Absolute continuity and Fokker-Planck equation for the law of Wong-Zakai approximations of Itô's stochastic differential equations
aa r X i v : . [ m a t h . P R ] J a n Absolute continuity and Fokker-Planck equation forthe law of Wong-Zakai approximations of Itˆo’sstochastic differential equations
Alberto Lanconelli ∗ Abstract
We investigate the regularity of the law of Wong-Zakai-type approximations forItˆo stochastic differential equations. These approximations solve random differen-tial equations where the diffusion coefficient is Wick-multiplied by the smoothedwhite noise. Using a criteria based on the Malliavin calculus we establish ab-solute continuity and a Fokker-Planck-type equation solved in the distributionalsense by the density. The parabolic smoothing effect typical of the solutions of Itˆoequations is lacking in this approximated framework; therefore, in order to proveabsolute continuity, the initial condition of the random differential equation needsto possess a density itself.
Key words and phrases: stochastic differential equations, Wong-Zakai approxima-tion, Malliavin calculusAMS 2000 classification: 60H10; 60H07; 60H30
The celebrated Wong-Zakai theorem [22],[23], extended to the multidimensional caseby Stroock and Varadhan [20], provides a crucial insight in the theory of stochasticdifferential equations. It asserts that, given a suitable smooth approximation { B εt } t ∈ [0 ,T ] of the Brownian motion { B t } t ∈ [0 ,T ] , the solution { X εt } t ∈ [0 ,T ] of the random ordinarydifferential equation ˙ X εt = b ( t, X εt ) + σ ( t, X εt ) · ˙ B εt , (1.1)converges, as ε goes to zero, to the solution of the Stratonovich stochastic differentialequation (SDE, for short) dX t = b ( t, X t ) dt + σ ( t, X t ) ◦ dB t instead to the more popular Itˆo’s interpretation of the corresponding stochastic equation, dX t = b ( t, X t ) dt + σ ( t, X t ) dB t . ∗ Dipartimento di Scienze Statistiche ”Paolo Fortunati”, Alma Mater Studiorum Universit di Bologna,Via Belle Arti 41 - 40126 Bologna - Italia. E-mail: [email protected] { B t } t ∈ [0 ,T ] is approximated but rather the way we multiply the diffusion coefficient σ ( t, X εt ) withthe smoothed white noise ˙ B εt in (1.1). The following example clarifies this point. Example 1.1
Consider the random ordinary differential equation ˙ X εt = X εt · ˙ B εt , X ε = x which corresponds to (1.1) for b ( t, x ) = 0 and σ ( t, x ) = x . Its solution X εt = x exp { B εt } , t ∈ [0 , T ] converges to X t := x exp { B t } , t ∈ [0 , T ] (1.2) whenever for each t ∈ [0 , T ] the random variable B εt converges to B t in probability as ε tends to zero. A direct verification shows that (1.2) is the unique solution of dX t = X t / dt + X t dB t , X = x which is equivalent to dX t = X t ◦ dB t , X = x. In fact, it is well known (see for instance Karatzas and Shreve [14]) that the StratonovichSDE dX t = b ( t, X t ) dt + σ ( t, X t ) ◦ dB t , X = x is equivalent to the Itˆo SDE dX t = h b ( t, X t ) + σ ( t, X t ) ∂ x σ ( t, X t ) / i dt + σ ( t, X t ) dB t , X = x. Therefore, in order to recover the Itˆo interpretation of the limiting SDE one may startwith the modified equation˙ Y εt = h b ( t, Y εt ) − σ ( t, Y εt ) ∂ y σ ( t, Y εt ) / i + σ ( t, Y εt ) · ˙ B εt , Y ε = x to obtain in the limit dY t = h b ( t, Y t ) − σ ( t, Y t ) ∂ x σ ( t, Y t ) / i dt + σ ( t, Y t ) ◦ dB t , Y = x which corresponds to dY t = b ( t, Y t ) dt + σ ( t, Y t ) dB t , Y = x. However, this procedure has some drawbacks. For instance, certain probabilistic prop-erties of the exact solution may be lost in the approximated solution.2 xample 1.2
Suppose we wish to approximate dY t = Y t dB t , Y = x (1.3) according to the previous procedure; then, we should consider the random ordinary dif-ferential equation ˙ Y εt = − Y εt / Y εt · ˙ B εt , Y ε = x whose solution is Y εt = x exp { B εt − t/ } , t ∈ [0 , T ] . However, on one hand we have E [ Y t ] = x for all t ∈ [0 , T ] while on the other E [ Y εt ] = x unless E [( B εt ) ] = t for all ε > . (1.4)The problem of finding a version of equation (1.1) having in the limit the Itˆo interpre-tation of the SDE (by-passing the Stratonovich interpretation involved in the proceduredescribed above) was partially solved by Hu and Øksendal [12]: they proved that thesolution of ˙ Y εt = b ( t, Y εt ) + σ ( t ) Y εt ⋄ ˙ B εt , (1.5)where ⋄ stands for the Wick product, converges as ε goes to zero to the solution of theItˆo SDE dY t = b ( t, Y t ) dt + σ ( t ) Y t dB t . (1.6)Here σ is a deterministic function; the assumption of a linear diffusion coefficient isutilized in connection with a reduction method to solve equation (1.5). Example 1.3
