Estimates for the density of functionals of SDE s with irregular drift
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1 Estimates for the density of functionals of SDE s with irregular drift Arturo KOHATSU-HIGA a, Azmi MAKHLOUF a, a Ritsumeikan University and Japan Science and Technology Agency, Japan Abstract We obtain upper and lower bounds for the density of a functional of a diffusion whose drift is bounded and measurable The argument consists of using Girsanov s theorem together with an Itô-Taylor expansion of the change of measure One then applies Malliavin calculus techniques in a non-trivial manner so as to avoid the irregularity of the drift An integration by parts formula for this set-up is obtained Keywords: stochastic differential equations, density, Malliavin calculus, irregular drift 1 MSC: 6H7, 6H1 1 Introduction It is well known that the regularity of the fundamental solution of a linear second order parabolic PDE is related to the regularity of its coefficients Malliavin calculus or stochastic calculus of variations has arised as a prob- Corresponding author addresses: arturokohatsu@gmailcom Arturo KOHATSU-HIGA, azmimakhlouf@yahoofr Azmi MAKHLOUF Research supported by grants of the Japanese goverment The authors thank the referees for comments that improved the presentation and the generality of the paper Preprint submitted to Stochastic Processes and their Applications January, 13
2 abilistic tool for studying the existence and the smoothness of densities of diffusions, through the celebrated integration by parts formula Although this formula requires regularity on the coefficients of the associated stochastic differential equation, it applies to a wide range of stochastic equations that do not necessarily have a PDE counterpart In this article, we consider a two-component multidimensional SDE, where the second component depends only on the first one: X t = x + Y t = y + bx u du + ψx u, Y u du + σ 1 X u db u + σ X u db u, 1 ϕx u, Y u db u, t T, where B, B denotes a m = m 1 +m -dimensional Brownian motion, x R d 1, y R d, b : R d 1 R d 1, σ 1 : R d 1 R d 1 m 1, σ : R d 1 R d 1 m, ψ : R d 1+d R d, ϕ : R d 1+d R d m, and the stochastic integrals are Itô stochastic integrals The drift b is supposed to be only measurable and bounded, while the other coefficients are smooth and bounded The diffusions σ and ϕ are uniformly elliptic Our goal is to obtain upper and lower bounds for the density of Y t, thus extending the use of Malliavin calculus to situations where the drift b is irregular This type of results are useful in applications as they indicate that the behavior of the process Y is asymptotically similar to a Gaussian process As a by-product of our analysis, we obtain a general integration by parts formula for this setting The motivation for this problem comes from finance where Y is related to Asian options in the case that ϕ = and filtering problems, where Y is the observed signal process and X is the state process to be inferred see eg
3 Øksendal [11] For results on the weak existence and uniqueness of solutions of SDE s with non-lipschitz coefficients, see eg [5] or [8] and the references therein On the other hand, for results on the weak existence and uniqueness of fundamental solutions of the associated PDE, the literature is vast, although most of it deals with weak existence, see eg Stroock [13] where a bounded fundamental solution in W 1, is obtained One exception is Friedman [3], where the parametrix method is used for a diffusion with Hölder coefficients This method was slightly improved by McKean and Singer [4] We also note that bounds for the density of solution of SDE s have been obtained in several works under several regularity assumptions on the coefficients: for example Kusuoka and Stroock [9], Kohatsu-Higa [6] and Bally [] use Malliavin Calculus, for