A strong order 3/4 method for SDEs with discontinuous drift coefficient
aa r X i v : . [ m a t h . P R ] A p r A STRONG ORDER / METHOD FOR SDES WITH DISCONTINUOUSDRIFT COEFFICIENT
THOMAS M ¨ULLER-GRONBACH AND LARISA YAROSLAVTSEVA
Abstract.
In this paper we study strong approximation of the solution of a scalar stochasticdifferential equation (SDE) at the final time in the case when the drift coefficient may havediscontinuities in space. Recently it has been shown in [14] that for scalar SDEs with a piecewiseLipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuitypoints of the drift coefficient the classical Euler-Maruyama scheme achieves an L p -error rate ofat least 1 / p ∈ [1 , ∞ ). Up to now this was the best L p -error rate available in theliterature for equations of that type. In the present paper we construct a method based onfinitely many evaluations of the driving Brownian motion that even achieves an L p -error rateof at least 3 / p ∈ [1 , ∞ ) under additional piecewise smoothness assumptions on thecoefficients. To obtain this result we prove in particular that a quasi-Milstein scheme achievesan L p -error rate of at least 3 / Introduction
Consider a scalar autonomous stochastic differential equation (SDE)(1) dX t = µ ( X t ) dt + σ ( X t ) dW t , t ∈ [0 , ,X = x with deterministic initial value x ∈ R , drift coefficient µ : R → R , diffusion coefficient σ : R → R ,1-dimensional driving Brownian motion W and assume that (1) has a unique strong solution X . In this paper we study L p -approximation of X based on finitely many evaluations of W atpoints in [0 ,
1] in the case when µ may have finitely many discontinuity points.Numerical approximation of SDEs with a drift coefficient that is discontinuous in space hasgained a lot of interest in recent years, see [2, 3] for results on convergence in probability andalmost sure convergence of the Euler-Maruyama scheme and [1, 4, 9, 10, 11, 15, 16, 17, 18] forresults on L p -approximation. Up to now the most far going results on L p -approximation havebeen achieved under the following two assumptions on the coefficients µ and σ .(A1) There exist k ∈ N and ξ , . . . , ξ k +1 ∈ [ −∞ , ∞ ] with −∞ = ξ < ξ < . . . < ξ k < ξ k +1 = ∞ such that µ is Lipschitz continuous on the interval ( ξ i − , ξ i ) for all i ∈ { , . . . , k + 1 } ,(A2) σ is Lipschitz continuous on R and σ ( ξ i ) = 0 for all i ∈ { , . . . , k } .Note that under the assumptions (A1) and (A2) the equation (1) has a unique strong solution,see [9, Theorem 2.2]. In [9, 10] a numerical method has been constructed which is based on asuitable transformation of the solution X and which achieves, under the assumptions (A1) and(A2), an L -error rate of at least 1 / W . In [11] ithas been shown that the Euler-Maruyama scheme achieves an L -error rate of at least 1 / − in terms of the number of evaluations of W if (A1) and (A2) are satisfied and, additionally,the coefficients µ and σ are bounded. In [15] an adaptive Euler-Maruyama scheme has beenconstructed, which achieves, under the assumptions (A1) and (A2), an L -error rate of at least1 / − in terms of the average number of evaluations of W . Finally, in [14] it has been shownthat, under the assumptions (A1) and (A2), the Euler-Maruyama scheme in fact achieves for all p ∈ [1 , ∞ ) an L p -error rate of at least 1 / W as in thecase of SDEs with globally Lipschitz continuous coefficients.It is well known that if the coefficients µ and σ are differentiable and have bounded andLipschitz continuous derivatives, then the Milstein scheme achieves for all p ∈ [1 , ∞ ) an L p -errorrate of at least 1 in terms of the number of evaluations of W , see e.g. [6]. It is therefore naturalto ask whether an L p -error rate better than 1 / W also in the case of coefficients µ and σ that satisfy (A1) and (A2) andhave additional piecewise smoothness properties. To the best of our knowledge the answer tothis question was not known in the literature up to now. In the present paper we answer thisquestion in the positive. More precisely, we show that if the coefficients µ and σ satisfy (A1)and (A2) and, additionally, the assumption(A3) µ and σ are differentiable on the interval ( ξ i − , ξ i ) with Lipschitz continuous derivativesfor all i ∈ { , . . . , k + 1 } then an L p -error rate of at least 3 / p ∈ [1 , ∞ ) can be achieved by a method based onevaluations of W at a uniform grid. More formally, we have the following result, which is animmediate consequence of Theorem 4 in Section 4. Theorem 1.
