Under some regularity conditions on $b$, $\sigma$ and $\alpha$, we prove that
the following perturbed stochastic differential equation \begin{equation}
X_t=x+\int_0^t b(X_s)ds+\int_0^t \sigma(X_s) dB_s+\alpha \sup_{0 \le s \le t}
X_s, \ \ \ \alpha<1 \end{equation} admits smooth densities for all $0 \le t \le
t_0$, where $t_0>0$ is some finite number

Xu, Lihu

Yue, Wen

Zhang, Tusheng

English

arXiv

Lihu Xu

Wen Yue

Tusheng Zhang

Crossref

Statistics & Probability Letters

Smooth densities of the laws of perturbed diffusion processes

Under some regularity conditions on b, σ and α, we prove that the solution of the following perturbed stochastic differential equation Xt=x+∫0 tb(Xs)ds+∫0 tσ(Xs)dBs+αsup0≤s≤tXs,α0, where t0&gt;0 is some finite number.</p

The University of Manchester - Institutional Repository

The University of Manchester ResearchSmooth densities of the laws of perturbed diffusionprocessesDOI:10.1016/j.spl.2016.07.016Document VersionAccepted author manuscriptLink to publication record in Manchester Research ExplorerCitation for published version (APA):Xu, L., Yue, W., & Zhang, T. (2016). Smooth densities of the laws of perturbed diffusion processes. Statistics andProbability Letters, 119, 55-62. https://doi.org/10.1016/j.spl.2016.07.016Published in:Statistics and Probability LettersCiting this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:21. Oct. 2022SMOOTH DENSITIES OF THE LAWS OF PERTURBED DIFFUSIONPROCESSESLIHU XU, WEN YUE, AND TUSHENG ZHANGABSTRACT. Under some regularity conditions on b, σ and α, we prove that the solutionof the following perturbed stochastic differential equation(0.1) Xt = x+∫ t0b(Xs)ds+∫ t0σ(Xs)dBs + α sup0≤s≤tXs, α < 1admits smooth densities for all 0 < t ≤ t0, where t0 > 0 is some finite number.Keywords: Perturbed diffusion processes, Malliavin differentiability, Smooth density.Mathematics Subject Classification (2000): 60H07.1. INTRODUCTIONThere have been a considerable body of literatures devoted to the study of perturbedstochastic differential equations(SDEs), see [1]-[7],[9], [11], [12]. Let (Ω,F , {Ft}t≥0,P)be a filtered probability space with filtration{Ft}t≥0, let {Bt}t≥0 be a one-dimensionalstandard {Ft}t≥0-Brownian Motion. Suppose that σ(x), b(x) are Lipschitz continuousfunctions on R. It was proved in [5] that the following perturbed stochastic differentialequation:(1.1) Xt = x+∫ t0b(Xs)ds+∫ t0σ(Xs)dBs + α sup0≤s≤tXs, ∀ α < 1,admits a unique solution. If |σ(x)| > 0, it was shown in [12] that the law of Xt isabsolutely continuous with respect to Lebesgue measure, i.e. the law of Xt admits adensity for t > 0.There seem no results on the smoothness of the density of the law of a perturbeddiffusion process. This paper aims to partly fill in this gap. The smoothness of densitiesis a popular topic in stochastic analysis and has been intensively studied for severaldecades. We