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Approximation for non-smooth functionals of stochastic differential equations with irregular drift

This paper aims at developing a systematic study for the weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to obtain the rates of approximation for the expectation of various non-smooth functionals of both stochastic differential equations and killed diffusion. We also apply our method to the study of the weak approximation of reflected stochastic differential equations whose drift is H\"older continuous.


Introduction
Let T > 0 be fixed and (X t ) 0≤t≤T be the solution to where W is a d-dimensional Brownian motion. The diffusion (X t ) 0≤t≤T is used to model many random dynamical phenomena in many fields of applications. In practice, one often encounters the problem of evaluating functionals of the type E[f (X)] for some given function f : C[0, T ] → R. For example, in mathematical finance the function f is commonly referred as a payoff function. Since they are rarely analytically tractable, these expectations are usually approximated using numerical schemes. One of the most popular approximation methods is the Monte Carlo Euler-Maruyama method which consists of two steps: 1. The diffusion process (X t ) 0≤t≤T is approximated using the Euler-Maruyama scheme (X h t ) 0≤t≤T with a small time step h > 0:

The expectation E[f (X)] is approximated using
This approximation procedure is influenced by two sources of errors: a discretization error and a statistical error We say that the Euler-Maruyama approximation (X h ) is of weak order κ > 0 for a class H of functions f if there exists a constant K(T ) such that for any f ∈ H, The effect of the statistical error can be handled by the classical central limit theorem or large deviation theory. Roughly speaking, if f (X h T ) has a bounded variance, the L 2 -norm of the statistical error is bounded by N −1/2 V ar(X h T ) 1/2 . Hence, if the Euler-Maruyama approximation is of weak order κ, the optimal choice of the number of Monte Carlo iterations should be N = O(h −2κ ) in order to minimize the computational cost. Therefore, it is of both theoretical and practical importance to understand the weak order of the Euler-Maruyama approximation.
It has been shown that under sufficient regularity on the coefficients b and σ as well as f , the weak order of the Euler-Maruyama approximation is 1. This fact is proven by writing the discretization error Err(f, h) as a sum of terms involving the solution of a parabolic partial differential equation (see [2,39,32,20,11]). It should be noted here that besides the Monte Carlo Euler-Maruyama method, there are many other related approximation schemes for E[f (X T )] which have either higher weak order or lower computational cost. For example, one can use Romberg extrapolation technique to obtain very high weak order as long as Err(h) can be expanded in terms of powers of h (see [39]). When f is a Lipschitz function and the strong rate of approximation is known, one can implement a Multi-level Monte Carlo simulation which can significantly reduce the computation cost of approximating E[f (X)] in many cases (see [9]). It is also worth looking at some algebraic schemes introduced in [25]. However, all the accelerated schemes mentioned above require sufficient regularity conditions on the coefficients b, σ and the test function f .
The stochastic differential equations with non-smooth drift appear in many applications, especially when one wants to model sudden changes in the trend of a certain random dynamical phenomenon (see e.g., [20]). There are many papers studying the Euler-Maruyama approximations in this context. In [13] (see also [3]), it is shown that when the drift is only measurable, the diffusion coefficient is non-degenerate and Lipschitz continuous then the Euler-Maruyama approximations converges to the solution of stochastic differential equation. The weak order of the Euler-Maruyama scheme when both coefficients b and σ as well as payoff functions f are Hölder continuous has been studied in [20,31]. In the papers [21] and [33,34], the authors studied the weak and strong convergent rates of the Euler-Maruyama scheme for specific classes of stochastic differential equations with discontinuous drift. We would like to note here some very recent papers on the unbiased simulation [16] and exact simulation [1,5,6,36] methods for stochastic differential equations with irregular coefficients.
The aim of the present paper is to investigate the weak order of the Euler-Maruyama approximation for stochastic differential equations whose diffusion coefficient σ is constant, whereas the drift coefficient b may have a very low regularity, or could even be discontinuous. More precisely, we consider a class A of functions which contains not only smooth functions but also some discontinuous one such as indicator function. The drift b will then be assumed to be either in A or α-Hölder continuous. It should be noted that no smoothness assumption on the payoff function f is needed in our framework. As a by product of our method, we establish the weak order of the Euler-Maruyama approximation for some particular functionals f which include the path-wise maximum of the diffusion, integral of diffusion with respect to time as well as the approximation of a diffusion processes killed when it leaves an open set. We also apply our method to study the weak approximation of reflected stochastic differential equation whose drift is Hölder continuous.
The remainder of this paper is organized as follows. In the next section we introduce some notations and assumptions for our framework together with the main results. All proofs are deferred to Section 3.