Assume that µ and σ satisfy (A1) to (A3). Then there exists a sequence of mea-surable functions ϕ n : R n → R , n ∈ N , such that for all p ∈ [1 , ∞ ) there exists c ∈ (0 , ∞ ) suchthat for all n ∈ N , (2) E (cid:2) | X − ϕ n ( W /n , W /n , . . . , W ) | p (cid:3) /p ≤ c/n / . We illustrate the statement of Theorem 1 by the SDE(3) dX t = (1 + X t ) 1 [0 , ∞ ) ( X t ) dt + dW t , t ∈ [0 , ,X = x . Clearly, the assumptions (A1) to (A3) are satisfied with k = 1 and ξ = 0. For the SDE (3),the strongest result on L p -approximation of X which was available in the literature so far isprovided by [14, Theorem 1], which states that the Euler-Maruyama scheme achieves an L p -errorrate of at least 1 / p ∈ [1 , ∞ ). However, by Theorem 1 we see that for this SDE in factan L p -error rate of at least 3 / p ∈ [1 , ∞ ) can be achieved by a method based on finitelymany evaluations of W .We believe that the upper error bound (2) in Theorem 1 can not be improved in general bya method based on n evaluations of W . See also Conjecture 2 in Section 5. We furthermorebelieve that an L p -error rate better than 3 / µ and σ satisfy (A1) and (A2) and are, additionally, infinitely often differentiable onthe interval ( ξ i − , ξ i ) with Lipschitz continuous derivatives of all orders for all i ∈ { , . . . , k + 1 } .A study of these conjectures will be the subject of future work. METHOD OF ORDER 3 / Similarly to the approach taken in [9, 10], the proof of Theorem 1 is based on applying asuitable bi-Lipschitz mapping G : R → R to the solution X of (1). Under the assumptions (A1)to (A3) it is possible to construct G in such a way that the transformed solution G ◦ X =( G ( X t )) t ∈ [0 , is the unique strong solution of a new SDE with coefficients that are both globallyLipschitz continuous and piecewise differentiable with Lipschitz continuous derivatives. For thelatter SDE we introduce a quasi-Milstein scheme ( b X n,ℓ/n ) ℓ =0 ,...,n and prove that b X n, achievesfor all p ∈ [1 , ∞ ) an L p -error rate of at least 3 / W forapproximating G ( X ). Using the Lipschitz continuity of G − yields the statement of Theorem 1with ϕ n ( W /n , W /n , . . . , W ) = G − ( b X n, ).To be more precise we introduce the following three assumptions on the coefficients µ and σ of the SDE (1), which are stronger than the assumptions (A1) to (A3).(B1) µ and σ are Lipschitz continuous on R ,(B2) there exist k µ , k σ ∈ N and ξ , . . . , ξ k µ +1 ∈ [ −∞ , ∞ ] with −∞ = ξ < ξ < . . . <ξ k µ < ξ k µ +1 = ∞ as well as η , . . . , η k σ +1 ∈ [ −∞ , ∞ ] with −∞ = η < η < . . . <η k σ < η k σ +1 = ∞ such that µ is differentiable on the interval ( ξ i − , ξ i ) with Lipschitzcontinuous derivative for all i ∈ { , . . . , k µ + 1 } and σ is differentiable on the interval( η i − , η i ) with Lipschitz continuous derivative for all i ∈ { , . . . , k σ + 1 } ,(B3) σ ( ξ i ) = 0 for all i ∈ { , . . . , k µ } and σ ( η i ) = 0 for all i ∈ { , . . . , k σ } .Furthermore, for all n ∈ N we define the quasi-Milstein scheme ( b X n,ℓ/n ) ℓ =0 ,...,n with step-size1 /n associated to the SDE (1) by b X n, = x and b X n, ( ℓ +1) /n = b X n,ℓ/n + µ ( b X n,ℓ/n ) · /n + σ ( b X n,ℓ/n ) · ( W ( ℓ +1) /n − W ℓ/n )+ 12 σδ σ ( b X n,ℓ/n ) · (( W ( ℓ +1) /n − W ℓ/n ) − /n )for ℓ = 0 , . . . , n −
1, where δ σ ( x ) = σ ′ ( x ) if σ is differentiable at x and δ σ ( x ) = 0 otherwise.We then have the following result, which is an immediate consequence of Theorem 3 in Sec-tion 3. Theorem 2.