Referring to Example 1.2, we now utilize (1.5) to approximate equation(1.3), i.e. ˙ Y εt = Y εt ⋄ ˙ B εt , Y ε = x. The solution to the previous equation can be computed using the Wick calculus as Y εt = x exp ⋄ { B εt } = x exp (cid:8) B εt − E [ | B εt | ] / (cid:9) . Therefore, lim ε → Y εt = x exp { B t − t/ } as long as B εt → B t in L (Ω) which is the solution of equation (1.3). Moreover, in this case E [ Y εt ] = E [ Y t ] = x for any t ∈ [0 , T ] independently of the particular approximation of { B t } t ∈ [0 ,T ] utilized (in contrast with(1.4)). Z T γ t dB t = Z T γ t ⋄ ˙ B t dt (1.7)which connects the Itˆo-Skorohod integral of a stochastic process { γ t } t ∈ [0 ,T ] (on the lefthand side) with a standard integral of the Wick product between γ t and the white noise˙ B t . We refer the reader to the book Holden et al. [8] for a proof of this identity.The aim of the present paper is to investigate the probabilistic properties of theWong-Zakai-type approximation (1.5), as suggested in [12], of the Itˆo SDE (1.6). Inparticular, we will focus on the absolute continuity of the law of the solution Y εt findingsufficient conditions for the existence of a density with respect to the one dimensionalLebesgue measure. We will also write a Fokker-Planck-type equation solved in the dis-tributional sense by the density of Y εt . This analogy with exact solutions of Itˆo SDEs ishowever conditioned by a restriction on the initial data of equation (1.5) that is requiredto possess a density. Our approach relies on the criteria for absolute continuity based onthe Malliavin calculus. We stress that Wong-Zakai approximations, contrary to manyother approximation schemes like for instance the Euler discretization (see e.g. Ballyand Talay [1] for an investigation on the convergence of the densities of such approxi-mations), are defined in terms of non adapted stochastic processes.There is a vast literature on Wong-Zakai approximations for Stratonovich stochastic(partial) differential equations driven by different types of noise: one may look at Brez-niak and Flandoli [4], Gy¨ongy and A. Shmatkov [6], Hu et al. [10], Hu and Nualart[11], Konecny [15], Tessitore and Zabczyk [21] just to mention a few. We also mentionthe remarkable paper Hairer and Pardoux [7] where a Wong-Zakai theorem for a generalnonlinear Itˆo-type stochastic heat equation driven by a space-time white noise is proved.Wong-Zakai approximations for Itˆo SDEs are quite rare in the literature. As the insightof [12] shows, one has to deal in this case with equations involving the Wick product,which corresponds to treat Skorohod SDEs (see Section 2). This type of equations pos-sesses a global solution only in some particular cases making even the existence of theWong-Zakai approximation a tough issue. We mention the paper Da Pelo et al. [5]dealing with Wong-Zakai approximations for Itˆo-Stratonovich interpolations and BenAmmou and Lanconelli [2] investigating the rate of convergence for Wong-Zakai approx-imations of Itˆo SDEs in the spirit of the present paper.To state our main result we need to introduce a few notation. Let (Ω , F , P ) be theclassical Wiener space over the time interval [0 , T ], where T is an arbitrary positiveconstant, and denote by { B t } t ∈ [0 ,T ] the coordinate process, i.e. B t : Ω → R ω B t ( ω ) = ω ( t ) . By construction, the process { B t } t ∈ [0 ,T ] is under the measure P a one dimensionalBrownian motion. Now, let π be a finite partition of the interval [0 , T ], that means π = { t , t , ..., t n − , t n } with0 = t < t < · · · < t n − < t n = T polygonal approximation of the Brownian motion { B t } t ∈ [0 ,T ] relative tothe partition π : B πt := (cid:18) − t − t k t k +1 − t k (cid:19) B t k + t − t k t k +1 − t k B t k +1 if t ∈ [ t k , t k +1 [ (1.8)and B πT := B T . It is well known that for any ε > p ≥ C p,T,ε such that E " sup t ∈ [0 ,T ] | B πt − B t | p /p ≤ C p,T,ε | π | / − ε where | π | := max k ∈{ ,...,n − } ( t k +1 − t k ) stands for the mesh of the partition π . We referthe reader to Lemma 2.1 in Hu et al. [10] and Lemma 11.8 in Hu [9] for sharper esti-mates. We assume that the finite partition π is fixed throughout the present paper.Given the partition π , let h π be a function in L ([0 , T ]) such that h π = 0 almost every-where and 1 t k +1 − t k Z t k +1 t k h π ( u ) du = 0 for all k ∈ { , ..., n − } . The crucial role of the function h π will be made clear in Proposition 2.4 below. Thefollowing is our main result: the symbols ⋄ , D ,p and D h π denote the Wick product,the Sobolev-Malliavin space and the directional Malliavin derivative in the direction h π ,respectively. Definitions and useful properties are postponed to Section 2. Theorem 1.4
