | |≤ tm3/2 | | (2.52) 1 −p k ∂αpt(y) C 1+ y , α 1,...,m , | |≤ k,p,T tm(2k+3)/2 | | ∈ { } for every 0 Proof. (a) We no longer need to distinguish between y0 and the (close) point y where the density is evaluated; hence we just set y = y0 and consider SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 23 1 ′ suitable radii. For y > η, we set Ry = 10 dist(y, Bη(0)) 1. By (H1 ), b and σ ∞ | | ∧ are of class Cb on B5Ry (y) and satisfy (H2) on B3Ry (y). From Theorem 2.4 it follows that Xt admits a smooth density on BRy (y). This holds true for Rm every ball BRy (y) with center y in Bη(0); hence statement (a) follows. (b) Without loss of generality, we take\ R = 1. As a consequence of (2.4), the constants introduced before Theorem 2.4 can be bounded as follows, for 0 t T and y > η + 5: ≤ ≤ | | P (t,y) c(1)(1+( y + 5)q) c(1) y q, k ≤ k | | ≤ k | | P σ(t,y) P Z (t,y) c(1) y 2q, k ∨ 1 ≤ k | | (2.53) P C (t,y) c(1)( y 4 + y 2)2(m+1) c(1) y 8(m+1), m ≤ | | | | ≤ | | c(1) C (t,y) y 2q(m+1) y 8(m+1) m ≤ 2(m+1) | | | | C0 c(1) y 2(q+4)(m+1), ≤ | | (1) (1) for some constants c and ck depending also on m,q and on the bounds on b, σ and their derivatives in (2.4) and (2.5). The exponential factors e and eZ must be treated on a specific basis. Z Indeed, e·(t,y) and e· (t,y) may explode when y + . Nevertheless, ex- plosion can be avoided stepping further into the| optimizati|→ ∞ on procedure set up in the proof of Theorem 2.4. More precisely, we restart from step 4 and force the state variable y to appear in the choice of δ, setting δ(ξ,y)= t/2 1 ξ −a y −4q. ∧ ∧ | | ∧ | | Z Now, whatever the value of p is, ep and ep are reduced to e (δ(ξ,y)) e (1 y −4q,y) p ≤ p ∧ | | exp (c(1)(1 y −2qp)(1 + 1 y −2q)p y qp) ≤ 1 ∧ | | ∧ | | | | exp (c (1 y −2qp) y qp) ≤ p ∧ | | | | exp (c ) ≤ p and eZ (δ(ξ,y)) eZ (1 y −4,y) p ≤ p ∧ | | = exp (c(1)(1 y −2qp)(1 + 1 y −2q)p y 2qp) 1 ∧ | | ∧ | | | | exp (c ). ≤ p We then perform the integration over ξ as done in the last step of the proof of Theorem 2.4, and employing (2.53) we obtain estimate (2.51) for Λk, ′ for y > η +5. The value of qk(q) is obtained from the definition of Θk and| (2.53| ). 24 S. DE MARCO (c) From boundedness of moments of sups≤t Xs , for any interval It [0,t] one can easily deduce the estimate | | ⊆ P(inf X y : s I 3) {| s − | ∈ t}≤ (2.54) P(sup X : s I y 3) ≤ {| s| ∈ t} ≥ | |− 1 E r (2) 1 r sup Xs cr r , ≤ ( y 3) s≤t | | ≤ y | |− h i | | for any r> 0, 0 t T and y > 3. It is then easy to obtain the desired ≤ ≤ | | estimate on pt with Theorem 2.4: for a given p> 0, we employ (2.33) with ′ y0 = y and (2.54) with r>p + q3(q). Similarly, to obtain the estimate on ′ derivatives, one employs (2.34) and (2.54) with r>p + q2k+3(q). 3. A square root-like (CIR/CEV) process with local coefficients. We ap- ply our results to the solution of (1.1). We will be able to refine the polyno- mial estimate on the density at + giving exponential-type upper bounds. Under some additional assumptions∞ on the coefficients, we also study the asymptotic behavior of the density at zero, that is, the point where the diffusion coefficient is singular. We first collect some basic facts concerning the solution of (1.1). Let us recall the SDE dX = (a(X ) b(X )X ) dt + γ(X )Xα dW , t 0, α [1/2, 1), (3.1) t t t t t t t X = x 0. − ≥ ∈ 0 ≥ When α = 1/2 and a, b and γ are constant, the solution to (3.1) is the cele- brated Cox–Ingersoll–Ross process (see [5]), appearing in finance as a model for short interest rates. It is well known that, in spite of the lack of globally Lipschitz-continuous coefficients, existence and uniqueness of strong solu- tions hold for the equation of a CIR process. If a 0, the solution stays ≥ a.s. in R+ = [0, + ); furthermore, a solution starting at x> 0 stays a.s. in ∞ 2 R> = (0, + ) if the Feller condition 2a γ is achieved (cf. [14] for details). The following∞ proposition gives the (straightforward)≥ generalization of the previous statements to the case of coefficients a, b, γ that are functions of the underlying process. The proof is left to Appendix A.2. Proposition 3.1. Assume: 1 2 (s0) α [1/2, 1); a, b and γ Cb with a(0) 0 and γ(x) > 0 for every x>∈ 0. ∈ ≥ Then, for any initial condition x 0 there exists a unique strong solution ≥ to (3.1) which is such that P(Xt 0; t 0) = 1. Let then x> 0 and τ0 = inf t 0 : X = 0 , with inf ∅ = ≥. ≥ { ≥ t } { } ∞ SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 25 If α> 1/2 and • (s1)′ a(0) > 0 and z 1 is integrable at 0+, 7→ γ2(z)z2α−1 then (3.2) P(τ = ) = 1. 0 ∞ If α = 1/2 and • 1 (s1) γ2 is integrable at zero, (s2) there exists x> 0 such that 2a(x) 1 for 0 When X is a CIR process, the moment-generating function of Xt can be computed explicitly, leading to the knowledge of the density. Setting −bt 2 Lt = (1 e )γ /4b, then Xt/Lt follows a noncentral chi-square law with δ = 4a/γ−2 degrees of freedom and parameter ζ = 4xb/(γ2(ebt 1)) (recall t − that x is here the initial condition). The density of Xt is then given by (cf. [14]) −ζt/2 δ/2−1 ∞ n e −y/(2Lt) y (y/(4Lt)) n pt(y)= e ζ , y> 0. 2δ/2L L n!Γ(δ/2+ n) t t t n=0 X δ/2−1 We incidentally remark that pt is in general unbounded, since y diverges at zero when δ/2 1 = 2a/γ2 1 is negative (in fact, fixed a value of δ/2 1, n there exists a n −0 such that− d p is unbounded). − ≥ dyn t The standard techniques of Malliavin calculus cannot be directly applied to study the existence of a smooth density for the solution of (3.1), as the diffusion coefficient in general is not (depending on γ) globally Lipschitz con- tinuous. Actually, Alos and Ewald [1] have shown that if X is CIR process, 1,2 2 then Xt, t> 0, belongs to D when the Feller condition 2a γ is achieved. Higher order of differentiability (in the Malliavin sense) ca≥n be proven, re- quiring a stronger condition on a and γ, and the authors apply these results to option pricing within the Heston model. If we are interested in density estimation, the results of the previous sections allow us to overcome the problems related to the singular behavior of the diffusion coefficient and to directly establish the existence of a smooth density, independently from any Feller-type condition [provided that (s0) is satisfied]. More precisely, we can give the following preliminary result: Proposition 3.2 (Preliminary result). Assume (s0) and let a, b, γ be of class C∞. Let X = (X ; t 0) be the strong solution of (3.1) starting at b t ≥ x 0. For any t> 0, Xt admits a smooth density pt on (0, + ). pt is such ≥ p ∞ that limy→∞ pt(y)y = 0 for any p> 0. Proof. It is easy so see that, under the current assumptions, the drift and diffusions coefficients of (3.1) satisfy (H1′) with η = 0 and (H4) with 26 S. DE MARCO q = 1. (H3) holds as well, by Proposition 3.1. As the coefficients have sub- linear growth, for any t> 0, sups≤t Xs has finite moments of any order. The conclusion follows from Theorem 2.5(c). 