Assume that µ and σ satisfy (B1) to (B3) and let p ∈ [1 , ∞ ) . Then there exists c ∈ (0 , ∞ ) such that for all n ∈ N , (4) E (cid:2) | X − b X n, | p (cid:3) /p ≤ c/n / . If, additionally, k σ = 0 then there exists c ∈ (0 , ∞ ) such that for all n ∈ N , (5) E (cid:2) | X − b X n, | p (cid:3) /p ≤ c/n. Note that if k σ = 0 then σ is differentiable on R and thus b X n coincides with the classicalMilstein scheme. However, the upper error bound (5) was known in the literature so far only inthe case of k µ = k σ = 0, see e.g. [6].For illustration of the statement of Theorem 2 we consider the SDEs(6) dX (1) t = X (1) t · [0 , ∞ ) ( X (1) t ) dt + (1 + X (1) t · [0 , ∞ ) ( X (1) t )) dW t , t ∈ [0 , ,X (1)0 = x and(7) dX (2) t = X (2) t · [0 , ∞ ) ( X (2) t ) dt + dW t , t ∈ [0 , ,X (2)0 = x , Clearly, the assumptions (B1) to (B3) are satisfied for the coefficients of the SDE (6) with k µ = k σ = 1 and ξ = η = 0 and for the coefficients of the SDE (7) with k µ = 1, k σ = 0 and ξ = 0. The best possible L p -error rate for approximation of X (1)1 and X (2)1 which was availablein the literature so far is equal to 1 / p ∈ [1 , ∞ ) the associated quasi-Milstein schemeachieves an L p -error rate of at least 3 / X (1)1 and X (2)1 , respectively.We briefly describe the content of the paper. In Section 2 we introduce some notation. Section 3contains our result Theorem 3 on the quasi-Milstein scheme under the assumptions (B1) to (B3).In Section 4 we construct the bi-Lipschitz transformation G that is then used to construct amethod of order 3 / G . 2. Notation
For A ⊂ R and a function f : A → R we put k f k ∞ = sup x ∈ A | f ( x ) | . For a function f : R → R we define δ f : R → R by δ f ( x ) = ( f ′ ( x ) , if f is differentiable in x, , otherwise.3. A quasi-Milstein scheme for SDEs with Lipschitz continuous coefficients
Let (Ω , F , P ) be a probability space with a normal filtration ( F t ) t ∈ [0 , and let W : [0 , × Ω → R be an ( F t ) t ∈ [0 , -Brownian motion on (Ω , F , P ). Moreover, let x ∈ R and let µ, σ : R → R befunctions that satisfy the following three assumptions.(B1) µ and σ are Lipschitz continuous on R ,(B2) there exist k µ , k σ ∈ N and ξ , . . . , ξ k µ +1 ∈ [ −∞ , ∞ ] with −∞ = ξ < ξ < . . . <ξ k µ < ξ k µ +1 = ∞ as well as η , . . . , η k σ +1 ∈ [ −∞ , ∞ ] with −∞ = η < η < . . . <η k σ < η k σ +1 = ∞ such that µ is differentiable on the interval ( ξ i − , ξ i ) with Lipschitzcontinuous derivative for all i ∈ { , . . . , k µ + 1 } and σ is differentiable on the interval( η i − , η i ) with Lipschitz continuous derivative for all i ∈ { , . . . , k σ + 1 } (B3) σ ( ξ i ) = 0 for all i ∈ { , . . . , k µ } and σ ( η i ) = 0 for all i ∈ { , . . . , k σ } .We consider the SDE(8) dX t = µ ( X t ) dt + σ ( X t ) dW t , t ∈ [0 , ,X = x , which has a unique strong solution due to the assumption (B1). METHOD OF ORDER 3 / Moreover, for every p ∈ (0 , ∞ ),(9) E (cid:2) k X k p ∞ (cid:3) < ∞ , see, e.g. [12, Thm. 2.4.4].For n ∈ N we use b X n = ( b X n,t ) t ∈ [0 , to denote a time-continuous quasi-Milstein scheme withstep-size 1 /n associated to the SDE (8), which is defined recursively by b X n, = x and b X n,t = b X n,i/n + µ ( b X n,i/n ) · ( t − i/n ) + σ ( b X n,i/n ) · ( W t − W i/n )+ 12 σδ σ ( b X n,i/n ) · (cid:0) ( W t − W i/n ) − ( t − i/n ) (cid:1) for t ∈ ( i/n, ( i + 1) /n ] and i ∈ { , . . . , n − } . Note that for all x
6∈ { η , . . . , η k σ } we have δ σ ( x ) = σ ′ ( x ).We have the following error estimates for b X n . Theorem 3.