Let { X πt } t ∈ [ s,T ] be the unique solution of the random Cauchy problem (cid:26) ˙ X πt = b ( t, X πt ) + σ ( t ) X πt ⋄ ˙ B πt , t ∈ ] s, T ] X πs = Y (1.9) where s ∈ [0 , T [ and we assume that • b : [0 , T ] × R → R is a continuous function with bounded first and second partialderivatives with respect to the second variable; • σ : [0 , T ] → R belongs to L ([0 , T ]) ; • Y ∈ D ,p for all p ≥ and E [ | D h π Y | − q ] is finite for some q > .Then,1. for any t ∈ [ s, T ] the law of X πt is absolutely continuous with respect to the onedimensional Lebesgue measure with a bounded and continuous density;2. the density ( t, x ) p π ( t, x ) of the random variable X πt solves in the sense ofdistributions the Fokker-Planck equation ( ∂ t + L x ) ∗ u ( t, x ) = 0 where ( ∂ t + L x ) ∗ stands for the formal adjoint of the operator ∂ t + L x and L x := b ( t, x ) ∂ x + σ ( t ) xg ( t, x ) ∂ xx . (1.10) for a suitable measurable function g : [ s, T ] × R → R . Moreover, Z R | g ( t, x ) | q p π ( t, x ) dx is finite for all q ≥ and t ∈ [ s, T ] . emark 1.5 We observe that the assumptions on Y entail the absolute continuity ofits law with respect to the one dimensional Lebesgue measure. In fact, contrary to theassumptions usually adopted for the study of the absolute continuity for exact solutionsof SDEs, here we need the initial condition to have a density. The stochastic equation(1.9), being an approximated version of an Itˆo SDE, does not possess the same smoothingproperties of the original equation. Remark 1.6
The way we prove the existence of the function g appearing in the operator L x does not give us information about its regularity and sign (see formula (2.16) below).Therefore, we don’t know whether the existence of a density for the law of X πt can bededuced from the properties of the operator L x in (1.10). From this point of view, thecriteria for absolute continuity based on the Malliavin calculus as utilized in this paperturns out to be crucial. The paper is organized as follows: in Section 2 we begin recalling definitions andauxiliary results from the Malliavin calculus and analysis on the Wiener space whichwill be employed in the proof of Theorem 1.4; Section 2.1 deals with the proof of theabsolute continuity of the law of the Wong-Zakai approximation while Section 2.2 ad-dresses the derivation of the Fokker-Planck-type equation solved by the density. Here,an anticipating Itˆo formula for the Wong-Zakai approximation (see Theorem 2.5 below)plays a major role. An example illustrating the results previously obtained closes thesection.
The proof of our main theorem will be divided in two parts: the existence of thedensity in Section 2.1 and the Fokker-Planck equation in Section 2.2. We first set thenotation and recall few auxiliary results from the Malliavin Calculus. For more detailswe refer the reader to one of the books Bogachev [3], Hu [9], Janson [13] and Nualart[19]. Here we adopt the presentation of [19].Let S denote the class of smooth random variables of the form F = ϕ (cid:18)Z T h ( u ) dB u , ..., Z T h n ( u ) dB u (cid:19) (2.1)where ϕ ∈ C ∞P ( R n ) (the space of infinitely differentiable functions having, together withall their partial derivatives, polynomial growth), the functions h , ..., h n are elements of L ([0 , T ]) and n ≥ H := L ([0 , T ]) anddenote by k · k p the norm in L p (Ω). We note that S is dense in L p (Ω) for all p ≥ derivative of a smooth random variable F of the form (2.1) is the H -valued randomvariable DF given by D t F = n X i =1 ∂ i ϕ (cid:18)Z T h ( u ) dB u , ..., Z T h n ( u ) dB u (cid:19) h i ( t ) , t ∈ [0 , T ] . It is easy to see that for any h ∈ H and smooth random variables F and G we have thefollowing integration-by-parts formula E [ GD h F ] = E (cid:20) − F D h G + F G Z T h ( t ) dB t (cid:21) . E [ · ] denotes the expectation on the probability space (Ω , F , P ) while D h Z := R T D u Z · h ( u ) du stands for the directional derivative of Z in the direction h ∈ H .By means of the previous identity one can prove that the operator D is closable from L p (Ω) to L p (Ω; H ) for any p ≥