3.1. Exponential decay at . In order to further develop our study of the density, we could take advantage∞ of some of the generalized-chaining tools settled by Viens and Vizcarra in [17]. In particular notice that, in order to estimate the density by means of Theorem 2.5, we need to deal with the probability term Pt(y) appearing therein. For our present purposes, we can rely on alternative strategies involving time-change arguments and the existence of quadratic exponential moments for suprema of Brownian motions (Fernique’s theorem) in the current section, and a detailed analysis of negative moments of the process X in Section 3.2. From now on, condition (s0) is assumed, the coefficients a, b, γ are of ∞ class Cb and (Xt; t T ) denotes the unique strong of (3.1) on [0, T ], T > 0. We make explicit the≤ dependence with respect to the initial condition de- noting pt(x, ) the density at time t of X starting at x 0. The following result improves· Proposition 3.2 in the estimate of the density≥ for y . →∞ Proposition 3.3. Assume that (3.3) lim b(x)x1−α > . x→∞ −∞ Then there exist positive constants γ and C (T ), k 3, such that 0 k ≥ 1 (y x)2(1−α) (3.4) pt(x,y) C3(T ) 1+ exp γ0 − ≤ t3/2 − 2Ct and 2(1−α) (k) 1 (y x) (3.5) p (x,y) Ck(T ) 1+ exp γ0 − t ≤ t(2k+3)/2 − 2Ct 3−2α 2 2 for every y>x + 1, with C = 2 + 2 γ 0(1 α) . The Ck(T ) also depend on α and on the coefficients a, b and γ.| | − Remark 3.1. In the case of constant coefficients and a = b = 0, the bound (3.4) can be compared to the density of the CEV process as pro- vided, for example, in [7], Theorem 1.6 (see also the references therein). The comparison shows that our estimate is in the good range on the log-scale. Proof of Proposition 3.3. In the spirit of Lamperti’s change-of-scale argument (cf. [12], page 294), let ϕ C2((0, )) be defined by ∈ ∞ x 1 1 ϕ(x)= dy = x1−α, γ yα γ (1 α) Z0 | |0 | |0 − SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 27 ′ 1 ∞ R so that ϕ (x)= α . Let moreover θ C ( ) be such that 1[2,∞) θ 1[1,∞) |γ|0x ∈ b ≤ ≤ and θ′ 1. We set ≤ θ(x)ϕ(x), x> 0, ρ(x)= 0, x 0, ≤ so that ρ is of class C2(R). We define the auxiliary process Y = X x, t t − t 0, which is such that P(Yt x)=1, t 0. An application of Itˆo’s formula≥ yields ≥− ≥ t ρ(Yt)= f(Ys) ds + Mt, Z0 where f(y)= ρ′(y)(a(y) b(y)y)+ 1 ρ′′(y)γ(y)2y2α − 2 and t ′ α Mt = ρ (Ys)γ(Ys)Ys dWs. Z0 The key point is the fact that f is bounded from above on ( x, ) and, on the other hand, M is a martingale with bounded quadratic− variation.∞ Indeed, f is continuous, it is zero for y 1 and for y> 2 one has ≤ a(y) b(y) α γ(y)2 f(y)= y1−α yα−1, γ yα − γ − 2 γ | |0 | |0 | |0 and hence, recalling that a is bounded, lim f(y) < is ensured by y→∞ ∞ condition (3.3). Then we set C1 = supy≥0 f(y). For M, one has t M = ρ′(Y )2γ(Y )2Y 2α ds h it s s s Z0 t (3.6) (ϕ(2) + ϕ′(Y ))2γ(Y )2Y 2α ds ≤ s s s Z0 2(ϕ(2)2 + 1)t, ≤ 2 22(1−α) and hence we set C2 = 2(ϕ(2) +1) = 2( 2 2 +1). Now, since ρ is strictly |γ|0(1−α) increasing, Y >y = ρ(Y ) > ρ(y) for any y > 0. Moreover, ρ(Y ) > { t } { t } { t ρ(y) M + C t > ρ(y) 2M 2 + 2C2t2 > ρ(y)2 . We set I = [(t } ⊆ { t 1 } ⊆ { t 1 } t − 1) t/2,t] and τ = inf s 0 : Ys 3/2 . The quadratic