the locally elliptic case From the PDE theory point of view see Aronson [1] uniformly elliptic case The methodology used in this article is related to Kohatsu-Higa and Tanaka [8] In this reference, the authors have studied the system 1- with bounded irregular drift b, and have established the existence and smoothness of the density of Y t, through the study of its characteristic function In order to obtain this result, one first performs a change of probability measure, using Girsanov s theorem, in order to get rid of the irregular drift in 1, which leads to an SDE with smooth coefficients, driven by a new Brownian motion W Then, since the change of measure is non-smooth, they expand it using Itô-Taylor expansion up to a sufficiently high order Finally, in order to regularize the drift, they take the conditional expectation over some subinterval of the time interval in each multiple stochastic integral, and then 3
4 apply conditional Malliavin calculus to derive the smoothness of the density This result relies on the truncation of the Itô-Taylor formula and the explicit form of the definition of characteristic function, therefore it does not lead to an integration by parts formula which is essential in order to obtain upper and lower bounds for the density The goal of the present article is to show a variation of the argument which allows to obtain an integration by parts for any function and as result obtain upper and lower bounds for the density of Y t Our proof follows the first two steps as in Kohatsu-Higa and Tanaka [8], but with a simpler and more general proof which leads to an integration by parts formula that could not be derived in [8] This will not only imply the existence and smoothness of the density of Y t, but also implies upper bounds for the density of Y t and its derivatives Moreover, the general integration by parts formula that we state here enables us to derive at the same time a lower bound for the density, using the well-known lower bound of the law of smooth SDE s The present compact version of the integration by parts formula will clearly lead to various other applications to other stochastic equations Preliminaries, assumptions and notations As in [8], a general problem which generalizes the system 1- will be considered We describe that set-up in detail here although we will not be back into the setting of 1- until Section 4 Let W be a Brownian motion defined on the probability space Ω, F, Q 4
5 Define the following change of measure G t := exp bxu dw u 1 b bxu du Here b : R d 1 R m is a bounded and measurable function The system we will discuss in the next sections is X t = x + Y t = y + βx u du + ψx u, Y u du + σx u dw u, 3 ϕx u, Y u dw u 4 where all the coefficients β, σ, ψ, ϕ are smooth, bounded functions with bounded derivatives These assumptions are in force throughout the article In the next section, we work under the probability measure Q In particular, E denotes expectations under the probability measure Q In order to consider the system 1-, we will define a probability measure P by dp = G dq T Then Girsanov s theorem will lead to the needed conclusions in Section 4 when β = and b = σ 1σ 1 σ1 1 b and B = W + bx s ds Therefore, we remark that we start our discussion from a general system before the Girsanov change of measure is to be performed, and G t represents the change of measure that will lead the system 3-4 to become 1- with the measure dp dq = G T G t Throughout the paper, we use the following estimate for the moments of Lemma 1 For all p R, E[G p t ] exp 1 p p b t 5