1. Therefore, one can define the space D ,p as the closureof S with respect to the norm k F k ,p = ( E [ | F | p ] + E [ | DF | pH ]) p . Iterating the action of the operator D in such a way that for a smooth random variable F the iterated derivative D k F is a random variable with values in H ⊗ k , we introduceon S for every p ≥ k ≥ k F k k,p = E [ | F | p ] + k X j =1 E (cid:2) | D j F | pH ⊗ j (cid:3)! p . We will denote by D k,p the completion of S with respect to the norm k · k k,p .We now introduce the divergence operator δ which is the adjoint of the operator D . Itis a closed and unbounded operator on L (Ω; H ) with values in L (Ω) such that: • the domain of δ is the set of H -valued square integrable random variables u ∈ L (Ω; H ) such that | E [ h DF, u i H ] | ≤ c k F k , for all F ∈ D , , where c is a constant depending only on u • if u belongs to the domain of δ , then δ ( u ) is the element of L (Ω) characterized by E [ F δ ( u )] = E [ h DF, u i H ]for any F ∈ D , .One can prove that D , ( H ) is included in the domain of δ . In particular, if F ∈ D , and h ∈ H , then F h ∈ D , ( H ) and F ⋄ Z T h ( u ) dB u := δ ( F h ) ∈ L (Ω) (2.2)is called the Wick product of F and R T h ( u ) dB u . This definition can be generalized toinclude more general random variables in the place of R T h ( u ) dB u . Moreover, it followsfrom the properties of D and δ that F ⋄ Z T h ( u ) dB u = F · Z T h ( u ) dB u − D h F. emark 2.1 There are different ways to define the Wick product of two random vari-ables: via the Wiener-Itˆo chaos expansion or specifying its action on stochastic exponen-tials (see below) or through the so-called S -transform. Here, we preferred to use (2.2)since it clearly shows the duality between Wick product and Malliavin derivative. Otherapproaches can be found in [8] and [9]. We also remark that, in addition to its key rolein the theory of Itˆo-Skorohod integration (1.7), the Wick product has important proba-bilistic interpretations in the Gaussian and Poissonian analysis. See [17], [18], [16] andthe references quoted there. For h ∈ H we define the stochastic exponential to be a random variable of the form E ( h ) := exp (cid:26)Z T h ( u ) dB u − Z T h ( u ) du (cid:27) . We remark that the span of such family of elements is dense in L p (Ω) and D k,p for any p ≥ k ∈ N . For g ∈ H we also define the translation operator T g as the operatorthat shifts the Brownian path by the function R · g ( u ) du ; more precisely, the action of T g on a stochastic exponential is given by T g E ( h ) := E ( h ) · exp {h h, g i H } . There is a close relationship between Wick product and translation operators; it is theso called Gjessing formula: F ⋄ E ( h ) = T − h X · E ( h ) (2.3)which is valid for any h ∈ H and F ∈ L p (Ω) for some p >
1. We refer to Holden et al.[8] and Janson [13] for more details on Wick product and translation operators.We complete this preliminary part observing that the polygonal approximation { B πt } t ∈ [0 ,T ] of the Brownian motion { B t } t ∈ [0 ,T ] defined in (1.8) can be written also in the compactform B πt = Z T K πt ( u ) dB u (2.4)with K πt ( u ) := n − X k =0 (cid:18) [0 ,t k [ ( u ) + t − t k t k +1 − t k [ t k ,t k +1 [ ( u ) (cid:19) [ t k ,t k +1 [ ( t ) . (2.5)It is straightforward to see that 0 ≤ K πt ( u ) ≤ t, u ∈ [0 , T ] and˙ B πt = B t k +1 − B t k t k +1 − t k if t ∈ [ t k , t k +1 [ . Moreover, in analogy with (2.4) we set˙ B πt = Z T ∂ t K πt ( u ) dB u with ∂ t K πt ( u ) = n − X k =0 t k +1 − t k [ t k ,t k +1 [ ( u )1 [ t k ,t k +1 [ ( t ) . (2.6)8 .1 Existence of the density The aim of the present section is to prove the first statement in Theorem 1.4, i.e.the absolute continuity of the law of X πt with respect to the one dimensional Lebesguemeasure. We will assume that the function σ in the diffusion coefficient of equation(1.9) is identically equal to one. The general case can be recovered with straightforwardmodifications.As a first step we investigate the Malliavin regularity of X πt and write an equation for DX πt . Theorem 2.2
For any t ∈ [ s, T ] the random variable X πt belongs to D ,p for all p ≥ .Moreover, D u X πt = D u Y + Z ts b x ( r, X πr ) D u X πr dr + Z ts D u X πr ⋄ ˙ B πr dr (2.7)+ Z ts X πr ∂ r K πr ( u ) dr Remark 2.3
The need for checking that X πt belongs to D ,p in order to write the equationfor D u X πt is due to the fact that integrand in Z ts D u X πr ⋄ ˙ B πr dr requires a control on the second order Malliavin derivative of X πr to be well defined. Infact, from (2.2) we have D u X πr ⋄ ˙ B πr = δ ( D u X πr ∂ r K πr ( · )) implying that D u X πr ∈ D , is a sufficient conditions for the membership of D u X πr ∂ r K πr ( · ) to the domain of δ (see Proposition 1.3.1 in [19]). Proof.