Assume (B1) to (B3). Let p ∈ [1 , ∞ ) . Then there exists c ∈ (0 , ∞ ) such that forall n ∈ N , (10) E (cid:2) k X − b X n k p ∞ (cid:3) /p ≤ cn / . If, additionally, k σ = 0 then there exists c ∈ (0 , ∞ ) such that for all n ∈ N , (11) E (cid:2) k X − b X n k p ∞ (cid:3) /p ≤ cn . The proof of Theorem 3 is postponed to Section 6.
Remark 1.
Note that if (B1) to (B3) are satisfied with k σ = 0 then b X n coincides with theclassical time-continuous Milstein scheme. We add that in case of k µ = k σ = 0 and undermuch stronger smoothness assumptions on µ and σ than stated in (B1) and (B2), the errorestimate (11) is known, see e.g. [7, Thm. 10.6.3]. Remark 2.
In [8] a randomized Milstein scheme is constructed that is based on evaluationsof W at the grid points ℓ/n , ℓ = 1 , . . . , n , and randomly chosen intermediate points s ℓ ∈ (( ℓ − /n, ℓ/n ), ℓ = 1 , . . . , n . This scheme is shown to achieve for all p ∈ [1 , ∞ ) an L p -error rateof at least 1 in terms of n under assumptions that are, in comparison with (B1) to (B3), weakerwith respect to µ and stronger with respect to σ , namely the assumptions that µ is Lipschitzcontinuous on R , σ is differentiable on R with a bounded Lipschitz continuous derivative σ ′ and σσ ′ is Lipschitz continuous on R .4. A strong order / method for SDEs with discontinuous drift coefficient As in Section 3 we consider a probability space (Ω , F , P ) with a normal filtration ( F t ) t ∈ [0 , and we assume that W : [0 , × Ω → R is an ( F t ) t ∈ [0 , -Brownian motion on (Ω , F , P ). In contrastto Section 3 we now turn to SDEs with a drift coefficient µ that may be only piecewise Lipschitzcontinuous.Let x ∈ R and let µ, σ : R → R be functions that satisfy the following three assumptions.(A1) There exist k ∈ N and ξ , . . . , ξ k +1 ∈ [ −∞ , ∞ ] with −∞ = ξ < ξ < . . . < ξ k < ξ k +1 = ∞ such that µ is Lipschitz continuous on the interval ( ξ i − , ξ i ) for all i ∈ { , . . . , k + 1 } , M ¨ULLER-GRONBACH AND YAROSLAVTSEVA (A2) σ is Lipschitz continuous on R and σ ( ξ i ) = 0 for all i ∈ { , . . . , k } ,(A3) µ and σ are differentiable on the interval ( ξ i − , ξ i ) with Lipschitz continuous derivativesfor all i ∈ { , . . . , k + 1 } .For later purposes we note that (A1) implies the existence of the one-sided limits µ ( ξ i − ) and µ ( ξ i +) for all i ∈ { , . . . , k } .We consider the SDE(12) dX t = µ ( X t ) dt + σ ( X t ) dW t , t ∈ [0 , ,X = x , which has a unique strong solution, see [9, Theorem 2.2].Our goal is to show that the solution of (12) at the final time X can be approximated in p -thmean sense by means of a method based on W /n , W /n , . . . , W at least with order 3 / n of equidistant evaluations of the driving Brownian motion W , see Theorem 4.To achieve this goal we adopt the transformation strategy used in [10] and [14]. We show that X can be obtained by applying a Lipschitz continuous transformation to the solution of anSDE with coefficients satisfying the assumptions (B1) to (B3) in Section 3, and then we employTheorem 3.We start by introducing the transformation procedure. For k ∈ N , z ∈ T k = { ( z , . . . , z k ) ∈ R k : z < · · · < z k } and α = ( α , . . . , α k ) ∈ R k we put ρ z,α = ( | α | , if k = 1 , min (cid:0)(cid:8) | α i | : i ∈ { , . . . , k } (cid:9) ∪ (cid:8) z i − z i − : i ∈ { , . . . , k } (cid:9)(cid:1) , if k ≥ , where we use the convention 1 / ∞ . Let φ : R → R be given by(13) φ ( x ) = (1 − x ) · [ − , ( x ) . For all k ∈ N , z ∈ T k , α ∈ R k and ν ∈ (0 , ρ z,α ) we define a function G z,α,ν : R → R by(14) G z,α,ν ( x ) = x + k X i =1 α i · ( x − z i ) · | x − z i | · φ (cid:16) x − z i ν (cid:17) . The following two technical lemmas provide the properties of the mappings G z,α,ν that arecrucial for our purposes. The proofs of both lemmas are postponed to Section 7. Lemma 1.