6 Proof E[G p t ] = E[exp p = E[exp p p bxu dw u p bt bxu du exp bt bxu du] p bx u dw u 1 exp 1 p p b te[exp p bx u dw u 1 = exp 1 p p b t p bt bxu du] p bt bxu du] We shall use the following assumptions: Assumption A σ : σ is uniformly elliptic, ie there exists a nonnegative constant σ such that σσ x σi, for all x R d 1 Assumption A ϕ : ϕ is uniformly elliptic, ie there exists a nonnegative constant ϕ such that ϕϕ x ϕi, for all x R d 1+d Assumption A H : The system 3-4 satisfies the uniform Hörmander condition of order k That is, define the vector fields d 1 d V := β i x i + ψ i y i 1 i=1 i=1 d d 1 m σ jl x jσ il x i i=1 j=1 l=1 d 1 d V i := σ ji x j + ϕ ji y j, 1 i m j=1 j=1 Here we denote σ ij = ϕ i d1,j if i > d 1 and x j denotes the partial derivative operator wrt to the j-th coordinate, j = 1,, d 1 + d and with some abuse of notation we let x j+d 1 = y j 6
7 We now define A := {V 1,, V m }; A n := {[V i, Z]; i =,, m, Z A n 1 }, n 1 We assume that there exists k N and c > such that, for all ξ R d and x R d1, < V x, ξ > c ξ V k l=1 A l Our objective is the study of the density of Y t This density under the initial probability P at the point y is denoted by p Yt y = E P [δ y Y t ] = E [δ y Y t G t ], where δ y denotes the Dirac function at the point y This is a slight abuse of notation, but one can take a sequence of functions approximating the Dirac function and pass to the limit, as we already know that the density of Y t is smooth from [8] For more details, see the proof of Proposition 16 in [1] E Fs denotes the conditional expectation, under Q, wrt F s F s s being the filtration generated by the Brownian motion W mf denotes the space of F-measurable functions σ{x a, a A} denotes the smallest σ-field generated by the family of random variables {X a, a A}, A [, T ] The constants that appear troughout the text, only depend on the data x, y, β, σ, ψ, ϕ and T and may change from one line to another and whenever we want to emphasize the dependence of a constant C on a parameter p, we write it either as Cp or as C p The high order partial mixed derivative of a function fy = fy 1,, y d wrt y α 1,, y α k with α1,, α k {1,, d } is denoted by f α y where α = α 1,, α k The length k of α is denoted by α and the set of all 7
8 multi-indices of length less or equal than k is denoted by A k From now on, all these index manipulations will refer to the variables associated with the process Y and this index set is partially ordered using the inclusion operator That is, we say that β 1,, β j α 1,, α k if {β 1,, β j } {α 1,, α k } and the j l=1 1β l = a k l=1 1α l = a for any a {1,, d } In this sense we associate the identity operator to the zero-length empty index That is, f = f, = and α for any index α C α b denotes the space of bounded functions which are α -times continuously differentiable wrt y α 1,, y α k with bounded derivatives We denote the indicator function of the set [y 1, + [y d, + where y 1,, y d = y by 1 [y,+ Function spaces and their notation may change dimension parameter from d 1 to d 1 + d without any further specific mention N := N\{} Vector components are denoted with superscripts and matrix components by subscripts We use strongly the integration by parts formula of Malliavin Calculus We refer the reader to [1] for results and notation used in this article 3 Integration by parts formula The following formula gives the basic decomposition that will be used in order to obtain our integration by parts formula Lemma Let {t n, n } be a strictly decreasing sequence of strictly positive real numbers with t = t Then, G t L = G tn+1 Zt n n= 8
9 Here s sn Zt n = bxsn 1 dw sn 1 bx s dw s, t 1 t t n Zt 1 Proof First note that dg s = bx s G s dw s Using the SDE satisfied by G iteratively, we obtain : where R N = G t = G t1 + = G t1 + = bxs G s dw s t 1 bxs G t + t 1 s bxs3 G s3 dw s3 t s dw s = G t1 + G t bxs dw s + bxs bx s3 G s3 dw s3 dw s t 1 t 1 t N s sn G tn+1 bxsn+1 dw sn+1 bx s dw s + R N, t t n n= s t 1 t t 1 sn+1 t N+1 bxsn+ G sn+ dw sn+ bxsn+1 dw sn+1 bx s dw s Clearly using Lemma 1, E R N expt b b t N+1 N + 1! N + Our aim now is to derive an integration by parts formula that will enable us to estimate p Yt derivatives the density of Y t under the probability P and its Let us first recall the following classic result on the integration by parts for SDE s with smooth coefficients see [9] 9