variation of M is ∨ { ≥ ≥ } α 2 strictly increasing after τ, since (ρ(Yτ∨t)γ(Yτ∨t)Yτ∨t) > 0: hence, by Du- bins and Schwarz’s theorem (cf. Theorem 3.4.6 in [12]) there exists a one- dimensional Brownian motion (b ; t 0) such that M = b . Clearly t ≥ τ∨t hMiτ∨t 28 S. DE MARCO one has Y > 2 τ 3.2. Asymptotics at 0. We have established conditions under which the solution of (3.1) admits a smooth density p on (0, + ). According to Propo- t ∞ sition 3.1, the process remains almost surely in R+: this trivially means that for any t> 0, Xt has an identically zero density on ( , 0), which can be extended to 0 when τ = a.s. We are now wondering−∞ what are sufficient 0 ∞ conditions for pt to converge to zero at the origin, hence providing the exis- tence of a continuous (eventually differentiable, eventually C∞) density on the whole real line. What we have in mind is the application of Theorem 2.5 to the inversed process Y = 1 (considered on the event τ = ). An application of Itˆo’s t Xt 0 formula yields { ∞} (3.7) dY = J (Y ) dt γ(Y )Y 2−α dW , t α t − t t t b SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 29 where J (Y )= a(Y )Y 2 + b(Y )Y + γ(Y )2Y 3−2α α t − t t t t t t with the notation f(y)= f(1/y), y> 0, for f = a, b, σ. Equation (3.7) has b super-linear coefficients, inb particular conditionb (2.4) of Theorem 2.2 holds with q = 2. Willingb to apply Theorem 2.5(c), we first need some preliminary results on the moments of Y . The proof of the next statement is based on the techniques employed in [3], proof of Lemma 2.1, that we adapt to our framework. Lemma 3.1. (1) If α> 1/2, assume (s1)′. Then for any initial condition x> 0, for any t> 0 and p> 0 1 (3.8) E sup C, Xp ≤ s≤t s for some positive constant C depending on x,p,α,t and on the coeffi- cients of (3.1). (2) If α = 1/2, then assume (s1) and (s2) and let ∗ 2a(x) (3.9) l = lim 2 > 1. x→0 γ(x) Then (3.8) holds for p> 0 such that (3.10) p + 1 < l∗. Proof. Let τn be the stopping time defined by τn = inf t 0 : Xt 1/n . The application of Itˆo’s formula to Xp , p> 0, yields { ≥ ≤ } t∧τn 1 1 t∧τn (3.11) E = + E ϕ(X ) ds, Xp xp s t∧τn Z0 where b(x) p p + 1 ϕ(x)= p + γ(x)2x2α−1 a(x) , x> 0. xp xp+1 2 − g(x) It is easy to see that, if | {z } (3.12) lim g(x) < 0, x→0 |b|0 there exists a positive constant C such that ϕ(x) 0. If (3.12) holds, from (3.11) we get 1 1 t 1 E sup + Ct + p b E sup ds, Xp ≤ xp | |0 Xp s≤t s∧τn Z0 u≤s u∧τn 30 S. DE MARCO and hence, by Gronwall’s lemma, 1 1 (3.13) E sup + Ct ep|b|0t. Xp ≤ xp s≤t s∧τn We verify (3.12), distinguishing the two cases. Case α> 1/2. We simply observe that limx→0 g(x)= a(0) < 0. Esti- mate (3.8) then follows by taking the limit n in (3.13) and− using Propo- sition 3.1 under assumption (s1)′. →∞ Case α = 1/2. We have g(x) = p+1 γ2(x) a(x). If p satisfies (3.10), 2 − then (3.9) ensures that limx→0 g(x) < 0. We conclude again taking the limit n and using Proposition 3.1 under assumptions (s1) and (s2). →∞ We are now provided with the tools to prove the following: Proposition 3.4. (1) If α> 1/2, assume (s1)′. Then for every t> 0, every p 0 and every k> 0 the density p of X on (0, + ) is such that ≥ t t ∞ −p lim y pt(y) = 0, y→0+ | | (3.14) −p (k) lim y pt (y) = 0. y→0+ | | (2) If α = 1/2, then assume (s1) and (s2) and define l∗ as in Lemma 3.1. If ∗ ′ (3.15) l > 3+ q3(2) [where q′( ) has been defined in