Notations
In particular we denote g c (x, y) = g cI (x, y) for c ∈ R where the matrix I is the identity matrix.
Let A be a class of exponentially bounded functions ζ : R d → R such that there exists a sequence of functions (ζ N ) N ∈N ⊂ C 1 (R d ) satisfying: for some positive constant K. We call (ζ N ) N ∈N an approximation sequence of ζ in A.
The following propositions shows that this class is quite large.
It is easy to verify that the class A contains all C 1 (R d ) functions which has all first order derivatives polynomially bounded. Furthermore, the class A contains also some non-smooth functions of the type ζ(x) = (x 1 − a) + or ζ(x) = I a<x<b for some a, b ∈ R d . Moreover, we call a function ζ : R d → R monotone in each variable separately if for each i = 1, . . . , d, the map Proposition 2.2. Class A contains all exponentially bounded functions which are monotone in each variable separately.
The proofs of Propositions 2.1 and 2.2 and further properties of class A were presented in [22] and [33].
We recall that a function ζ : R d → R d is called α-Hölder continuous for some α ∈ (0, 1] if there exists a positive constant C such that |ζ(x) − ζ(y)| ≤ C|x − y| α for all x, y ∈ R d . We denote M(R d , R d ) the class of all measurable functions b : R d → R d . Let α ∈ (0, 1] and let B(α) be the class fo functions given by A function b ∈ B(α) could be very irregular since its trajectory may both change continuously very fast because of the Hölder part b H , and be discontinuous at some points because of the part b A ∈ A. Moreover, the class A does not contain any class of Hölder continuous functions H β for any β ∈ (0, 1) (see [34]).