10 Theorem 1 Let U s := X s, Y s defined by the system 3-4, and assume A H and the smoothness of the coefficients β, σ, ψ, ϕ Let s 1 < s t Then, U s D, and, for every V D and α any multi-index of length k, there exists a random variable H α [s 1,s ] U s, V such that, for any φ C α b, [ E Fs 1 [φ α U s V ] = E Fs 1 φus H[s α 1,s ]U s, V ] Moreover, there exist positive constants C k depending on the norms V k,p and µ k such that for p > 1 H α [s1,s ]U s, V L p C k1 + x, y µ k s s 1 µ k 5 As a natural extension we define H [s 1,s ] U s, V 1 The following lemma is the important element in the proofs of our main results Lemma 3 Let s 1 < s and k N, and assume A H We have the following integration by parts formula that holds for any measurable function f C α b, and any random variable Zs wrt F s := σ{x s, W r W s, r [s, ]}, [ [ ] E Fs 1 α y fy s3 Z s = E F s1 fy s3 Z s where R β s, p>1 Lp Ω for any β α β α L Ω that is measurable R β s, H β [s 1,s ] X s, Y s, 1 Proof For the sake of clarity, we first deal with the one-dimensional case and the first derivative: we want an integration by parts for E Fs 1 [f Y t Z s ] Y s3 s, x, y denotes the second component of the stochastic flow defined by X, Y starting from x, y at time s We define gs, x, y := E Fs [fys3 s, x, y y Y s3 s, x, y 1 Z s X s = x] 1 ],
11 Then y gs, x, y = E Fs [f Y s3 s, x, yz s X s = x] E Fs [fys3 s, x, y y Y s3 s, x, y yy s3 s, x, yz s X s = x] This leads to the following IBP: E Fs 1 [f Y s3 Z s ] = E Fs 1 [E F s [f Y s3 Z s ]] = E Fs 1 [ y gs, X s, Y s ] + E Fs 1 [fys3 y Y s3 s, x, y yy s3 s, x, y Z s x=xs,y=y s = E Fs 1 [fys3 y Y s3 s, x, y 1 Z x=xs s H [s1,s ]X s, Y s, 1],y=Y s + E Fs 1 [fys3 y Y s3 s, x, y yy s3 s, x, y Z s ] x=xs,y=y s Therefore defining R s 1, = y Y s3 s, x, y 1 x=xs,y=y s Rs, = y Y s3 s, x, y yy s3 s, x, y finishes the proof in this particular case and x=xs,y=y s In the multidimensional case, we first state the following identity for multiple chain-rule f α denotes the multiple derivative of f wrt the y variable, with multi-index α = α 1,, α α : Lemma 4 Let α be any multi-index Let f : R d R, ϕ : R d R d be functions which are α = k N times continuously differentiable, such that ϕ is invertible Then, f α ϕ = β A k { f ϕdet ϕ γβ P β ϕ } β, 6 where γ β are positive integer exponents and P β ϕ are polynomial functions of {D j ϕ; j k} 11 ]
12 Proof of Lemma 4 First, one shows by induction on the length of α Before this, one proves two basic properties These are we use the notation i :=,, 1,,,, 1 being in the i-th coordinate f i ψ = fψ i fψ i 7 d f i ϕ = f ϕk k=1 In fact for α = j one easily proves that f ϕ j = d k=1 Therefore 6 follows from ϕ 1 ik { f ϕ ϕ 1 jk k f ϕ ϕ 1 jk k} 8 ϕ 1 = det ϕ 1 Adj ϕ, where Adj ϕ denotes the adjoint of ϕ Now we assume that 6 is valid for any α with α k Then in order to prove 6 for a multi-index of length k + 1 we apply the formula 6 for f α 1 instead of f Then applying successively 8 and 7, one obtains the result Remark 1 Much more detailed information may be obtained on the polynomials appearing in 6 but as this will make the proof longer we do not mention it here We want to apply Lemma 4 with d = d and ϕ := Y s3 s, x, therefore all derivatives are wrt the variables in y It is known that det y Y s3 s, x, y 1 L p Ω for any p > 1 In fact, det y Y s3 s, x, y satisfies a linear SDE with intial condition equal to 1: see [1] page 15 For β A k, define g β s, x, y := E Fs [fys3 s, x, yr β,α Y s3 s, x, yz s X s = x] 1