Employing a standard reduction method in combination with the proper-ties of the Wick product, we can represent the solution { X πt } t ∈ [ s,T ] of equation (1.9) as X πt = Z πt ⋄ E ( K πs,t ) where K πs,t ( · ) := K πt ( · ) − K πs ( · ) and { Z πt } t ∈ [ s,T ] is the unique solutionof the equation (cid:26) ˙ Z πt = b ( t, Z πt ⋄ E ( K πs,t )) ⋄ E ( − K πs,t ) Z πs = Y (2.8)(see [2] for the details of this technique). Moreover, according to the Gjessing formula(2.3) we have Z πt ⋄ E ( K πs,t ) = (cid:16) T − K πs,t Z πt (cid:17) · E ( K πs,t ) . Therefore, since E ( K πs,t ) belongs to D k,p for all p ≥ k ∈ N and T − K πs,t maps D k,p into D k,q for any q < p and k ∈ N , the first part of the statement will follow from Proposition1.5.6 in [19] once we prove that Z πt belongs to D ,p for all p ≥ Z πt = Y + Z ts b ( r, Z πr · E ( − K πs,r ) − ) · E ( − K πs,r ) dr. (2.9)9e first derive an upper bound for | Z πt | which will be useful to control the norms in D ,p : | Z πt | ≤ | Y | + Z ts | b ( r, Z πr · E ( − K πs,r ) − ) | · E ( − K πs,r ) dr ≤ | Y | + M Z ts (1 + | Z πr | · E ( − K πs,r ) − ) · E ( − K πs,r ) dr = | Y | + M Z ts E ( − K πs,r ) dr + M Z ts | Z πr | dr ≤ | Y | + M Z Ts E ( − K πs,r ) dr + M Z ts | Z πr | dr. The positive constant M utilized above comes from the inequality | b ( t, x ) | ≤ M (1 + | x | ) , t ∈ [0 , T ] , x ∈ R which follows from the assumptions on b (in particular the boundedness of the firstpartial derivative with respect to x ). By the Gronwall inequality we deduce that | Z πt | ≤ (cid:18) | Y | + M Z Ts E ( − K πs,r ) dr (cid:19) e M ( t − s ) and hence sup t ∈ [ s,T ] | Z πt | ≤ (cid:18) | Y | + M Z Ts E ( − K πs,r ) dr (cid:19) e M ( T − s ) . Computing the L p (Ω)-norms we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup t ∈ [ s,T ] | Z πt | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ (cid:18) k Y k p + M Z Ts kE ( − K πs,r ) k p dr (cid:19) e M ( T − s ) = (cid:18) k Y k p + M Z Ts exp { ( p − | K πs,r | H / } dr (cid:19) e M ( T − s ) ≤ ( k Y k p + C ) e M ( T − s ) (2.10)where C is a positive constant depending on p , T and M (recall that 0 ≤ K πt ( u ) ≤ | K πs,t | ≤ Z πt ; from equation (2.9) we obtain D u Z πt = D u Y + Z ts D u [ b ( r, Z πr · E ( − K πs,r ) − ) · E ( − K πs,r )] dr = D u Y + Z ts b x ( r, Z πr · E ( − K πs,r ) − )( D u Z πr + Z πr K πs,r ( u )) dr − Z ts b ( r, Z πr · E ( − K πs,r ) − ) · E ( − K πs,r ) K πs,r ( u ) dr which gives | D u Z πt | ≤ | D u Y | + L Z ts | D u Z πr | dr + L Z ts | Z πr || K πs,r ( u ) | dr M Z ts (1 + | Z πr · E ( − K πs,r ) − | ) · E ( − K πs,r ) | K πs,r ( u ) | dr ≤ | D u Y | + L Z ts | D u Z πr | dr + L Z ts | Z πr | dr + M Z ts E ( − K πs,r ) dr + M Z ts | Z πr | dr ≤ | D u Y | + L Z ts | D u Z πr | dr + ( L + M ) Z Ts | Z πr | dr + M Z Ts E ( − K πs,r ) dr. Here, L stands for the Lipschitz constant of b with respect to the variable x . Then, bythe Gronwall inequality we obtain | D u Z πt | ≤ (cid:18) | D u Y | + ( L + M ) Z Ts | Z πr | dr + M Z Ts E ( − K πs,r ) dr (cid:19) e L ( t − s ) which in turn implies | DZ πt | H ≤ e L ( t − s ) (cid:18) | DY | H + √ T ( L + M ) Z Ts | Z πr | dr + √ T M Z Ts E ( − K πs,r ) dr (cid:19) ≤ e L ( t − s ) | DY | H + √ T ( L + M )( T − s ) sup r ∈ [ s,T ] | Z πr | + √ T M Z Ts E ( − K πs,r ) dr ! . With the help of estimate (2.10) we can conclude that k| DZ πt | H k p ≤ C k Y k D ,p + C where C and C are positive constants depending on p , T , M and L . This proves that Z πt belongs to D ,p . For the second order Malliavin derivative of Z πt one proceeds asbefore differentiating twice the identity (2.9) and resorting to the Grownwall inequalityfor the estimation of the norm of D v D u Z πt .To get identity (2.7) one has simply to differentiate the equality X πt = Y + Z ts b ( r, X πr ) dr + Z ts X πr ⋄ ˙ B πr dr in connection with the following chain rule for the Wick product: D u ( X πr ⋄ ˙ B πr ) = ( D u X πr ) ⋄ ˙ B πr + X πr ⋄ D u ˙ B πr = ( D u X πr ) ⋄ ˙ B πr + X πr ⋄ ∂ r K πr ( u )= ( D u X πr ) ⋄ ˙ B πr + X πr · ∂ r K πr ( u )The proof is complete.Equation (2.7) shows that D u X πr solves a linear differential equation containing the nonhomogeneous term Z ts X πr ∂ r K πr ( u ) dr.
11n the next proposition we will find a direction h π ∈ H along which the quantity abovewill be identically zero and this will produce an explicit expression for the resultingdirectional derivative of X πr . Proposition 2.4
There exists a function h π ∈ H such that h π = 0 almost everywhereand R T ∂ r K πr ( u ) h π ( u ) du = 0 for all r ∈ [0 , T ] . Moreover, D h π X πt = T − K πs,t D h π Y · exp (cid:26)Z ts b x ( r, T − K πs,t T K πs,r X πr ) dr (cid:27) · E ( K πs,t ) . (2.11) Proof.