Weak approximation of stochastic differential equations
Let (Ω, F , (F t ) t≥0 , P) be a filtered probability space and (W t ) t≥0 be a d-dimensional standard Brownian motion. We consider a d-dimensional stochastic differential equation where σ is a d × d deterministic, uniformly elliptic matrix, that is for the matrix a := σσ * , there exist 0 < a < a < ∞ such that for any ξ ∈ R d , and b : R d → R d is a Borel measurable function. Let X h , h > 0, denote the Euler-Maruyama approximation of X, where η h (s) = kh if kh ≤ s < (k + 1)h for some nonnegative integer k. In this paper, we study the convergent rates of the error as h → 0 for some payoff function f : Remark 2.3. Note that the uniformly elliptic condition plays an important role in establishing the convergence of the Euler-Maruyama approximation for SDEs with non-Lipschitz coefficients. In fact, Hairer, Hutzenthaler and Jentzen [15, Theorem 5.1] constructed a class of 4-dimensional SDEs whose drift coefficient is a smooth, bounded and non-Lipschitz function, and diffusion coefficient is a deterministic non-uniformly elliptic matrix for which the Euler-Maruyama scheme does not converges with any polynomial rate, that is A Borel measurable function ζ : R d → R d is called sub-linear growth if ζ is bounded on compact sets and ζ(y) = o(|y|) as y → ∞. ζ is called linear growth if |ζ(y)| < c 1 |y| + c 2 for some positive constants c 1 , c 2 .
Remark 2.4. (i) It has been shown recently in [17] that when b is of super-linear growth, i.e., there exist constants C > 0 and θ > 1 such that |ζ(y)| ≥ |y| θ for all |y| > C, then the Euler-Maruyama approximation (6) converges neither in the strong mean square sense nor in weak sense to the exact solution at a finite time point.
Thus, in this paper we will consider the case that b is of at most linear growth.
(ii) In the one-dimensional case, d = 1, it is well-known that if σ = 0 and b is of linear growth, then the strong existence and path-wise uniqueness hold for the equation (5) (see [4]).
(iii) In the multidimensional case, d > 1, it has been shown in [41] that if b is bounded then the equation (5) has a strong solution and the solution of (5) is strongly unique. Moreover, if σ is the identity matrix, then the equation (5) has a unique strong solution in the class of continuous processes such that P T 0 |b(X s )| 2 ds < ∞ = 1 provided that R d |b(y)| p dy < ∞ for some p > d ∨ 2 (see [24]).
Throughout this paper, we suppose that equation (5) has a weak solution which is unique in the sense of probability law (see Chapter 5 [19]).
The following results requires no assumption on the smoothness of f .
Theorem 2.5. Let X be a solution of the SDE (5). Suppose that b ∈ B(α) and b is of linear growth. Moreover, assume that f : If b is of sub-linear growth, we can obtain the rate of weak convergence as follows.
Theorem 2.6. Let X be a solution of the SDE (5) and h > 0. Suppose that b ∈ B(α) and b is of sub-linear growth. Moreover, assume that f : The following result concerns with the approximation of maximum of SDEs.
Corollary 2.7. Assume the hypotheses of Theorem 2.6. Moreover, suppose that g : R → R is β-Hölder continuous with β ∈ (0, 1]. Then there exists a constant C which does not depend of h such that For an integral type functional, we obtain the following corollary.
is bounded, then for any Lipschitz continuous function f and g ∈ B(β) with β ∈ (0, 1], there exists a constant C which does not depend of h such that Remark 2.9. In the paper [29], the author considered the weak rate of convergence of the Euler-Maruyama scheme for equation (5) in the case of a one-dimensional diffusion. It was claimed that if b was Lipschitz continuous, the weak rate of approximation is of order 1. However, we would like to point out that the given proof contains several gaps (see for instance Lemma 2 of [29] and Remark 3.3 below) which leave us unsure about the claim.
Remark 2. 10. It has been shown in [20,31] that for a stochastic differential equation with α-Hölder continuous drift and diffusion coefficients with α ∈ (0, 1), one has where f ∈ C 2 b and the second derivative of f is α-Hölder continuous. On the other hand, in [14], Gyöngy and Rásonyi have obtained the strong rate of convergence for a one-dimensional stochastic differential equation whose drift is the sum of a Lipschitz continuous and a monotone decreasing Hölder continuous function, and its diffusion coefficient is Hölder continuous. In [33], the authors improve the results in [14]. More precisely, we assume that the drift coefficient b is a bounded and one-sided Lipschitz function, i.e., there exists a positive constant L such that for any x, y ∈ R d , Therefore, if the payoff function f is Lipschitz continuous, it is straightforward to verify that In the following we consider a special case of the functional f . More precisely, we are interested in the law at time T of the diffusion X killed when it leaves an open set. Let D be an open subset of R d and denote τ D = inf{t > 0 : X t ∈ D}. Quantities of the type E[g(X T )1 (τD>T ) ] appear in many domains, e.g. in financial mathematics when one computes the price of a barrier option on a d-dimensional asset price random variable X t with characteristics f, T and D (see [10,12] and the references therein for more detail). We approximate τ D by τ h D = inf{kh > 0 : X h kh ∈ D, k = 0, 1, . . .}.
Corollary 2.11. Assume the hypotheses of Theorem 2.6. Furthermore, we assume (i) D is of class C ∞ and ∂D is a compact set (see [8] and [10]); Then for any p > 1, there exist constants C and C p independent of h such that Finally, we consider the approximation for the density of SDE (5). Let p t (x 0 , ·) and p h t (x 0 , ·) be the density functions of X t and X h t respectively. Then we have the following rate of convergence.
Theorem 2.12. Suppose that b ∈ B(α) and bounded. Then for any p > d and r > 1, there exists constants C p,r and c p such that for any y ∈ R d and h ∈ (0, T /2), where g cpT (x 0 , y) is the density function of normal distribution defined in (2). Remark 2.13. (i) Note that if d = 1, we can choose p ∈ (1, 2) and r = 2/p, and then (ii) Konakov and Menozzi [23] obtain a better rate of convergence under a further assumption that the drift coefficient is piecewise smooth (see Theorem 2 in [23]). However, in our setting, the drift coefficient may have infinite number of discontinuous points.