Theorem 2.5], then · · −p (3.16) lim y pt(y) = 0 y→0+ for every 0 p < l∗ (3 + q′ (2)). Moreover, if ≤ − 3 ∗ ′ (3.17) l > 2k +3+ q2k+3(2), then −p (k) (3.18) lim y pt (y) = 0 y→0+ | | for every 0 p < l∗ (2k +3+ q′ (2)). ≤ − 2k+3 Proof. We apply Theorem 2.2 to Y = 1/X. For simplicity of notation, we write p for pt and pY for the density of Yt. As Y satisfies (3.7), from ′ Theorem 2.2(b) it follows that the bound (2.51) on Λk holds with qk(q)= ′ qk(2). Hence, from Theorem 2.2(c) it follows that ′ ′ p′ (3.19) lim pY (y ) y = 0 y′→+∞ | | SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 31 and, for a given k 0, ≥ ′ (3.20) lim p(k)(y′) y′ p = 0, y′→+∞ Y | | ′ ′ if sups≤t Ys has a finite moment of order r>p + q2k+3(2). Now, it is easy to see that 1 1 (3.21) p(y)= p , y2 Y y and hence, after some rather straightforward computations, k j 1 2(k+1) dν 1 (3.22) p(k)(y) C p , 0 Remark 3.2. Proposition 3.3 states that pt decays exponentially at infinity for any value of α, as far as condition (3.3) holds true. When α> 1/2, Proposition 3.4 states that pt and all its derivatives tend to zero at the origin, while the price to pay for the same conclusion to hold is higher when α = 1/2 [cf. conditions (3.15) and (3.17), which become rapidly strong for growing values of k]. With regard to this behavior at zero, we recall that Proposi- tion 3.4 only provides sufficient conditions for estimates (3.16) and (3.18) to hold. We do not give any conclusion on the behavior of the density at zero when condition (3.15) [or (3.17) for the derivatives] fail to hold. APPENDIX We collect here the proofs of some of the more technical results. A.1. Proofs of Lemmas 2.1 and 2.2. Proof of Lemma 2.1. We refer to the notation introduced in the proof of [16], Theorem 2.2.2, allowing us to write the equation satisfied by the kth Malliavin derivative in a compact form. This is stated as follows: for any subset K = h ,...,h of 1,...,k , one sets j(K)= j ,...,j and { 1 η} { } h1 hη 32 S. DE MARCO r(K)= rh1 , . . . , rhη . Then, one defines i j1,...,jk i αl,j1,...,jk (s, r1, . . . , rk):= Dr1,...,rk Al(Xs) = ∂ ∂ Ai(X ) k1 · · · kν l s X j(I1) j(Iν ) D Xk1 D Xkν × r(I1) s · · · r(Iν ) s and i j1,...,jk i βj1,...,jk (s, r1, . . . , rk):= Dr1,...,rk B (Xs) = ∂ ∂ Bi(X ) k1 · · · kν s X j(I1) j(Iν ) D Xk1 D Xkν , × r(I1) s · · · r(Iν ) s where in both cases the sum is extended to the set of all partitions of 1,...,k = I I . Finally, one sets αi (s)= Ai (X ). Making use of { } 1 ∪···∪ ν j j s this notation, it is shown that the equation satisfied by the kth derivative reads as k j1,...,jk i i Dr1,...,rk Xt = αjε,j1,...,jε−1,jε+1,...,jk (rε, r1, . . ., rε−1, rε+1, . . ., rk) ε=1 X t i (A.1) + (βj1,...,jk (s, r1, . . . , rk) ds Zr1∨···∨rk i l + αl,j1,...,jk (s, r1, . . ., rk) dWs), j1,...,jk i if t r r , and Dr ,...,r X = 0 otherwise. We prove (2.16) by in- ≥ 1 ∨···∨ k 1 k t duction. The estimate is true for k = 1, with γ1,p = 2C1,p: this simply fol- lows with an application of Burkholder’s inequality and Gronwall’s lemma to (A.1) taken for k =1. Let us suppose that (2.16) is true up to k 1. As done for k = 1, we apply Burkholder’s inequality to (A.1), and, setting− r = r r , we get 1 ∨···∨ k E[ Dj1,...,jk Xi p] | r1,...,rk t | k i p C E[ α (rε, r1, . . . , rε−1, rε+1, . . . , r ) ] ≤ k,p | jε,j1,...,jε−1,jε+1,...,jk k | (ε=1 X + (t r)p/2−1 − t