Let h π be a function in H such that h π = 0 almost everywhere and1 t k +1 − t k Z t k +1 t k h π ( u ) du = 0 for all k ∈ { , ..., n − } . Then, according to the second equation in (2.6) we have Z T ∂ r K πr ( u ) h π ( u ) du = n − X k =0 t k +1 − t k Z t k +1 t k h π ( u ) du · [ t k ,t k +1 [ ( r )= 0 for all r ∈ [0 , T ] . If now we multiply both sides of equation (2.7) by the function h π and integrate withrespect to u between zero and T , we see that the last term is zero and we are left with D h π X πt = D h π Y + Z ts b x ( r, X πr ) D h π X πr dr + Z ts D h π X πr ⋄ ˙ B πr dr. (2.12)Equation (2.12) is linear and homogeneous in D h π X π · . We now find its unique solution.First of all observe that employing the reduction method mentioned above we can write D h π X πt = V πt ⋄ E ( K πs,t )where { V πt } t ∈ [ s,T ] satisfies V πt = D h π Y + Z ts ( b x ( r, X πr )( V πr ⋄ E ( K πs,r ))) ⋄ E ( − K πs,r ) dr (2.13)We note that applying the Gjessing formula (2.3) twice we can write( b x ( r, X πr )( V πr ⋄ E ( K πs,r ))) ⋄ E ( − K πs,r ) = ( b x ( r, X πr ) · T − K πs,r V πr · E ( K πs,r )) ⋄ E ( − K πs,r )= T K πs,r ( b x ( r, X πr ) · T − K πs,r V πr · E ( K πs,r )) · E ( − K πs,r )= b x ( r, T K πs,r X πr ) · V πr · T K πs,r E ( K πs,r ) · E ( − K πs,r )= b x ( r, T K πs,r X πr ) · V πr . Therefore, equation (2.13) now reads V πt = D h π Y + Z ts ( b x ( r, X πr )( V πr ⋄ E ( K πs,r ))) ⋄ E ( − K πs,r ) dr = D h π Y + Z ts b x ( r, T K πs,r X πr ) · V πr dr V πt = D h π Y exp (cid:26)Z ts b x ( r, T K πs,r X πr ) dr (cid:27) . Therefore, D h π X πt = V πt ⋄ E ( K πs,t )= (cid:18) D h π Y exp (cid:26)Z ts b x ( r, T K πs,r X πr ) dr (cid:27)(cid:19) ⋄ E ( K πs,t )= T − K πs,t D h π Y · T − K πs,t exp (cid:26)Z ts b x ( r, T K πs,r X πr ) dr (cid:27) · E ( K πs,t )= T − K πs,t D h π Y · exp (cid:26)Z ts b x ( r, T − K πs,t T K πs,r X πr ) dr (cid:27) · E ( K πs,t ) . The proof is complete.We are now ready to prove that the law of X πt is absolutely continuous with respect tothe Lebesgue measure. Our strategy is based on the following criteria that generalizesto some extent Proposition 2.1.1 in [19]: Suppose that X ∈ D , and let h ∈ H be such that D h X = 0 almost surely and h/D h X belongs to the domain of δ . Then, X possesses a bounded and continuous density. First of all we note that the assumption E [ | D h π Y | − q ] is finite for some q > D h π Y = 0 almost surely (and also that T − K πs,t D h π Y = 0since Cameron-Martin shifts preserve negligible sets). This fact combined with Propo-sition 2.4 entails that D h π X πt = 0 almost surely. Hence, we are left with the verifi-cation that h π /D h π X πt belongs to the domain of δ ; a sufficient condition for that is h π /D h π X πt ∈ D , ( H ).We observe that E [ | h π /D h π X πt | H ] = | h π | H E [1 / | D h π X πt | ]and E [ | D ( h π /D h π X πt ) | H ⊗ H ] = | h π | H E [ | DD h π X πt | H / | D h π X πt | ] ≤ | h π | H E [ | DD h π X πt | pH ] /p · E [1 / | D h π X πt | q ] /q where we applied the H¨older inequality with 1 /p + 1 /q = 1. Therefore, h π /D h π X πt ∈ D , ( H ) if D h π X πt ∈ D ,p for all p ≥ E [1 / | D h π X πt | q ] is finite for some q > . (2.14)The first condition follows from the fact that X πt ∈ D ,p (see Theorem 2.2). We nowverify the validity of (2.14). Let q > E [ | D h π Y | − q ] is finite and let α > ε > q := ( q − ε ) /α >
4; then, recalling the identity (2.11) we canwrite | D h π X πt | − ˜ q = |T − K πs,t D h π Y | − ˜ q · exp (cid:26) − ˜ q Z ts b x ( r, T − K πs,t T K πs,r X πr ) dr (cid:27) · |E ( K πs,t ) | − ˜ q ≤ e ˜ qL ( t − s ) |T − K πs,t D h π Y | − ˜ q · |E ( K πs,t ) | − ˜ q = e ˜ qL ( t − s ) T − K πs,t | D h π Y | − ˜ q · |E ( K πs,t ) | − ˜ q . Therefore, E [ | D h π X πt | − ˜ q ] ≤ e ˜ qL ( t − s ) E [ T − K πs,t | D h π Y | − ˜ q · |E ( K πs,t ) | − ˜ q ] ≤ e ˜ qL ( t − s ) E [ T − K πs,t | D h π Y | − q + ε ] /α · E [ |E ( K πs,t ) | − ˜ qβ ] /β ≤ CE [ | D h π Y | − q ] ( q − ε ) /qα where 1 /α + 1 /β = 1 and C is a positive constant depending on L , α , ε , q and T . Inthe last inequality we utilized the fact that T − K πs,t maps L q (Ω) into L r (Ω) for all r < p and the membership of E ( K πs,t ) to all the spaces L p (Ω) for p ≥
1. The assumption on Y completes the proof of the claim (2.14) which in turn implies the absolute continuity ofthe law of X πt with respect to the one dimensional Lebesgue measure with a boundedand continuous density. We now prove the second part of Theorem 1.4. We will assume as before that thefunction σ in the diffusion coefficient of equation (1.9) is identically equal to one.As for exact solutions, the key ingredient to relate the law of the solution of the stochasticequation (1.9) with a Fokker-Planck type equation is the Itˆo formula. Theorem 2.5 (Itˆo formula)
Let { X πt } t ∈ [ s,T ] be the unique solution of equation (1.9)and let ϕ ∈ C , ([0 , T ] × R ) be such that ∂ x ϕ and ∂ xx ϕ have at most polynomial growthat infinity. Then, for s ≤ t ≤ T we have ϕ ( t, X πt ) − ϕ ( s, Y ) = Z ts [ ∂ t ϕ ( r, X πr ) + ∂ x ϕ ( r, X πr ) b ( r, X πr )] dr + Z ts ∂ xx ϕ ( r, X πr ) X πr D ∂ r K πr X πr dr + Z ts ( ∂ x ϕ ( r, X πr ) X πr ) ⋄ ˙ B πr dr. Proof.