Let k ∈ N , z ∈ T k , α ∈ R k , ν ∈ (0 , ρ z,α ) and put z = −∞ and z k +1 = ∞ . Thefunction G z,α,ν has the following properties. (i) G z,α,ν is differentiable on R with a Lipschitz continuous derivative G ′ z,α,ν that satisfies G ′ z,α,ν ( z i ) = 1 for all i ∈ { , . . . , k } and inf x ∈ R G ′ z,α,ν ( x ) > . In particular, G z,α,ν has aninverse G − z,α,ν : R → R that is Lipschitz continuous. Furthermore, there exists c ∈ (0 , ∞ ) such that for every x ∈ R with | x | > c , G ′ z,α,ν ( x ) = 1 . (ii) For every i ∈ { , . . . , k + 1 } , the function G ′ z,α,ν is two times differentiable on ( z i − , z i ) with Lipschitz continuous derivatives G ′′ z,α,ν and G ′′′ z,α,ν . METHOD OF ORDER 3 / (iii) For every i ∈ { , . . . , k } the one-sided limits G ′′ z,α,ν ( z i − ) and G ′′ z,α,ν ( z i +) exist and satisfy G ′′ z,α,ν ( z i − ) = − α i , G ′′ z,α,ν ( z i +) = 2 α i . Lemma 2.
Assume (A1) to (A3). Put ξ = ( ξ , . . . , ξ k ) , define α = ( α , . . . , α k ) ∈ R k by α i = µ ( ξ i − ) − µ ( ξ i +)2 σ ( ξ i ) for i ∈ { , . . . , k } , and let ν ∈ (0 , ρ ξ,α ) . Consider the function G ξ,α,ν and extend G ′′ ξ,α,ν : ∪ k +1 i =1 ( ξ i − , ξ i ) → R to the whole real line by taking G ′′ ξ,α,ν ( ξ i ) = 2 α i + 2 µ ( ξ i +) − µ ( ξ i ) σ ( ξ i ) for i ∈ { , . . . , k } . Then the functions (15) e µ = ( G ′ ξ,α,ν · µ + G ′′ ξ,α,ν · σ ) ◦ G − ξ,α,ν and e σ = ( G ′ ξ,α,ν · σ ) ◦ G − ξ,α,ν satisfy the assumptions (B1) to (B3). We turn to the transformation of the SDE (12). Take ξ, α, ν as in Lemma 2 and define astochastic process Z : [0 , × Ω → R by(16) Z t = G ξ,α,ν ( X t ) , t ∈ [0 , . Lemma 3.
Assume (A1) to (A3). Then the process Z given by (16) is the unique strong solutionof the SDE (17) dZ t = e µ ( Z t ) dt + e σ ( Z t ) dW t , t ∈ [0 , ,Z = G ξ,α,ν ( x ) with e µ and e σ given by (15) .Proof. Lemma 1(i) implies that G ′ ξ,α,ν is absolutely continuous. We therefore may apply Itˆo’slemma with G ξ,α,ν to obtain that Z is a solution of (17). According to Lemma 2, e µ and e σ areLipschitz continuous, which implies that the solution of (17) is unique. (cid:3) Remark 3.
The construction of the transformations G z,α,ν used here is similar to the construc-tion of the transformations used in [10] and [14]. In the latter works the transformations are alsogiven by (14), but with φ : R → R defined by(18) φ ( x ) = (1 − x ) · [ − , ( x )in place of (13). Note that using (18) in place of (13), the functions G ′′ z,α,ν may not be differ-entiable at the points z i ± ν for i ∈ { , . . . , k } , and therefore e µ may not be differentiable at thepoints ξ i ± ν for i ∈ { , . . . , k } .For every n ∈ N we use b Z n = ( b Z n,t ) t ∈ [0 , to denote the time-continuous quasi-Milstein schemewith step-size 1 /n associated to the SDE (17), see Section 3. Thus, b Z n, = G ξ,α,ν ( x ) and b Z n,t = b Z n,i/n + e µ ( b Z n,i/n ) · ( t − i/n ) + e σ ( b Z n,i/n ) · ( W t − W i/n )+ 12 e σ δ e σ ( b Z n,i/n ) · (cid:0) ( W t − W i/n ) − ( t − i/n ) (cid:1) M ¨ULLER-GRONBACH AND YAROSLAVTSEVA for t ∈ ( i/n, ( i + 1) /n ] and i ∈ { , . . . , n − } .We have the following error estimates for G − ξ,α,ν ◦ b Z n = ( G − ξ,α,ν ( b Z n,t )) t ∈ [0 , . Theorem 4.