refer readers to [8], [10] and references therein. Our approach to provingthe smoothness of densities is by Malliavin calculus, so let us first recall some wellknown results on Malliavin calculus [8] to be used in this paper.Let Ω = C0(R+) be the space of continuous functions on R+ which are zero atzero. Denote by F the Borel σ-field on Ω and P the Wiener measure, then the canonicalLihu Xu is supported by the following grants: Science and Technology Development Fund MacaoS.A.R FDCT 049/2014/A1, MYRG2015-00021-FST.12 L. XU, W. YUE, AND T.ZHANGcoordinate process {ωt, t ∈ R+} on Ω is a Brownian motion Bt. Define F0t = σ(Bs, s ≤t) and denote by Ft the completion of F0t with respect to the P-null sets of F .Let H := L2(R+,B, µ) where (R+,B) is a measurable space with B being the Borelσ-field of R+ and µ being the Lebesgue measure on R+. We denote the norm of H by∥.∥H . For any h ∈ H , W (h) is defined by(1.2) W (h) =∫ ∞0h(t)dBt.Note that {W (h), h ∈ H} is a Gaussian Process on H .We denote by C∞p (Rn) the set of all infinitely differentiable functions f : Rn → Rsuch that f and all of its partial derivatives have polynomial growth. Let S be the set ofsmooth random variables defined byS = {F = f(W (h1), ...,W (hn)); h1, ..., hn ∈ H,n ≥ 1, f ∈ C∞p (Rn)}.Let F ∈ S, define its Malliavin derivative DtF by(1.3) DtF =n∑i=1∂if(W (h1), ...,W (hn))hi(t),and its norm by||F ||1,2 = [E(|F |2) + E(||DF ||2H)]12 ,where ||DF ||2H =∫∞0|DtF |2µ(dt). Denote by D1,2 the completion of S under the norm∥.∥1,2. We further define the norm||F ||m,2 =[E(|F |2) +m∑k=1E(∥DkF∥2H⊗k)] 12.Similarly, Dm,2 denotes the completion of S under the norm ||.||m,2.We shall use the following two propositions:Proposition 1.1 (Proposition 1.2.3 of [8]). Let ϕ : Rd → R be a continuously differen-tiable function with bounded partial derivatives. Suppose that F = (F 1, · · · , F d) is arandom vector whose components belong to the space D1,2. Then ϕ(F ) ∈ D1,2, andD(ϕ(F )) =d∑i=1∂iϕ(F )DFi.Proposition 1.2 (Proposition 2.1.5 of [8]). If F ∈ D∞,2 with D∞,2 = ∩m≥1Dm,2 and∥DF∥−1H ∈ ∩p≥1Lp(Ω), then the density of F belongs to the space C∞(R) of infinitelycontinuously differentiable functions.Throughout this paper, for a bounded measurable function f , we shall denote∥f∥∞ = supx∈R|f(x)|.SMOOTH DENSITIES OF PERTURBED DIFFUSION PROCESSES 32. MAIN RESULTSThroughout this paper, we need to assume α < 1 to guarantee that Eq. (1.1) has aunique solution [5]. Furthermore, it is shown in [12] thatTheorem 2.1. ([12, Theorem 3.1]) Let (Xt)t≥0 be the unique solution to Eq. (1.1). ThenXt ∈ D1,2 for all t > 0.Theorem 2.2. ([12, Theorem 3.2]) Assume that σ and b are both Lipschitz continuous,and |σ(x)| > 0 for all x ∈ R. Then, for t > 0, the law of Xt is absolutely continuouswith respect to Lebesgue measure.In this paper, we shall prove the following results about the smoothness of densities:Theorem 