. . Acad. Sci. Paris, Ser. I 338 (2004) 77–80

Probability Theory Exponential divergence estimates and heat kernel tail

Eulalia Nualart

Laboratoire de probabilités, Université de Paris VI, 4, place Jussieu, 75252 Paris, France Received 15 July 2003; accepted 12 November 2003 Presented by Paul Malliavin

Abstract We obtain a lower bound for the density of a real random variable on the Wiener space under an exponential moment condition of the divergence. We apply this result to the solution of a non-linear SDE. To cite this article: . Nualart, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Estimations exponentielles de divergence et queue du noyau de la chaleur. Nous obtenons une minoration exponentielle de la densité ’une variable aléatoire réelle dans ’espace de Wiener sous une condition portant sur le moment exponentiel de la divergence. Nous appliquons ce résultat à la solution d’une EDS non-linéaire. Pour citer cet article : E. Nualart, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  2003 Académie des sciences. Published by Elsevier SAS. All rights reserved.

1. Minoration of density in terms of divergence

Consider a non-degenerate R-valued random variable on the Wiener space , such that F ∈ D3,∞(W).Let ∂ W  = ∂ A be a covering vector field of ∂ξ for F , that is, a process on such that, formally, DF,A ∂ξ . This means that for φ : R → R smooth,   ∂φ   E (F ) = E φ(F)δ(A) , ∂ξ where δ(A) denotes the divergence of A, which is supposed to exist.

Theorem 1.1. Assume that there exists γ  1 and c>0 such that      E exp cδ(A)γ < ∞. Then the law of F has a continuously differentiable density p(ξ) which, if γ>1, satisfies the minoration   γ/(γ−1) p(ξ) > cγ exp −cγ |ξ| , for all ξ ∈ R. (1)

E-mail address: [email protected] (E. Nualart).

1631-073X/$ – see front matter  2003 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/.crma.2003.11.015 78 E. Nualart / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 77–80

If γ = 1, we have p(ξ) > c exp(−c exp(ξ)).

Proof. Let φ : R → R be smooth. By the integration by parts formula in R,   ∂p   φ(ξ) (ξ) dξ = φ(ξ)E δ(A)|F = ξ p(ξ)dξ. ∂ξ R R

:= p = E[ | = ] Therefore, (ξ) p (ξ) δ(A) F ξ . Consider the function ψ : η → exp(c ηγ ), η > 0. By Jensen’ inequality,      

E ψ v (F )  E ψ δ(A) . (2)

= ξ | | Introduce the increasing function w(ξ) a v (u) du, ξ > a, where a is a point in the interior of the support of the law of F . In particular, p(a) > 0. We assume without loss of generality that a = 0. The case ξ<0 follows similarly. Note that   p(ξ)  p(0) exp −w(ξ) . (3) By (2) and (3), we obtain +∞        1 γ exp c(w )γ (ξ) − w(ξ) dξ  E exp cδ(A) < ∞. p(0) 0 The resolution of c(w )γ (ξ) − w(ξ) = 0 shows that the exponent appearing in (1) cannot be improved. We now define the increasing sequence αn = inf{ξ: w(ξ)  }. Consider the functional αn+1   γ Jn(w) := exp c(w ) (ξ) dξ.

αn

Lemma 1.2 (Variational Lemma). The minimum of Jn(w) among all increasing and continuously differentiable α + − functions w : [α ,α + ] → R such that n 1 w (ξ) dξ = 1, is reached for w(ξ) = l 1ξ,wherel := α + − α . n n 1 αn n n n 1 n

Proof. We apply the Lagrange multipliers method looking for the absolute minimum of J (w) − λ αn+1 w (ξ) dξ, n αn where λ is a constant. Denoting δ a variational of w , we must have at the minimum

αn+1       γ − γc w (ξ) 1 exp c(w )γ (ξ) − λ δ dξ = 0, for all δ.

αn − Therefore, γc(w(ξ))γ 1 exp(c(w )γ (ξ)) = λ, which implies w (ξ) equal a constant determined by the condition αn+1 w (ξ) dξ = 1. αn As a consequence,

αn+1       γ −γ exp c(w ) (ξ) − w(ξ) dξ  exp −(n + 1) ln exp cln .