Assume (A1) to (A3) and let p ∈ [1 , ∞ ) . Then there exists c ∈ (0 , ∞ ) such thatfor all n ∈ N , (19) E (cid:2) k X − G − ξ,α,ν ◦ b Z n k p ∞ (cid:3) /p ≤ cn / . Proof.
Using the Lipschitz continuity of G − ξ,α,ν , see Lemma 2(i), the fact that e µ and e σ satisfythe assumptions (B1) to (B3) and Theorem 3 we obtain that there exist c , c ∈ (0 , ∞ ) suchthat for all n ∈ N , E (cid:2) k X − G − ξ,α,ν ◦ b Z n k p ∞ (cid:3) /p ≤ c · E (cid:2) k Z − b Z n k p ∞ (cid:3) /p ≤ c n / , which completes the proof of the theorem. (cid:3) Discussion of the error bounds in Theorems 3 and 4
It is well known that, in general, the upper error bound (11) in Theorem 3 can not be improvedby any method that is based on n evaluations of the driving Brownian motion W , see [5, 13] forresults on matching lower error bounds.We believe that an analogue statement holds true with respect to the upper error bound (10)in Theorem 3. In particular, we conjecture that the following statement is true: Conjecture 1.
There exist x ∈ R , functions µ, σ : R → R that satisfy (B1) to (B3) and c ∈ (0 , ∞ ) such that the solution X of the corresponding SDE (8) satisfies for every n ∈ N , (20) inf t ,...,t n ∈ [0 , ψ : R n → R measurable E (cid:2) | X − ψ ( W t , . . . , W t n ) | (cid:3) ≥ cn / . We furthermore believe that in general the upper error bound (19) in Theorem 4 can not beimproved by any method that is based on n evaluations of the driving Brownian motion W . Inparticular, we conjecture that the following statement is true: Conjecture 2.
There exist x ∈ R , functions µ, σ : R → R that satisfy (A1) to (A3) and c ∈ (0 , ∞ ) such that the solution X of the corresponding SDE (12) satisfies for every n ∈ N , (21) inf t ,...,t n ∈ [0 , ψ : R n → R measurable E (cid:2) | X − ψ ( W t , . . . , W t n ) | (cid:3) ≥ cn / . Note that the assumptions (B1) to (B3) are stronger than the assumptions (A1) to (A3).Thus, if the Conjecture 1 is true then the Conjecture 2 is true as well.On the other hand side, the following example shows that the lower bound (21) does nothold true for all choices of the coefficients µ, σ : R → R that satisfy (A1) to (A3) such that µ isdiscontinuous. Example 1.
Let x = 0, take k = 1, z = 0, α = − / ν ∈ (0 , /
4) in (14), and considerthe functions µ, σ : R → R given by(22) µ = 1 [0 , ∞ ) , σ = 1 /G ′ , − / ,ν . METHOD OF ORDER 3 / Clearly, µ is Lipschitz continuous and differentiable on each of the intervals ( −∞ ,
0) and (0 , ∞ )with Lipschitz continuous derivative µ ′ = 0. Using Lemma 1(i) we see that σ is Lipschitzcontinuous on R . Moreover, by Lemma 1(ii) we obtain that on each of the intervals ( −∞ ,
0) and(0 , ∞ ), σ is differentiable with derivative σ ′ = − G ′′ , − / ,ν / ( G ′ , − / ,ν ) . According to Lemma 1(i)there exist c ∈ (0 , ∞ ) such that for all x ∈ R with | x | > c we have G ′′ , − / ,ν ( x ) = 0. EmployingLemma 9 in Section 7 we thus conclude that on each of the intervals ( −∞ ,
0) and (0 , ∞ ), σ ′ isLipschitz continuous. Moreover, by Lemma 1(i) we have σ (0) = 1. Hence, µ and σ satisfy (A1)to (A3) with k = 1 and ξ = 0.Let X denote the solution of (12) with µ and σ given by (22). Since σ (0) = 1 we obtain byLemmas 2, 3 that the process Z = G , − / ,ν ◦ X is the solution of the SDE (17) with coefficients e µ, e σ : R → R that satisfy (B1) to (B3). Note that e σ = 1 and thus one may take k e σ = 0 in (B2).Hence, using the second part of Theorem 3 and the Lipschitz continuity of G − , − / ,ν we concludethat for every p ∈ [1 , ∞ ) there exist c , c ∈ (0 , ∞ ) such that for every n ∈ N , E [ k X − G − , − / ,ν ◦ b Z n k p ∞ ] /p ≤ c · E [ k Z − b Z n k p ∞ ] /p ≤ c /n. Proof of Theorem 3