13 where using the same notations as in Lemma 4 R β,α Y s3 s, x, y := det y Y s3 s, x, y γ β P β Y s3 s, x, y Then the function g β is jointly measurable, regular in y and integrable wrt to the joint law of X s, Y s Therefore using a standard argument, one can approximate the function g β s,, s being fixed in the L µ sense wrt µ, the joint law of X s, Y s whose density under Q is smooth using functions in C b Therefore the argument that follows is applied first to the approximation and then the same result follows by taking limits We have E Fs 1 [f α Y s3 Z s ] = E Fs 1 [E F s [f α Y s3 Z s ]] = E Fs1 β [g β s, X s, Y s ] 9 β A k = E Fs1 [gβ s, X s, Y s H β [s 1,s ] X s, Y s, 1] 1 β A k = E Fs1 [fys3 R β,α Y s3 Zs s 3 H β [s 1,s ] X s, Y s, 1] β A k As mentioned before, notice that in order to apply Theorem 1, we need to approximate the function g β as noted above Using the approximation argument and by passing to the limit for the equality 9=1 the result follows in the general case From the above equality one defines R β s, = R β,α Y s3 Its moment properties follow from the regularity of the process X, Y The following result is the integration by parts formula Theorem Assume A H, and let k N i For any multi-index α of length α = k, there exists a random variable 13
14 H α t such that, for any f C α b Rd ; R, E [f α Y t G t ] = E [fy t H α t ], ii There exists a positive constant C k C k T and µ k such that, E H α t 1 C k t µ k It follows that, for any f C α b Rd ; R and multi-index α such that α = k, E [f α Y t G t ] C k t µ k E fyt 1 11 Proof of Theorem Let t = t > t 1 > > t n > > to be chosen later, and recall that Z n t n 1, and Z t = 1 It is clear that Z n t From Lemma, = s t 1 t s n bxsn+1 t n dw sn+1 bx s dw s, for is Ft tn -measurable [ ] E [f α Y t G t ] = E G tn+1 f α Y t Zt n n= In order to exchange E and n=, let us study n= e n, where e n := E [ ] [ G tn+1 f α Y t Zt n = E Gtn+1E Ft n+1 [f α Y t Zt n ] ] From Lemma 3 recall that Z n t are Ft tn -measurable, one has using Lemma 3, with = t, s = t n and s 1 = t n+1 [ E Ft n+1 [f α Y t Zt n ] = E Ft n+1 fy t Zt n β A k R β t n,th β [t n+1,t n] X t n, Y tn, 1 ], and then e n = E [ fy t Z n t β A k R β t n,th β [t n+1,t n] X t n, Y tn, 1G tn+1 ] 14
15 Furthermore, e n fy t L H β [t n+1,t X n] t n, Y tn, 1 L 6 Zt n R β t n,t L 6 G tn+1 L 6 β A k C k fy t L t n t n+1 µ k Ct b n/ 5 exp n 1! 1/6 t n+1 b, 1 where we have used, respectively, estimate 5 from Theorem 1, classical estimate BDG inequality for Z n t and Lemma 1 for G tn+1 By choosing t n := t, n, it is clear that n n= e n <, and E [f α Y t G t ] = e n e n C k t µ k fy t L n= Thus, we have established that n= E [f α Y t G t ] = E [fy t H α t ], with H α t := n= β A k G tn+1 Z n t R β t n,th β [t n+1,t n] X t n, Y tn, 1, E H α t 1 C k t µ k 4 Bounds for the density of Y t in In this section, we go back to the consideration of the system 1- To obtain the upper and lower bounds for the density of Y we will link this system to the system 3-4 by using the Girsanov s theorem In fact, let 15
16 β = and b = σ 1σ 1 σ1 1 b and σ = σ 1 Then by Girsanov s σ theorem and B = W + bx s ds B is a Brownian motion under P Then the solution of the system 3-4 under P is equal in law to the weak solution of the system 1-41 Upper bounds Theorem 3 Let Y be defined by, and assume A σ and A H with β Then 1 The density p Yt under the initial probability P of Y t exists and is smooth Assume moreover A ϕ and that ϕ and ψ are bounded Then, for any multi-index α of length α = k, p α Yt y Ck t µ k exp y y Proof of Theorem 3 The first statement follows from Theorem, using fy := e i<θ,y>, and the characterization of the existence and smoothness of the density of a random variable in terms of the integrability of its characteristic function this statement is the main result in [8] The second statement immediately follows from 11 with fy := 1 [y,+ y, using the classical exponential martingale inequality see A5 in [1] after a Girsanov s transform to get rid of the drift in 4 for which we need the extra ellipticity assumption on ϕ Ct 16