αn As αn+1 ∞  

exp c(w )γ (ξ) − w(ξ) dξ<∞, = n 0 αn E. Nualart / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 77–80 79

−γ −γ we must have exp(−(n + 1))ln exp(cln ) → 0, as n →∞. This implies that for all n  n0, (n + 1) − cln > 0, for some constant c>0. In particular, for >n0, + q q q 1 cγ cγ ln = αq+1 − αn > > dx. 0 (n + 1)1/γ x1/γ n=n0 n=n0 n0+1

Note that as ξ<αq+1,w(ξ)

2. Stochastic differential equations

Let = (x(t), t ∈[0,T]) be the solution of the following SDE:     dx(t)= a x(t) dW(t)+ x(t) dt, x(0) = 0, where W is a Brownian motion and a and b are C3(R)-functions with bounded derivatives. The derivative Dr (x(t)) satisfies the following linear SDE: t t           Dr x(t) = a x(r) + Dr a x(s) dW(s)+ Dr b x(s) . r r The Jacobian J = (J (t), t ∈[0,T]) is the solution of the linearized SDE      

dJ(t)= a x(t) dW(t)+ b x(t) dt J(t), J(0) = 1.

−1 One can easily check that Dr (x(t)) = J(t)J (r)a(x(r)). We associate to J the rescaled variation defined as (t) = J (t)/a(x(t)).

Proposition 2.1 [1]. The rescaled variation is a differentiable function of t and t   a 1 z(t) = z(0) exp λ x(s) ds , where λ = b − b − a a is called the feedback effect rate. a 2 0

Therefore, t       Dr x(t) = a x(t) exp λ x(s) ds . r We now fix t>0 and define the process t 1 1   u(r) = exp − λ x(s) ds ,r t. t a(x(t)) r

∈ 1 R Proposition 2.2. For any function φ Cb ( ) and for t>0, the following formula holds:        E φ x(t) = E φ x(t) δ(u) . 80 E. Nualart / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 77–80

Proof. By the duality relation between D and δ, and the definition of u, t            

E φ x(t) δ(u) = E φ x(t) Dr x(t) u(r) dr = E φ x(t) . 0

Theorem 2.3. Assume that λ, λ a, a are bounded and |a(ξ)|  c>0, for all ξ ∈ R. Then for all t  t > 0,x(t)admits a continuously differentiable density such that 0  2 pt (ξ) > c exp −c|ξ| , for all ξ ∈ R, (4) where c is a constant depending on t and the bounds on the coefficients.

Proof. Fix t  t0 > 0 and define the random variable and the adapted process t r 1 1     v(t) = exp − λ x(s) ds ,w(r)= exp λ x(s) ds ,r t. t a(x(t)) 0 0 By the properties of δ, t t   δ(u) = v(t) w(r)dW(r)− Dr v(t) w(r)dr. 0 0 We now compute the derivative Dr (v(t)): t   t   1 1   1 1   D v(t) = D exp − λ x(s) ds + D exp − λ x(s) ds , r t a(x(t)) r r t a(x(t)) 0 0 where   t 1 1 1 a (x(t))   D = exp λ x(s) ds , and r t a(x(t)) t a(x(t)) r t t t s          

Dr exp − λ x(s) ds =−exp − λ x(s) ds λ x(s) a x(s) exp λ x(s) ds ds. 0 0 0 r By the exponential martingale inequality, there exists c>0 depending on t such that E[exp(c|δ(u)|2)] < ∞. The conclusion follows from Proposition 2.2 and Theorem 1.1.

Corollary 2.4. Suppose a = 1.Thenifb ,b are bounded, the minoration (4) holds.

Remark 1. In one dimension by a change of variables we can always reduce to the case a = 1.

Acknowledgement I would like to thank Paul Malliavin for all the discussions and suggestions on the subject.

References

[1] E. Barucci, P. Malliavin, .E. Mancino, R. Renò, A. Thalmaier, The price-volatility feedback rate: an implementable mathematical indicator of market stability, Math. Finance 13 (2003) 17–35.