2.3. Assume that b is bounded smooth with ∥b′∥∞ < ∞ and that σ(x) ≡ σ.If α < 1, t0 > 0 and b satisfyθ(t0, α, b) < 1/2,with θ(t0, α, b) :=√2∥b′∥2∞t20 + 8α2+∥b′∥2∞t20 + 4α2, then the law of Xt in (1.1) admitsa smooth density for all t ∈ (0, t0].Theorem 2.4. Assume that b is bounded smooth with ∥b′∥∞ < ∞ and that σ is boundedsmooth with ∥σ′∥∞ < ∞, ∥σ′′∥∞ < ∞ and infx∈R |σ(x)| > 0. Let(2.1) F (y) =∫ yx1σ(u)du, y ∈ (−∞,∞)and b̃(x) = b(F−1(x))σ(F−1(x))− 12σ′(F−1(x)), then b̃ is bounded smooth with ∥b̃′∥∞ < ∞. Ifα < 1, t0 > 0 and b satisfyθ(t0, α, b̃) < 1/2with θ(t0, α, b̃) :=√2∥b̃′∥2∞t20 + 8α2+∥b̃′∥2∞t20 + 4α2, then the law of Xt in (1.1) admitsa smooth density for all t ∈ (0, t0].Proofs of Theorems 2.3 and 2.4: The main idea is to use Proposition 1.2 to prove thetwo theorems. To verify the conditions in Proposition 1.2, it suffices to prove that Xt ∈Dm,2 for all m ≥ 1 and ∥DXt∥H ≥ c > 0 a.s. for some constant c > 0.Theorem 2.3 immediately follows from Lemmas 3.1 and 3.4 below.Now we prove Theorem 2.4. Recall Yt =∫ Xtx1σ(u)du in Lemma 3.5 below, by thecondition of σ, F is a continuous and strictly increasing function with bounded deriva-tive and thus(2.2) ∥DYt∥H = ∥DF (Xt)∥H ≤1infx∈R |σ(x)|∥DXt∥H .Hence, by Lemmas 3.1 and 3.5 below, under the same condition as in Theorem 2.4 wehave(2.3)∥DXt∥H ≥ infx∈R|σ(x)| · ∥DYt∥H ≥ infx∈R|σ(x)| · [1− 2θ(t0, α, b̃)]t2(1 + 2∥b̃′∥2∞t2 + 2α2)t ∈ [0, t0].4 L. XU, W. YUE, AND T.ZHANGHence, Xt admits a smooth density for all t ∈ (0, t0]. 3. AUXILIARY LEMMASIt is well known that ∥DXt∥H has the following representation [12] for all t > 0:∥DXt∥H =(∫ t0|DrXt|2dr) 12with DrXt satisfying(3.1) DrXt = σ(Xr) +∫ trDrb(Xs)ds+∫ trDrσ(Xs)dBs + αDr(sup0≤s≤tXs).We shall often use the following fact ([12], [8])(3.2) DrXt = 0 if r > t,(3.3)∥∥∥∥D( sup0≤s≤tXs)∥∥∥∥H≤ sup0≤s≤t∥DXs∥H ,where∥∥∥∥D( sup0≤s≤tXs)∥∥∥∥2H=∫ t0∣∣∣∣Dr ( sup0≤s≤tXs)∣∣∣∣2 dr, ∥DXt∥2H = ∫ t0|DrXt|2dr.3.1. Xt is an element in Dm,2 for all t > 0 and m ≥ 1.Lemma 3.1. Let Xt be the solution of the perturbed stochastic differential equation(1.1), and suppose that the coefficients b and σ are smooth with bounded derivatives ofall orders. Then Xt belongs to Dm,2 for all t > 0 and all m ≥ 1.Proof. We shall use Picard iteration to prove the lemma. Letting X0t = x0 for all t > 0,define Xn+1t be the unique, adapted solution to the following equation:(3.4) Xn+1t = x0 +∫ t0σ(Xns )dBs +∫ t0b(Xns )ds+ α max0≤s≤t(Xn+1s),which obviously impliesmax0≤s≤t(Xn+1s)= x0 + max0≤s≤t(∫ t0σ(Xns )dBs +∫ t0b(Xns )ds)+ α max0≤s≤t(Xn+1s).Therefore,max0≤s≤t(Xn+1s)=x01− α+11− αmax0≤s≤t(∫ t0σ(Xns )dBs +∫ t0b(Xns )ds),SMOOTH DENSITIES OF PERTURBED DIFFUSION PROCESSES 5this and (3.4) further givesXn+1t =x01− α+∫ t0σ(Xns )dBs +∫ t0b(Xns )ds+α1− αmax0≤s≤t(∫ s0σ(Xnu )dBu +∫ s0b(Xnu )du).By the above representation of Xn+1t and a standard method [5], for every t > 0 wehave(3.5) limn→∞Xnt = Xt in