Weak approximation of reflected stochastic differential equations
We first recall the Skorohod problem.
Lemma 2.14 ( [19], Lemma III.6.14). Let z ≥ 0 be a given number and y : [0, ∞) → R be a continuous function with y 0 = 0. Then there exists a unique continuous function ℓ = (ℓ t ) t≥0 satisfying the following conditions: Let us consider the following one-dimensional reflected stochastic differential equation valued in [0, ∞) such that where (L 0 t (X)) 0≤t≤T is a non-decreasing continuous process stating at the origin and L 0 t (X) is called the symmetric local time of X up to time t at the level 0. Equation (8) will be studied under the assumptions that b is of sub-linear growth and Hölder continuous and σ is a non-zero constant. Moreover, we assume that the SDE (8) has a weak solution and the uniqueness in the sense of probability law holds (see [38,40]). Using Lemma 2.14, we have for the reflected stochastic differential equation (8). Let X h 0 := x 0 and define The existence of the pair (X h t , L 0 t (X h )) 0≤t≤T is deduced from Lemma 2.14. Moreover By the definition of the Euler-Maruyama scheme, we have the following representation. For each k = 0, 1, . . ., Though A k is defined by the supremum of a stochastic process, it can be simulated by using the following lemma.
Let U t be a centered Gaussian random variable with variance t and let V t be an exponential random variable with parameter 1/(2t) independent from U t . Define Then the vectors (W t , S t ) and (U t , Y t ) have the same law.
Under the Lipschitz condition for the coefficients of the reflected SDE (8), Lépingle [28] shows that for some constant C.
We obtain the following result on the weak convergence for the Euler-Maruyama scheme for a reflected SDE with non-Lipschitz coefficient.

Proofs
From now on, we will repeatedly use without mentioning the following elementary estimate sup x∈R |x| p e k|x|−x 2 < ∞, for any p ≥ 0, k ∈ R.
Throughout this section, a symbol C stands for a positive generic constant independent of the discretization parameter h, which nonetheless may depend on time T , coefficients b, σ and payoff function f .

Change of Measures
Let X be a solution of the SDE (5) defined in a certain probability space (Ω, F , P) with a Brownian motion (W t , F t ) t≥0 . The notations x 0 , b, σ, η h are defined as at the beginning of Section 2.2. From now on, we will use the following notations where we use Einstein's summation convention on repeated indices. We also use the following auxiliary stopping times Lemma 3.1. Let X be a solution of the SDE (5) and h > 0. Suppose that b is a function with at most linear growth, then we have the following representations and for all measurable functions f : C[0, T ] → R and g : R d → R provided that all the above expectations are integrable.
Proof. Let σ −1 be the inverse matrix of σ. Since b is of linear growth, so is σ −1 b. Thus, there exist constants c 1 , c 2 > 0 such that |b(x)| < c 1 |x| + c 2 for any x ∈ R d . For any 0 ≤ t ≤ t 0 ≤ T , Applying Gronwall's inequality for t ∈ [0, t 0 ], one obtains On the other hand, for each integer k ≥ 1, one has Hence, a simple iteration yields that Thus, for any t ∈ (0, T ], Therefore, for any t ∈ (0, T ], we have We define new measures Q and Q h as It follows from Corollary 3.5.16 in [19] together with estimates (13) and (14) that Q and Q h are probability measures. Furthermore, it follows from Girsanov theorem that processes are d-dimensional Brownian motions with respect to Q and Q h , respectively. Note that X s = x 0 + σB s and X h s = x 0 + σB h s . Therefore, Repeating the previous argument leads to , which implies (11). The proof of (12) is similar and will be omitted.
From now on, we will use the representation formulas in Lemma 3.1 to analyze the weak rate of convergence. We need the following estimates on the moments of Z and Z h . Lemma 3.2. Suppose that b is of sub-linear growth. Then for any p > 0, for some constant C which is not depend on h.
Proof. It suffices to proof the statement for p ≥ 1. Using Hölder's inequality, we have Since b is of linear growth, so is σ −1 b and it follows from Corollary 3.5.16 in [19] that On the other hand, since b is bounded on compact sets and b(y) = o(|y|) as y → ∞, for any δ > 0 sufficiently small, there exists a constant M > 0 such that |σ −1 b(x 0 + σy)| 2 ≤ δ|y| 2 + M for any y ∈ R d . Thus, Hence, where the last inequality follows from Hölder's inequality and the fact that Remark 3.3. In general, the conclusion of Lemma 3.2 is no longer correct if we only suppose that b is of linear growth or even Lipschitz. Indeed, consider the particular case that d = 1, σ = 1 and b(x) = x, which is a Lipschitz function. It follows from Hölder's inequality that Furthermore, for any p, T > 0 such that pT ≥ 2 and pT 2 < 1/2, we have E e pW 2 T /4 = ∞, whereas Therefore,