First of all we observe that the condition on the growth at infinity of ∂ x ϕ and ∂ xx ϕ together with the membership of X πr to D ,p for all p ≥ ∂ x ϕ ( r, X πr ) X πr ) ⋄ ˙ B πr = δ ( ∂ x ϕ ( r, X πr ) X πr ∂ r K πr )is well defined since ∂ x ϕ ( r, X πr ) X πr ∈ D , . Now, using equation (1.9) we get ϕ ( t, X πt ) − ϕ ( s, Y ) = Z ts h ∂ t ϕ ( r, X πr ) + ∂ x ϕ ( r, X πr ) ˙ X πr i dr Z ts [ ∂ t ϕ ( r, X πr ) + ∂ x ϕ ( r, X πr ) b ( r, X πr )] dr + Z ts ∂ x ϕ ( r, X πr ) · ( X πr ⋄ ˙ B πr ) dr. (2.15)Moreover, according to Proposition 1.3.3 in [19] we can write ∂ x ϕ ( r, X πr ) · ( X πr ⋄ ˙ B πr ) = ∂ x ϕ ( r, X πr ) · δ ( X πr ∂ r K πr )= δ ( ∂ x ϕ ( r, X πr ) X πr ∂ r K πr )+ Z T ( D u ∂ x ϕ ( r, X πr )) X πr ∂ r K πr ( u ) du = ( ∂ x ϕ ( r, X πr ) X πr ) ⋄ ˙ B πr + Z T ∂ xx ϕ ( r, X πr )( D u X πr ) X πr ∂ r K πr ( u ) du = ( ∂ x ϕ ( r, X πr ) X πr ) ⋄ ˙ B πr + ∂ xx ϕ ( r, X πr ) X πr D ∂ r K πr X πr . The substitution of the integrand in (2.15) with the last member of the previous chainof equalities provides the desired formula.Now, let ϕ ∈ C ∞ ([ s, T ] × R ); then, by Theorem 2.5 we obtain0 = ϕ ( T, X πT ) − ϕ ( s, Y )= Z Ts [ ∂ t ϕ ( r, X πr ) + ∂ x ϕ ( r, X πr ) b ( r, X πr )] dr + Z Ts ∂ xx ϕ ( r, X πr ) X πr D ∂ r K πr X πr dr + Z Ts ( ∂ x ϕ ( r, X πr ) X πr ) ⋄ ˙ B πr dr. Taking the expectation, recalling that for all X ∈ D , E [ X ⋄ ˙ B πr ] = E [ δ ( X∂ r K πr )] = 0and denoting by p π ( r, x ) the density of the random variable X πr , we get0 = E (cid:20)Z Ts ∂ t ϕ ( r, X πr ) + ∂ x ϕ ( r, X πr ) b ( r, X πr ) dr (cid:21) + E (cid:20)Z Ts ∂ xx ϕ ( r, X πr ) X πr D ∂ r K πr X πr dr (cid:21) = Z Ts E [ ∂ t ϕ ( r, X πr ) + ∂ x ϕ ( r, X πr ) b ( r, X πr )] dr + Z Ts E (cid:2) ∂ xx ϕ ( r, X πr ) X πr D ∂ r K πr X πr (cid:3) dr = Z Ts Z R ( ∂ t ϕ ( r, x ) + ∂ x ϕ ( r, x ) b ( r, x )) p π ( r, x ) dxdr Z Ts E (cid:2) ∂ xx ϕ ( r, X πr ) X πr E [ D ∂ r K πr X πr | σ ( X πr )] (cid:3) dr = Z Ts Z R ( ∂ t ϕ ( r, x ) + ∂ x ϕ ( r, x ) b ( r, x )) p π ( r, x ) dxdr + Z Ts Z R ∂ xx ϕ ( r, x ) xg ( r, x ) p π ( r, x ) dxdr where g : [ s, T ] × R → R is a measurable function such that g ( r, x ) = E [ D ∂ r K πr X πr | σ ( X πr )] | X πr = x . (2.16)We observe that g is uniquely defined up to sets that are negligible with respect to thelaw of X πr and hence negligible with respect to the Lesbegue measure. According to thisconstruction the function g is barely measurable with only some integrability propertiesagainst the density p π ( t, x ). In fact, since D ∂ r K πr X πr ∈ L q (Ω) for all q ≥ Z R | g ( r, x ) | q p π ( r, x ) dx = E (cid:2) | E [ D ∂ r K πr X πr | σ ( X πr )] | q (cid:3) ≤ E (cid:2) | D ∂ r K πr X πr | q (cid:3) . We have therefore proved for any ϕ ∈ C ∞ ([ s, T ] × R ) the identity Z Ts Z R ( ∂ t ϕ ( r, x ) + ∂ x ϕ ( r, x ) b ( r, x ) + ∂ xx ϕ ( r, x ) xg ( r, x )) p π ( r, x ) dxdr = 0which is equivalent to the statement that p π ( t, x ) is a distributional solution of theequation ( ∂ t + b ( t, x ) ∂ x + xg ( t, x ) ∂ xx ) ∗ u ( t, x ) = 0 . Example 2.6
Consider the case where b ( t, x ) = 0 and σ ( t ) = 1 . Then, equation (1.9)reads (cid:26) ˙ X πt = X πt ⋄ ˙ B πt , t ∈ ] s, T ] X πs = Y (2.17) The solution to this equation is given by the formula X πt = Y ⋄ E ( K πs,t ) . (2.18) If we take Y = X s where { X t } t ∈ [0 ,T ] is the solution of (cid:26) dX t = X t dB t , t ∈ ]0 , T ] X = x ∈ R (2.19) (i.e. the SDE we are approximating with (2.17)) then X s = x · E (1 [0 .s [ ) and by theproperties of the Wick product we can write (2.18) as X πt = X s ⋄ E ( K πs,t )= x · E (1 [0 .s [ ) ⋄ E ( K πs,t )16 x · E (1 [0 .s [ + K πs,t ) . We now aim at finding an explicit expression for g from formula (2.16) and hence for theFokker-Planck-type equation associated to (2.17). First of all we compute the Malliavinderivative of X πt : D u X πt = D u ( x · E (1 [0 .s [ + K πs,t ))= X πt · (1 [0 .s [ ( u ) + K πs,t ( u )) . Then, we get g ( t, x ) = E [ D ∂ t K πt X πt | σ ( X πt )] | X πt = x = E (cid:20) X πt Z T (1 [0 .s [ ( u ) + K πs,t ( u )) ∂ t K πt ( u ) du (cid:12)(cid:12)(cid:12) σ ( X πt ) (cid:21) (cid:12)(cid:12)(cid:12) X πt = x = x Z T (1 [0 .s [ ( u ) + K πs,t ( u )) ∂ t K πt ( u ) du. Denoting ξ π ( t ) := Z T (1 [0 .s [ ( u ) + K πs,t ( u )) ∂ t K πt ( u ) du we can write the Fokker-Planck-type equation for (2.17) as ( ∂ t + x ξ π ( t ) ∂ xx ) ∗ p π ( t, x ) = 0 to be compared with ( ∂ t + ( x / ∂ xx ) ∗ p ( t, x ) = 0 which is the one for the exact equation (2.19). References [1] V. Bally and D. Talay, The law of the Euler scheme for stochastic differentialequations. II. Convergence rate of the density,
Monte Carlo Methods Appl. (1996)93-128.[2] B. K. Ben Ammou and A. Lanconelli, Rate of convergence for Wong-Zakai-typeapproximations of Itˆo stochastic differential equations, J. Theor. Probab. (2018)https://doi.org/10.1007/s10959-018-0837-x[3] V. I. Bogachev,
Gaussian Measures , American Mathematical Society, Providence,1998.[4] Z. Brezniak and F. Flandoli, Almost sure approximation of Wong-Zakai type forstochastic partial differential equations,
Stoch. Proc. and their Appl. (1995) 329-358.[5] P. Da Pelo, A. Lanconelli and A. I. Stan, An Itˆo formula for a family of stochasticintegrals and related Wong-Zakai theorems, Stoch. Proc. and their Appl. (2013)3183-3200. 176] I. Gy¨ongy and A. Shmatkov, Rate of Convergence of WongZakai Approximationsfor Stochastic Partial Differential Equations,
Appl. Math. Optim. (2006) 315-341.[7] M. Hairer and E. Pardoux, A WongZakai theorem for stochastic PDEs, J. Math.Soc. Japan. (2015) 1551-1604.[8] H. Holden, B. Øksendal, J. Ubøe and T.-S. Zhang, Stochastic Partial DifferentialEquations - II Edition , Springer, New York, 2010.[9] Y. Hu,
Analysis on Gaussian spaces , World Scientific Publishing Co. Pte. Ltd.,Hackensack, NJ, 2017.[10] Y. Hu, G. Kallianpur and J. Xiong, An approximation for Zakai equation,
AppliedMath. Optimiz. (2002) 2344.[11] Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer.Math. Soc. (2009) 2689-2718.[12] Y. Hu and B. Øksendal, Wick approximation of quasilinear stochastic differentialequations,
Stochastic analysis and related topics V Birkh¨auser (1996) 203-231.[13] S. Janson,
Gaussian Hilbert spaces , Cambridge Tracts in Mathematics, 129. Cam-bridge University Press, Cambridge, 1997.[14] I. Karatzas and S. E. Shreve,
Brownian motion and stochastic calculus , Springer-Verlag, New York, 1991.[15] F. Konecny, A Wong-Zakai approximation of stochastic differential equations,
Jour-nal of Multivariate Analysis (1983) 605-611.[16] A. Lanconelli, Standardizing densities on Gaussian spaces, Statistics and ProbabilityLetters (2018) 243-250[17] A. Lanconelli and A. I. Stan, A H¨older inequality for norms of PoissonianWick products,
Inf. Dim. Anal. Quantum Prob. Related Topics (2013)https://doi.org/10.1142/S0219025713500227[18] A. Lanconelli and A. I. Stan, A note on a local limit theorem for Wiener spacevalued random variables, Bernoulli (2016) 2001-2112.[19] D. Nualart, Malliavin calculus and Related Topics - II Edition , Springer, New York,2006.[20] D. W. Stroock and S. R. S. Varadhan, On the support of diffusion processes withapplications to the strong maximum principle,
Proceedings -th Berkeley SymposiumMath. Statist. Probab. (1972) University of California Press, Berkeley, 333-359.[21] G. Tessitore and G. J. Zabczyk, Wong-Zakai approximations of stochastic evolutionequations, Journal Evol. Equ. (2006) 621-655.[22] E. Wong and M. Zakai: On the relation between ordinary and stochastic differentialequations, Intern. J. Engr. Sci. (1965) 213-229.1823] E. Wong and M. Zakai, Riemann-Stieltjes approximations of stochastic integrals, Z. Wahrscheinlichkeitstheorie verw. Geb.12