E 1/2 i (t r) ∂k1 ∂kν B (Xs) × r " − | · · · | I1∪···∪Iν Z card(XI)≤k−1 SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 33 d p i (A.2) + ∂k1 ∂kν Al(Xs) | · · · |! Xl=1 j(I1) k1 j(Iν ) kν p Dr(I )Xs Dr(I )Xs ds × | 1 · · · ν | # t + (t r)p/2−1 E (t r)1/2 ∂ Bi(X ) − − | k s | Zr " d p i + ∂kAl(Xs) | |! Xl=1 j1,...,jk k p Dr1,...,rk Xs ds , × | | # ) j1,...,jk where, in the last line, we have isolated the term depending on Dr1,...,rk X. To estimate the second term in (A.2) we notice that, for any partition I1 Iν of 1,...,k such that card(I) k 1, using (2.16) up to order k ∪···∪1 we have { } ≤ − − E 1/2 i (t r) ∂k1 ∂kν B (Xs) " − | · · · | d p i j(I1) k1 j(Iν ) kν p (A.3) + ∂k1 ∂kν Al(Xs) Dr(I )Xs Dr(I )Xs | · · · |! | 1 · · · ν | # Xl=1 λ(1) C A k (t1/2 B + A )χk pe (t) k,p , ≤ {| |k−2 | |k | |k } p where we have defined (1) λk,p := sup γcard(I1),p + + γcard(Iν ),p I1∪···∪Iν ={1,...,k}{ · · · } card(I)≤k−1 and ν 2 χk =1+ (card(Il) + 1) . Xl=1 It is easy to see that χ (k + 1)2, k ≤ since ν ν ν card(I )2 card(I ) card(I )2 = (k 1)2 l (k 1)2 l = (k 1)k, l − (k 1)2 ≤ − (k 1) − Xl=1 Xl=1 − Xl=1 − 34 S. DE MARCO so that ν ν 2 χk =1+ card(Il) + 2 card(Il)+ ν Xl=1 Xl=1 1 + (k 1)k + 2k + ν ≤ − 1+ k2 k + 3k = (k + 1)2. ≤ − To estimate the first term in (A.2), notice that we have as well E i p [ αjε,j1,...,jε−1,jε+1,...,jk (rε, r1, . . . , rε−1, rε+1, . . . , rk) ] (A.4) | | 2 λ(2) C A k (r1/2 B + A )k pe (t) k,p , ≤ {| |k−1 ε | |k−1 | |k−1 } p with (2) λk,p := sup γcard(I1),p + + γcard(Iν ),p . I1∪···∪Iν ={1,...,k−1}{ · · · } (1) (2) We remark that λk and λk are defined by means of the γ’s up to order k 1. − Collecting (A.2), (A.3) and (A.4), we get E[ Dj1,...,jk Xi p] | r1,...,rk t | 2 (2) kp 1/2 k p λk,p Ck,p A k−1(t B k−1 + A k−1) ep(t) ≤ (| | | | | | 2 λ(1) + (t r)p/2 A k (t1/2 B + A )(k+1) pe (t) k,p − | |k−2 | |k | |k p m t p/2−1 1/2 p E j1,...,jk i p + (t r) (t B 1 + A 1) Dr1,...,rk Xs ds − | | | | r | | ) Xk=1 Z kp 2 λ(1) ∨λ(2) C A (t1/2 B + A )(k+1) pe (t) k,p k,p ≤ k,p| |k−1 | |k | |k p (1 + C tp/2(t1/2 B + A )pe (t)Ck,p ) × k,p | |1 | |1 p kp 2 λ(1) ∨λ(2) +2C C A (t1/2 B + A )(k+1) pe (t) k,p k,p k,p , ≤ k,p| |k−1 | |k | |k p where we have applied Gronwall’s lemma to get the second inequality. The constant Ck,p may vary from line to line, but never depends on t nor on the bounds on B and A. We recursively define γ by setting γ := λ(1) k,p k,p k,p ∨ (2) λk,p + 2Ck,p, and we finally obtain (2.16). SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 35 Proof of Lemma 2.2. Step 1. We first use the decomposition DsXt = YtZsA(Xs) (see, e.g., [16]) and write t ∗ ∗ ∗ σXt = Yt ZsA(Xs)A(Xs) Zs ds Yt 0 (A.5) Z ∗ = YtUtYt , t ∗ ∗ where we have set Ut = 0 ZsA(Xs)A(Xs) Zs ds. Notice that Ut is a positive operator, and that for any ξ Rm we have R ∈ t d t ∗ ∗ ∗ ∗ 2 ξ, Utξ = A(Xs) Zs ξ, A(Xs) Zs ξ ds = ZsAj(Xs),ξ . h i 0 h i 0 h i Z Xj=1 Z 2 −2 From identity (A.5) it follows that det σXt = (det Yt) det Ut = (det Zt) det Ut. Hence, applying H¨older’s inequality, E −p E 4p E −2p 1/2 [ det σXt ] ( [ det Zt ] [(det Ut) ]) (A.6) | | ≤ | | C (eZ (t)mE[(det U )−2p])1/2, ≤ p,m 4p t where in the last step we have used bound (2.15) on the entries of Zt. Step 2. Let λ = inf ξ, U ξ be the smallest eigenvalue of U , so that t |ξ|=1h t i t E −2p E −2mp P [(det Ut) ] [λt ]. We evaluate (λt ε). For any ξ such≤ that ξ = 