Some auxiliary estimates
We recall that A is a class of functions satisfying (3) and W is a d-dimensional Brownian motion.
The following result plays a crucial role in our argument.
Lemma 3.4. For any ζ ∈ A, any p ≥ 1, t > s > 0, for some constant C p not depending on neither t nor s. On the other hand, if ζ is α-Hölder continuous then Proof. If ζ ∈ A, let (ζ N ) N ∈N be an approximating sequence of ζ in A. Since ζ N → ζ in L 1 loc (R d ) and ζ and ζ N are uniformly exponential bounded, we have Next, we will show that Indeed, we write From (18) and (19) we get (16). The proof of (17) is straightforward.
Then there exists C > 0 such that for any 0 < t 1 < t 2 < t 3 < t 4 ≤ T , Proof. Let (ζ A,N ) N ∈N be an approximating sequence of ζ A in A. Since ζ A,N → ζ A in L 1 loc (R d ) and ζ A and ζ A,N are uniformly exponential bounded, we have Next, we will show that Using the mean-value theorem, (26) is bounded by thus (27) is bounded by From (24) and (25) we get (20). The proofs of (21), (22) and (23) follow from similar arguments.
In order to prove some key estimations in Lemma 3.7, we need the following inequality. Recall that η h (s) is defined in (1). Lemma 3.6. Let n be a natural number such that (n − 1)h < T ≤ nh. Define t h i = ih for i = 0, . . . , n − 1 and t h n = T . Then it holds that Proof. We first note that if s ≥ t h 1 then η h (s) ≥ s/2. The first integral is estimated as follows The second integral is estimated as follows The third integral is estimated as follows This concludes the proof of the statement. Now by using Lemmas 3.4, 3.5 and 3.6, we are able to obtain the following estimates on the difference of Y t and Y h t , which are defined in Section 3.1.
(i) If b is bounded, then for any p ≥ 2, there exists a constant C > 0 such that (ii) If b is of linear growth, then there exists a constant C > 0 such that It follows from Burkholder-Davis-Gundy's inequality that, Furthermore, it follows from Proposition 2.1 ii) and Lemma 3.4 that Therefore, S 1 (p) ≤ C(h 1/2 + h pα ).
Proof of (i). We assume that b is bounded. Using Hölder's inequality, we obtain Since b is bounded, it holds that for any x, y ∈ R and j = 1, . . . , d, Thus, by dividing the integral into two parts: from 0 to h and from h to T , and applying a similar argument as above, we obtain Thus, Proof of (ii). We assume that b is of linear growth. We observe that Let n be a natural number such that (n − 1)h < T ≤ nh. Define t h i = ih for i = 0, . . . , n − 1 and t h n = T . Since b is of linear growth, we have Since that for any x, y ∈ R d , by using Lemma 3.5 with t 1 = η h (s), t 2 = s, t 3 = η h (u) and t 4 = u, we obtain Therefore from Lemma 3.6, we have

Proof of Theorem 2.5
We first recall that Z t , Z h t , Y t and Y h t were defined in Section 3.
On the other hand, since f is bounded, it follows from (11) that This estimate together with (28) implies the desired result.

Proof of Theorem 2.6
It is clear that |e x − e y | ≤ (e x + e y )|x − y|. This estimate and Hölder's inequality imply that Thanks to the integrability condition of f and Lemma 3.2, This together with Lemma 3.7 implies the desired result.