L2(Ω).Let m ≥ 1, it is standard to check that Xnt ∈ Dm,2 for every t > 0 and n ≥ 1 [12,Theorem 3.1]. By a similar argument as in [12, Theorem 3.1], we have(3.6) supn≥1E[∥DkXnt ∥2H⊗k]< ∞, k = 1, ...,m.Next we prove Xt ∈ Dm,2 by the argument of [8, Lemma 1.2.3]. Indeed, by (3.6),there exists some subsequence DkXnjt weakly converges to some αk in L2(Ω, H⊗k) fork = 1, ...,m. By (3.5) and the remark immediately below [8, Proposition 1.2.2], the pro-jections of DkXnjt on any Wiener chaos converge in the weak topology of L2(Ω), as njtends to infinity, to those of αk for k = 1, ...,m. Hence, Xt ∈ Dm,2 and DkXt = αk fork = 1, ...,m. Moreover, for any weakly convergent subsequence the limit must be equalto α1, ..., αm by the same argument as above, and this implies the weak convergence ofthe whole sequence. 3.2. Additive noise case. If σ(x) ≡ σ, then Eq. (3.1) reads as(3.7) DrXt = σ +∫ trDrb(Xs)ds+ αDr(sup0≤s≤tXs).Lemma 3.2. Let t > 0 be arbitrary and b be bounded smooth with ∥b′∥∞ < ∞. For all0 < t1 < t2 ≤ t, we have∣∣∥DXt2∥2H − ∥DXt1∥2H∣∣ ≤ 2 [√2∥b′∥2∞(t2 − t1)2 + 8α2 + ∥b′∥2∞(t2 − t1)2 + 4α2] sup0≤s≤t∥DXs∥2H .Proof. It is easy to see that∣∣∥DXt2∥2H − ∥DXt1∥2H∣∣ = ∣∣∣∣∫ t20(DrXt2)2dr −∫ t10(DrXt1)2dr∣∣∣∣ ≤ I1 + I2,whereI1 :=∫ t2t1(DrXt2)2dr, I2 :=∫ t10∣∣(DrXt2)2 − (DrXt1)2∣∣ dr.6 L. XU, W. YUE, AND T.ZHANGWe claim that∫ t20(DrXt2 −DrXt1)2dr ≤ 2[∥b′∥2∞(t2 − t1)2 + 4α2]sup0≤s≤t∥DXs∥2H .(3.8)and we will prove it in the last part of this proof.Let us now estimate I1 and I2 by (3.8). ObserveI1 =∫ t2t1(DrXt2 −DrXt1)2dr ≤∫ t20(DrXt2 −DrXt1)2dr,by (3.8) we haveI1 ≤ 2[∥b′∥2∞(t2 − t1)2 + 4α2]sup0≤s≤t∥DXs∥2H .(3.9)Further observeI2 ≤[∫ t10(DrXt2 −DrXt1)2dr] 12[∫ t10|DrXt2 +DrXt1 |2dr] 12≤√2[∫ t10(DrXt2 −DrXt1)2dr] 12[∫ t10|DrXt2 |2 + |DrXt1 |2dr] 12≤√2[∫ t10(DrXt2 −DrXt1)2dr] 12[∫ t20|DrXt2 |2dr +∫ t10|DrXt1 |2dr] 12≤ 2[∫ t10(DrXt2 −DrXt1)2dr] 12sup0≤s≤t∥DXs∥H≤ 2[∫ t20(DrXt2 −DrXt1)2dr] 12sup0≤s≤t∥DXs∥H ,this inequality and (3.8) givesI2 ≤ 2√2[∥b′∥2∞(t2 − t1)2 + 4α2] sup0≤s≤t∥DXs∥2H .Combining the estimates of I1 and I2, we immediately get the desired inequality in thelemma.It remains to prove (3.8). By (3.7), we have(DrXt2 −DrXt1)2 ≤ 2∣∣∣∣∫ t2t1Drb(Xs)ds∣∣∣∣2 + 2α2 ∣∣∣∣Dr ( sup0≤s≤t1Xs)−Dr(sup0≤s≤t2Xs)∣∣∣∣2≤ 2∣∣∣∣∫ t2t1Drb(Xs)ds∣∣∣∣2 + 4α2 ∣∣∣∣Dr ( sup0≤s≤t1Xs)∣∣∣∣2 + 4α2 ∣∣∣∣Dr ( sup0≤s≤t2Xs)∣∣∣∣2 .SMOOTH DENSITIES OF PERTURBED DIFFUSION PROCESSES 7By Hölder inequality, (3.2) and Proposition 1.1, we have∫ t20∣∣∣∣∫ t2t1Drb(Xs)ds∣∣∣∣2 dr ≤ ∥b′∥2∞ ∫ t20(t2 − t1)∫ t2t1|DrXs|2dsdr= ∥b′∥2∞(t2 − t1)∫ t2t1∫ s0|DrXs|2drds≤ ∥b′∥2∞(t2 − t1)2 sup0≤s≤t∥DXs∥2H .Moreover, by (3.3) and (3.2) we have∫ t20∣∣∣∣Dr ( sup0≤s≤t2Xs)∣∣∣∣2 dr ≤ sup0≤s≤t2∥DXs∥2H ≤ sup0≤s≤t∥DXs∥2H ,∫ t20∣∣∣∣Dr ( sup0≤s≤t1Xs)∣∣∣∣2 dr = ∫ t10∣∣∣∣Dr ( sup0≤s≤t1Xs)∣∣∣∣2 dr ≤ sup0≤s≤t∥DXs∥2H .Collecting the above four inequalities, we immediately get the desired (3.8). Lemma 3.3. Let b be bounded