Throughout this section we assume that µ, σ : R → R satisfy the assumptions (B1) to (B3).Moreover, we put t n = ⌊ n · t ⌋ /n for every n ∈ N and every t ∈ [0 , L p -estimates anda Markov property of the time-continuous quasi-Milstein scheme b X n . Section 6.2 contains occu-pation time estimates for b X n , which finally lead to the p -th mean estimate(23) max ξ ∈{ ξ ,...,ξ kµ }∪{ η ,...,η kσ } E (cid:20)(cid:12)(cid:12)(cid:12)Z ( b X n,t − b X n,t n ) · { ( b X n,t − ξ )( b X n,tn − ξ ) ≤ } dt (cid:12)(cid:12)(cid:12) p (cid:21) ≤ cn p/ , where c ∈ (0 , ∞ ) does not depend on n , see Proposition 1. The latter result is a crucial tool forthe error analysis of the quasi-Milstein scheme. The results in Sections 6.1 and 6.2 are then usedin Section 6.3 to derive the error estimates in Theorem 3.Throughout this section we will employ the following three facts, which are an immediateconsequence of the assumptions (B1) to (B3). Namely, the functions µ and σ satisfy a lineargrowth condition, i.e.(24) ∃ K ∈ (0 , ∞ ) ∀ x ∈ R : | µ ( x ) | + | σ ( x ) | ≤ K · (1 + | x | ) , the functions δ µ and δ σ are bounded, i.e.(25) k δ µ k ∞ + k δ σ k ∞ < ∞ , the function µ and σ satisfy(26) ∃ c ∈ (0 , ∞ ) ∀ i ∈ { , . . . , k µ + 1 } ∀ x, y ∈ ( ξ i − , ξ i ) ∀ j ∈ { , . . . , k σ + 1 } ∀ ˜ x, ˜ y ∈ ( η j − , η j ) : | µ ( y ) − µ ( x ) − µ ′ ( x )( y − x ) | ≤ c · | y − x | and | σ (˜ y ) − σ (˜ x ) − σ ′ (˜ x )(˜ y − ˜ x ) | ≤ c · | ˜ y − ˜ x | . L p -estimates and a Markov property for the time-continuous quasi-Milsteinscheme. For technical reasons we have to provide L p -estimates and further properties of thetime-continuous quasi-Milstein scheme for the SDE (8) dependent on the initial value x . To beformally precise, for every x ∈ R we let X x denote the unique strong solution of the SDE(27) dX xt = µ ( X xt ) dt + σ ( X xt ) dW t , t ∈ [0 , ,X x = x, and for all x ∈ R and n ∈ N we use b X xn = ( b X xn,t ) t ∈ [0 , to denote the time-continuous quasi-Milstein scheme with step-size 1 /n associated to the SDE (27). Thus, X = X x and b X n = b X x n for all n ∈ N , and for all x ∈ R (28) b X xn,t = x + Z t µ ( b X xn,s n ) ds + Z t (cid:0) σ ( b X xn,s n ) + σδ σ ( b X xn,s n ) · ( W s − W s n ) (cid:1) dW s holds P -a.s. for all n ∈ N and t ∈ [0 , L p -estimates for b X xn , n ∈ N , which follow from (28), thelinear growth property (24) of µ and σ and the boundedness of δ σ , see (25), by using standardarguments. Lemma 4.
Let p ∈ [1 , ∞ ) . Then there exists c ∈ (0 , ∞ ) such that for all x ∈ R , all n ∈ N , all δ ∈ [0 , and all t ∈ [0 , − δ ] , E h sup s ∈ [ t,t + δ ] | b X xn,s − b X xn,t | p i /p ≤ c · (1 + | x | ) · √ δ. In particular, there exists c ∈ (0 , ∞ ) such that for all x ∈ R and all n ∈ N , sup n ∈ N E (cid:2) k b X xn k p ∞ (cid:3) /p ≤ c · (1 + | x | ) . The following lemma provides a Markov property of the time-continuous quasi-Milsein scheme b X xn relative to the gridpoints 0 , /n, /n, . . . , Lemma 5.
For all x ∈ R , all n ∈ N , all j ∈ { , . . . , n − } and P b X xn,j/n -almost all y ∈ R we have P ( b X xn,t ) t ∈ [ j/n, |F j/n = P ( b X xn,t ) t ∈ [ j/n, | b X xn,j/n as well as P ( b X xn,t ) t ∈ [ j/n, | b X xn,j/n = y = P ( b X yn,t ) t ∈ [0 , − j/n ] . Proof.