17 4 Lower bound Theorem 4 Assume A σ and A ϕ, and let T > Then, there exists a positive constant C CT such that, for all t, T ], p Yt y Ct d exp y y Ct Proof of Theorem 4 Let t = t, t 1 [ t, t to be chosen later, and t n := t n, n Besides, we use the same notations as in the proof of Theorem, with f := 1 [y,+ and α := 1,, 1 One has p Yt y = E [δ y Y t G t ] = E [δ y Y t G t1 ] + By the triangular inequality, - First term: E [δ y Y t G t ] E [δ y Y t G t1 ] e n n=1 e n 13 n=1 Let p > 1 to be chosen later too, and q such that 1 p + 1 q inequality, one has [ E F E t1 [δy Y t ] ] [ 1 p E F = E t1 [δyy t ] ] 1 p G 1 p t 1 G 1 p t 1 E [δ y Y t G t1 ] 1 p [ E = 1 Using Hölder s G q p t 1 Hence, using twice the classical lower bound for the transition density of SDE with uniformly elliptic diffusion coefficient see eg [9], [6], [] and [7], and ] 1 q 17
18 using Lemma 1, [ E [δ y Y t G t1 ] E G q p t 1 exp q p + 1 t 1 b C = exp q p + 1 t 1 b exp q p + 1 t b ] p [ q E F E t1 [δy Y t ] ] 1 p p [ E πt t 1 d p exp y ] Y t1 p pct t 1 E [πt t 1 d exp y ] Y t1 p Ct t 1 p 1d t t 1 p 1d Ct pd exp y y here, we have used the formula for the convolution of two Gaussian densities By choosing p := 1+ 1 d, one obtains we bound exp below by exp q + 1T p b - Second term: Ct E [δ y Y t G t1 ] t t Ct 1 4 d exp y y q + 1t p b from Ct 14 From 1 with fy := 1 [y,+ y and α := 1,, 1, and for n 1, one has note that µ d = d in the elliptic case e n 1 [y,+ Y t L C exp y y Ct Hence, - Conclusion: n=1 H α [t n+1,t n],y C1 t n+1 d L 6 Zn t Rt n L 6 G tn+1 L 6 t t 1 1 Ct b n 1 n 1! 1/6 3 n exp t n+1 pct t 1 b e n t t 1 1 Ct d exp y y 15 Ct From 14 and 15, one can choose t 1 [ t, t such that e n 1 E [δ y Y t G t1 ] n=1 18
19 Indeed, this is achieved by setting t 1 := t min{ t, ht}, where ht := 1 16 C C Hence, from 13, one has 4 t 1 exp 4C 1 C 1 y y E [δ y Y t G t ] 1 E [δ y Y t G t1 ] t Using the lower bound in 14, E [δ y Y t G t ] 1 C t 1/4 t 1/4 d/ exp y y, if t Ct 1 = t ; 1 C 1 1 C C 4 t 1/4 exp C 1 C 1 y y t 1/4 d/ exp 16 t if t 1 = t ht y y Ct, Therefore, E [δ y Y t G t ] Ct d exp C y y t References [1] DG Aronson, Bounds for the fundamental solution of a parabolic equation, Bull Amer Math Soc [] V Bally, Lower bounds for the density of locally elliptic Itô processes, Annals of Probability
20 [3] A Friedman, Partial differential equations of parabolic type, Prentice- Hall, 1964 [4] J H P McKean, IM Singer, Curvature and the eigenvalues of the Laplacian, J Differential Geom [5] I Karatzas, S Shreve, Brownian motion and stochastic calculus nd ed, Graduate Texts in Mathematics, 113 New York: Springer-Verlag, 1991 [6] A Kohatsu-Higa, Lower bounds for densities of uniformly elliptic nonhomogeneous diffusions, Proceedings of the Stochastic Inequalities Conference in Barcelona Progress in Probability [7] A Kohatsu-Higa, Lower bounds for densities of uniformly elliptic random variables on Wiener space, Probability Theory and Related Fields [8] A Kohatsu-Higa, A Tanaka, A Malliavin Calculus method to study densities of additive functionals of SDE s with irregular drifts 11 To appear in Annales de l Institut Henri Poincare [9] S Kusuoka, D Stroock, Applications of Malliavin calculus, part III, J Faculty Sci Univ Tokyo Sect 1A Math [1] D Nualart, The Malliavin calculus and related topics Second edition Probability and its Applications, Springer, 6 [11] B Øksendal, Stochastic differential equations: an introduction with applications, Springer, 1998
21 [1] I Shigekawa, Stochastic analysis, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 4 [13] DW Stroock, Diffusion semigroups corresponding to uniformly ellip tic divergence form operators in séminaire de probabilités, xxii, Lecture Notes in Math
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