smooth with ∥b′∥∞ < ∞, we have(3.10) sup0≤s≤t∥DXs∥2H ≥σ2t2(1 + 2∥b′∥2∞t2 + 2α2), t > 0.Proof. By (3.7) and using (a+ b)2 ≥ 12a2 − b2, we have(DrXt)2 ≥ 12σ2 −[∫ trDrb(Xs)ds+ αDr(sup0≤s≤tXs)]2≥ 12σ2 − 2(∫ trDrb(Xs)ds)2− 2α2[Dr(sup0≤s≤tXs)]2.Further observe∫ t0(∫ trDrb(Xs)ds)2dr ≤∫ t0(t− r)∫ tr|Drb(Xs)|2dsdr≤∫ t0(t− r)∥b′∥2∞∫ tr|DrXs|2dsdr≤ t∥b′∥2∞∫ t0∫ tr|DrXs|2dsdr= t∥b′∥2∞∫ t0∥DXs∥2Hds≤ t2∥b′∥2∞ sup0≤s≤t∥DXs∥2H ,(3.11)8 L. XU, W. YUE, AND T.ZHANGwhere the second inequality is by Proposition 1.1. Hence,∥DXt∥2H ≥σ2t2− 2∥b′∥2∞t2 sup0≤s≤t∥DXs∥2H − 2α2∥D( sup0≤s≤tXs)∥2H≥ σ2t2− 2∥b′∥2∞t2 sup0≤s≤t∥DXs∥2H − 2α2 sup0≤s≤t∥DXs∥2H ,where the last inequality is by (3.3).This clearly impliessup0≤s≤t∥DXs∥2H ≥σ2t2− 2∥b′∥2∞t2 sup0≤s≤t∥DXs∥2H − 2α2 sup0≤s≤t∥DXs∥2H ,which immediately yields the desired bound. Lemma 3.4. Let b is bounded smooth with ∥b′∥∞ < ∞ and σ(x) ≡ σ with σ ̸= 0. Ifα < 1, t0 > 0 and b satisfyθ(t0, α, b) < 1/2with θ(r, α, b) :=√2∥b′∥2∞r2 + 8α2 + ∥b′∥2∞r2 + 4α2 for r > 0, then(3.12) ∥DXt∥2H ≥[1− 2θ(t0, α, b)]σ2t2(1 + 2∥b′∥2∞t2 + 2α2), t ∈ [0, t0].Proof. Let t ∈ [0, t0]. For all 0 ≤ t1 ≤ t2 ≤ t, by Lemma 3.2, we have∣∣∥DXt2∥2H − ∥DXt1∥2H∣∣ ≤ 2θ(t2 − t1, α, b) sup0≤s≤t∥DXs∥2H .Hence, for all s ∈ [0, t],∥DXs∥2H ≤∣∣∥DXs∥2H − ∥DXt∥2H∣∣+ ∥DXt∥2H≤ 2θ(t− s, α, b) sup0≤s≤t∥DXs∥2H + ∥DXt∥2H ,and consequentlysup0≤s≤t∥DXs∥2H ≤ 2θ(t, α, b) sup0≤s≤t∥DXs∥2H + ∥DXt∥2H .The above inequality and (3.10) further give∥DXt∥2H ≥ [1− 2θ(t, α, b)] sup0≤s≤t∥DXs∥2H≥ [1− 2θ(t0, α, b)] sup0≤s≤t∥DXs∥2H .Combining the above inequality and Lemma 3.3 immediately gives the desired inequal-ity. SMOOTH DENSITIES OF PERTURBED DIFFUSION PROCESSES 93.3. Multiplicative noise case. By the condition of σ, we have supx∈R σ(x) < 0 orinfx∈R σ(x) > 0. Without loss of generality, we assume thatinfx∈Rσ(x) > 0.Let us consider the following well known transform(3.13) F (Xt) =∫ Xtx1σ(u)du,it is easy to see that F is a strictly increasing function with bounded derivative. Hence,(3.14) sup0≤s≤tF (Xs) = F(sup0≤s≤tXs).By Itô formula, we have(3.15) F (Xt) =∫ t0(b(Xs)σ(Xs)− 12σ′(Xs))ds+Bt + α∫ t01σ(Xs)dMswhere Mt = sup0≤s≤tXs. It is easy to see that Mt is an increasing function of t and that1σ(Xs)has a contribution to the related integral only when Xs = Ms. Hence,(3.16) F (Xt) =∫ t0(b(Xs)σ(Xs)− 12σ′(Xs))ds+Bt + α∫ t01σ(Ms)dMs.Since Mt is a continuous increasing function with respect to t, we have(3.17) F (Xt) =∫ t0(b(Xs)σ(Xs)− 12σ′(Xs))ds+Bt + α∫ Mt01σ(u)du.By (3.14),(3.18) F (Xt) =∫ t0(b(Xs)σ(Xs)− 12σ′(Xs))ds+Bt + α sup0≤s≤tF (Xs).Denote Yt = F (Xt), it solves the following perturbed SDE:(3.19) Yt =∫ t0b̃(Ys)ds+Bt + α sup0≤s≤tYswhere b̃(x) = b(F−1(x))σ(F−1(x))− 12σ′(F−1(x)). Applying Lemma 3.4, we get the followinglemma about the dynamics Yt:Lemma 3.5. Assume that b is bounded smooth and that σ is bounded smooth with∥σ′∥∞ < ∞, ∥σ′′∥∞ < ∞ and infx≥0 |σ(x)| > 0. Then b̃ is bounded