Proof of Corollary 2.7
Since g is β-Hölder continuous, it holds that E [|g (max 0≤s≤T |x 0 + σW s |) | r ] < ∞ for any r > 2. Thanks to Theorem 2.6, it remains to estimate Since g is β-Hölder continuous and b is sub-linear growth, we have Thus from (29), we obtain which implies the proof of the statement.

Proof of Corollary 2.8
We first note that if b is bounded, then it holds from Theorem 2.1 in [26] (see also Corollary 3.2 in [33]) that there exists a density function p h t of X h t for t ∈ (0, T ] and it satisfies the following Gaussian upper bound, i.e., for some positive constants C and c. is finite for any r > 2. Since |g(x)| ≤ Ke K|x| , it follows from Jensen's inequality that for any r > 2, Since Thanks to Theorem 2.6, it remains to prove that Since f is a Lipschitz continuous function, we have If s ∈ (0, h], then by using the Gaussian upper bound for p h s (x), we have On the other hand, for s ∈ [h, T ], using the Gaussian upper bound for p h η h (s) and following the proof of Lemma 3.4 (see also Lemma 3.5 in [33]), we have Therefore, we conclude the proof of the statement.

Proof of Corollary 2.11
Recall that τ W D , τ W,h D , Z, Z h were defined at the beginning of Section 3.1. It suffices to proof the statement for the case that g is positive. It follows from (12) It follows from the proof of Theorem 2.6 that Applying Hölder's inequality, we have where q is the conjugate of p. Thanks to Lemma 3.2 and the fact τ W,h D ≥ τ W D , we have It follows from Theorem 2.4 in [10] that there exists a constant K(T ) such that Combining this estimate with (30) completes the proof.

Proof of Theorem 2.12
The proof is the based on the perturbation or Levi's parametrix method (see [7]) for the density functions p t (x 0 , y) and p h t (x 0 , y). It is known that when the drift coefficient b is bounded and Hölder continuous, it holds that for any t ∈ (0, T ] where a = σσ * and g ta (x 0 , y) is called the parametrix (see, Chapter 1 in [7]). We first consider a representation which is similar to (31) for the density functions p t (x 0 , y) and p h t (x 0 , y) under the assumption that the drift coefficient is bounded measurable. Recently, Makhlouf [30,Theorem 3.1] proved that the representation (31) also holds for a Brownian motion with random drift b = (b t ) 0≤t≤T under the suitable growth condition. For the convenience of the reader, we give a proof below.
Proposition 3.8 (Makhlouf [30]). Let W = (W t ) 0≤t≤T be a d-dimensional (F t ) 0≤t≤T -Brownian motion. Suppose that the stochastic process b = (b t ) 0≤t≤T is adapted to (F t ) 0≤t≤T and there exists a constant K > 0 such that sup 0≤t≤T |b t | ≤ K almost surely. Then, for any t ∈ (0, T ] and x ∈ R d , the stochastic process Y t := x + t 0 b s ds + σW t admits a density γ t (x, ·) with respect to Lebesgue measure and for any y ∈ R d , Proof. It suffices to prove that for any infinitely differentiable functions f : It is well-known that the function u(s, x) := E [f (x + σW t−s )] is a solution to the following partial differential equation: Hence we have By using Itô's formula and (33), it holds that for any ε ∈ (0, T ), Since for any i = 1, . . . , d, s ∈ [0, T ) and x ∈ R d , and ∂ ∂x i u(s, x) ≤ C f ∞ (t − s) 1/2 , for some constant C > 0, the stochastic integral in (36) is martingale. By taking the expectation and Fubini's theorem, we have from (34) Taking ε → 0 and using the dominated convergence theorem, we have from (35) and (37), This concludes the proof.
Note that for any q > 1, there exist C q and c q such that E |g c(T −s) (x 0 + σW s , y)| q 1/q ≤ C q T T − s d(q−1) 2q g cqT (x 0 , y).
We first consider the upper bounded for A s . By using |e x − e y | ≤ (e x + e y )|x − y| and Hölder's inequality, for any p, q > 1 with 1/p + 1/q = 1 and r > 1, we have .
Since b is bounded, we have By the assumption h ∈ (0, T /2), it holds that dQ h dP = exp −