1, using the elementary≤ inequality (a + b)2 a2/2 b2 we get | | ≥ − d d d 1 Z A (X ),ξ 2 A (x),ξ 2 Z A (X ) A (x),ξ 2 h s j s i ≥ 2 h j i − h s j s − j i Xj=1 Xj=1 Xj=1 d 1 c Z A (X ) A (x) 2, ≥ 2 ∗ − | s j s − j | Xj=1 where in the last step we have used the ellipticity assumption (E). For any ε> 0 and a> 0 such that aε < t, the previous inequality gives d 1 2 P(λt ε) P ac∗ε sup aε ZsAj(Xs) Aj(x) ε , ≤ ≤ 2 − s≤aε( | − | ) ≤ ! Xj=1 and thus, if we take a = 4/c∗ in order to have ac∗/2 = 2 and apply Markov’s inequality, we obtain d c P(λ ε) P sup Z A (X ) A (x) 2 ∗ t ≤ ≤ | s j s − j | ≥ 4 s≤aε(j=1 ) ! (A.7) X q d q−1 4 E 2q d q sup ZsAj(Xs) Aj(x) , ≤ c∗ s≤aε| − | Xj=1 h i 36 S. DE MARCO where the last holds for all q> 1. Now, to estimate the last term we claim that, for all j = 1,...,d, E 2q q 1/2 3 2 2q Z C (A.8) sup ZsAj(Xs) Aj(x) Ct (t B 0 A 2 + A 1) e2q(t) , s≤t | − | ≤ | | | | | | h i for a constant C depending on q,m,d but not on the bounds on B and A. From (A.7) and this last estimate, it follows that εq P 1/2 3 2 2q Z Cq,m,d (λt ε) Cq,m,d 2q (t B 0 A 2 + A 1) e2q(t) , ≤ ≤ c∗ | | | | | | for any ε such that 4ε/c∗ < 1 t. ∧E −2mp Step 3. We finally estimate [λt ]. We write ∞ E −2mp E −2mp E −2mp [λt ]= [λt 1{λt>1}]+ [λt 1{1/(k+1)<λt≤1/k}] k=1 ∞ X 1+ (k + 1)2mpP(1/(k + 1) < λ 1/k), ≤ t ≤ Xk=1 and separate the contribution of the sum over k> 4 to obtain tc∗ E[λ−2mp] 1+ (k + 1)2mpP(1/(k + 1) < λ 1/k) t ≤ t ≤ 1≤k≤X4/(tc∗) + (k + 1)2mpP(λ 1/k) t ≤ k>X4/(tc∗) 4 2mp 1+ + 1 P(λ 1) ≤ tc t ≤ ∗ 1 Z Cq,m,d 1/2 3 2 2q + Cq,m,de2q(t) 2q (t B 0 A 2 + A 1) c∗ | | | | | | 1 (k + 1)2mp . × kq k>X4/(tc∗) We finally take q = 2mp + 2 in order to get convergent series. This last estimate, together with (A.6), gives the desired result. Proof of (A.8). We apply Itˆo’s formula to the product ZtAj(Xt) and get d d(ZtAj(Xt)) = Zt (∂AjB ∂BA)+ ∂Al(∂AlAj ∂AjAl) (Xt) dt ( − − ) Xl=1 1 (A.9) + Z ∂ ∂ A Ak1 Ak2 (X ) dt t 2 k1 k2 j l l t d + Z (∂A A ∂A A )(X ) dW l(t). t j l − l j t Xl=1 SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 37 Hence, by Burkholder’s inequality, i 2q sup E sup (ZsAj(Xs) Aj(x)) i=1,...,m s≤t | − | h i C t2q−1( B A + B A + A 2 A + A A 2)2q ≤ { | |0| |1 | |1| |0 | |1| |0 | |2| |0 + tq−1( A A )2q | |1| |0 } t 2q E sup (Zu)h,k du × 0 h,k=1,...,m | | Z h i Ctq(t1/2 B A 3 + A 2)2qeZ (t)C , ≤ | |0| |2 | |1 2q where the constant C depends on q,m and d, but not on t and on the bounds on B and A and their derivatives. In the last step, we have once again used bound (2.15) on the entries of Z. A.2. Proof of Proposition 3.1. We first collect the basic facts we need to give the proof of Proposition 3.1. We will start by proving existence and uniqueness of strong solutions for the following equation: t Xt = x + (a(Xs) b(Xs)Xs) ds 0 − (A.10) Z t + γ(X ) X α dW , t 0, α [1/2, 1), s | s| s ≥ ∈ Z0 whose coefficients are defined on the whole real line [a, b and γ are the functions appearing in (3.1)]. Once we have established that the unique strong solution of (A.10) is a.s. positive, then (A.10) will coincide with the original equation (3.1). The proof of Proposition 3.1 is split in the following two short lemmas. Lemma A.1. Assume condition (s0) of Proposition 3.1. Then existence and uniqueness of strong solutions hold for (A.10). Moreover, for any initial condition x 0 the solution is a.s. positive, P(X 0; t 