smooth. If α < 1,t0 > 0 and b satisfyθ(t0, α, b̃) < 1/210 L. XU, W. YUE, AND T.ZHANGwith θ(r, α, b̃) :=√2∥b̃′∥2∞r2 + 8α2 + ∥b̃′∥2∞r2 + 4α2 for r > 0, then(3.20) ∥DYt∥2H ≥[1− 2θ(t0, α, b̃)]t2(1 + 2∥b̃′∥2∞t2 + 2α2), t ∈ (0, t0].Proof. It is easy to check that under the conditions in the lemma b̃ is bounded smoothwith ∥b̃′∥∞ < ∞. Hence, the lemma immediately follows from applying Lemma 3.4 toYt. REFERENCES1. Ph. Carmona, F. Petit, M. Yor, Beta variables as times spent in [0,∞) by certain perturbed Brownianmotions, J. London Math. Soc. 58 (1998), 239-256.2. L. Chaumont, R.A. Doney, Pathwise uniqueness for perturbed versions of Brownian motion andreflected Brownian motion, Probab. Theory Related Fields 113 (1999), 519-534.3. B. Davis, Brownian motion and random walk perturbed at extrema, Probab. Theory Related Fields113(1999)501-518.4. R.A. Doney, Some calculations for perturbed Brownian motion. Séminaire de Probabilités, XXXII,231-236, Lecture Notes in Math., 1686, Springer, Berlin, 1998.5. R.A. Doney, T.S. Zhang, Perturbed Skorohod equations and perturbed reflected diffusion processes,Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 1, 107-121.6. J.F. Le Gall, M. Yor, Excursions browniennes et carrés de processus de Bessel, C.R. Acad. Sci. ParisSér. I 303(1986), 73-76.7. J.F. Le Gall, M. Yor, Enlacements du mouvement brownien autour des courbes de l’espace, Trans.Amer. Math. Soc. 317 (1990), 687-722.8. D. Nualart, The Malliavin calculus and related topics. Second edition. Probability and its Applica-tions (New York). Springer-Verlag, Berlin, 2006. xiv+382 pp.9. M. Perman, W. Werner, Perturbed Brownian motions, Probab. Theory Related Fields 108 (1997), no.3, 357-383.10. M. Sanz-Sole, Malliavin calculus. With applications to stochastic partial differential equations.Fundamental Sciences. EPFL Press, Lausanne; distributed by CRC Press, Boca Raton, FL, 2005.viii+162 pp.11. W. Werner, Some remarks on perturbed reflecting Brownian motion. Séminaire de Probabilités,XXIX, 37-43, Lecture Notes in Math., 1613, Springer, Berlin, 1995.12. W. Yue, T.S. Zhang, Absolute continuity of the laws of perturbed diffusion processes and perburbedreflected diffusion processes, J. Theoret. Probab. 28 (2015), no. 2, 587-618.DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE AND TECHNOLOGY, UNIVERSITY OFMACAU, E11 AVENIDA DA UNIVERSIDADE, TAIPA, MACAU, CHINAE-mail address: lihuxu@umac.moINSTITUTE OF ANALYSIS AND SCIENTIFIC COMPUTING, VIENNA UNIVERSITY OF TECHNOLOGY,WIEDNER HAUPTSTRASS 8-10, 1040 WIEN, AUSTRIAE-mail address: wenyue@hotmail.co.ukSCHOOL OF MATHEMATICS,UNIVERSITY OF MANCHESTER OXFORD ROAD, MANCHESTER M139PL, UNITED KINGDOME-mail address: tusheng.zhang@manchester.ac.uk

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Smooth densities of the laws of perturbed diffusion processes

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