0) = 1. ≥ t ≥ ≥ Proof. Existence of nonexplosive weak solutions for (A.10) follows from continuity and sub-linear growth of drift and diffusion coefficients. The ex- istence of weak solutions together with pathwise uniqueness imply the exis- tence of strong solutions (cf. [12], Proposition 5.3.20 and Corollary 5.3.23). Pathwise uniqueness follows in its turn from a well-known theorem of unique- ness of Yamada and Watanabe (cf. [12], Proposition 5.2.13). Indeed, as 1 a, b, γ Cb the diffusion coefficient of (A.10) is locally H¨older-continuous of exponent∈ α 1/2 and the drift coefficient is locally Lipschitz-continuous. ≥ 38 S. DE MARCO We apply the standard localization argument for locally Lipschitz coeffi- cients and the Yamada–Watanabe theorem to establish that solutions are pathwise unique up to their exit time from a compact ball, and hence path- wise uniqueness holds for (A.10). Lemma A.2 deals with the second part of Proposition 3.1, that is, the behavior at zero. The proof is based on Feller’s test for explosions of solutions of one-dimensional SDEs (cf. [12], Theorem 5.5.29). Letting τ denote the exit time from (0, ), that is, τ = inf t 0 : Xt / (0, ) with inf ∅ = , we have to verify that∞ { ≥ ∈ ∞ } ∞ (A.11) lim pc(x)= x→0 −∞ with pc defined by x y a(z) b(z)z (A.12) p (x) := exp 2 − dz dy, x> 0, c − γ(z)2z2α Zc Zc for a fixed c> 0. Property (A.11) implies that P(τ = )=1, then τ τ ∞ ≡ 0 with τ0 as defined in Proposition 3.1, because the solution of (A.10) does not explode at (cf. Lemma A.1). The inner integral in (A.12) is well defined and finite for∞ any y> 0 because γ(z)2 > 0 for any z> 0 and γ is continuous. Remark A.1. The conclusion does not depend on the choice of c (0, ). ∈ ∞ Lemma A.2. Assume (s0), and let X = (Xt; t 0) denote the unique strong solution of (A.10) for initial condition x> 0.≥ Then the statements of Proposition 3.1 on the stopping time τ0 hold true. Proof. We prove (A.11), for c = 1. We assume without restriction that x< 1 and distinguish the two cases. Case α> 1/2. We have a(z) a(0) a z, z> 0. Then ≥ − | |1 a(z) b(z)z a(0) ( a + b(z))z − − | |1 γ(z)2z2α ≥ γ(z)2z2α a(0) a + b | |1 | |0 , ≥ γ 2z2α − γ(z)2z2α−1 | |0 1 ′ γ(z)2z2α−1 is integrable at zero by (s1) , and then there exists a positive constant K such that y a(z) b(z)z 2a(0) 1 dz 2 − + K − γ(z)2z2α ≥ γ 2 z2α Z1 | |0 Zy 2a(0) 1 = 1 + K (2α 1) γ 2 y2α−1 − − | |0 SMOOTHNESS OF DENSITIES AND ASYMPTOTIC ESTIMATES 39 hence 1 2a(0) 1 p (x) C exp dy 1 ≤− (2α 1) γ 2 y2α−1 Zx − | |0 1/x 1 2a(0) = C exp t2α−1 dt . − t2 (2α 1) γ 2 x→→0+ −∞ Z1 − | |0 Case α = 1/2. By (s2), 2a(z) 1 , γ(z)2z ≥ z for z< x. Hence a(z) b(z)z 1 1 2 − 2 b , γ(z)2z ≥ z − | |0 γ(z)2 1 and thus, γ2 being integrable at zero, for x< x we have 1 1 1 p (x) C exp dz dy 1 ≤− z Zx Zy 1 1 = C dy . − y x→→0+ −∞ Zx Acknowledgments. I am grateful to Vlad Bally of University Paris-Est Marne-la-Vall´ee for introducing me to this subject and for many stimulating discussions and to Alexander Yu. Veretennikov of University of Leeds for useful insights. 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MR2329681 Laboratoire d’Analyse et Mathematiques´ Appliquees´ UMR 8050 Universite´ Paris-Est 5 bd Descartes, Champs-sur-Marne 77454 Marne-la-Vallee´ Cedex 2 France and Scuola Normale Superiore di Pisa Piazza dei Cavalieri 7 56126 Pisa Italy E-mail: [email protected]