The lemma is an immediate consequence of the fact that, by definition of b X xn , for every ℓ ∈ { , . . . , n } there exists a measurable mapping ψ : R × C ([0 , ℓ/n ]) → C ([0 , ℓ/n ]) such that forall x ∈ R and all i ∈ { , , . . . , n − ℓ } ,( b X xn,t + i/n ) t ∈ [0 ,ℓ/n ] = ψ (cid:0) b X xn,i/n , ( W t + i/n − W i/n ) t ∈ [0 ,ℓ/n ] (cid:1) . (cid:3) METHOD OF ORDER 3 / Occupation time estimates for the time-continuous quasi-Milstein scheme.
Wefirst provide an estimate for the expected occupation time of a neighborhood of a non-zero of σ by the time-continuous quasi-Milstein scheme b X xn . Lemma 6.
Let ξ ∈ R satisfy σ ( ξ ) = 0 . Then there exists c ∈ (0 , ∞ ) such that for all x ∈ R , all n ∈ N and all ε ∈ (0 , ∞ ) , (29) Z P ( {| b X xn,t − ξ | ≤ ε } ) dt ≤ c · (1 + x ) · (cid:16) ε + 1 √ n (cid:17) . Proof.
Let x ∈ R and n ∈ N . For t ∈ [0 ,
1] putΣ xn,t = σ ( b X xn,t n ) + σδ σ ( b X xn,t n ) · ( W t − W t n ) . Using (24), (25), (28) and Lemma 4 we conclude that b X xn is a continuous semi-martingale withquadratic variation(30) h b X xn i t = Z t (Σ xn,s ) ds, t ∈ [0 , . For a ∈ R let L a ( b X xn ) = ( L at ( b X xn )) t ∈ [0 , denote the local time of b X xn at the point a . Thus, for all a ∈ R and all t ∈ [0 , | b X xn,t − a | = | x − a | + Z t sgn( b X xn,s − a ) · µ ( b X xn,s ) ds + Z t sgn( b X xn,s − a ) · Σ xn,s dW s + L at ( b X xn ) , where sgn( y ) = 1 (0 , ∞ ) ( y ) − ( −∞ , ( y ) for y ∈ R , see, e.g. [19, Chap. VI]. Hence, for all a ∈ R and all t ∈ [0 , L at ( b X xn ) ≤ | b X xn,t − x | + Z t | µ ( b X xn,s ) | ds + (cid:12)(cid:12)(cid:12)Z t sgn( b X xn,s − a ) · Σ xn,s dW s (cid:12)(cid:12)(cid:12) ≤ Z t | µ ( b X xn,s ) | ds + (cid:12)(cid:12)(cid:12)Z t Σ xn,s dW s (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)Z t sgn( b X xn,s − a ) · Σ xn,s dW s (cid:12)(cid:12)(cid:12) . Using (24), (31), the H¨older inequality and the Burkholder-Davis-Gundy inequality we obtainthat there exists c ∈ (0 , ∞ ) such that for all x ∈ R , all n ∈ N , all a ∈ R and all t ∈ [0 , E (cid:2) L at ( b X xn ) (cid:3) ≤ c · Z (cid:0) E (cid:2) | b X xn,s | (cid:3)(cid:1) ds + c (cid:16)Z E (cid:2) (Σ xn,s ) (cid:3) ds (cid:17) / . By (24), (25) and the fact that for all s ∈ [0 ,
1] the random variables b X xn,s n and W s − W s n areindependent we obtain that there exist c , c ∈ (0 , ∞ ) such that for all s ∈ [0 , x ∈ R andall n ∈ N ,(33) E (cid:2) (Σ xn,s ) (cid:3) ≤ c · E (cid:2) (1 + | b X xn,s n | ) · (1 + | W s − W s n | ) (cid:3) ≤ c · (cid:0) E (cid:2) ( b X xn,s n ) (cid:3)(cid:1) . By employing Lemma 4 we conclude from (32) and (33) that there exist c , c ∈ (0 , ∞ ) suchthat for all x ∈ R , all n ∈ N , all a ∈ R and all t ∈ [0 , E (cid:2) L at ( b X xn ) (cid:3) ≤ c · (cid:16) E (cid:2) k b X xn k ∞ (cid:3) / (cid:17) ≤ c · (1 + | x | ) . Using (30), (34) and the occupation time formula it follows that there exists c ∈ (0 , ∞ ) suchthat for all x ∈ R , all n ∈ N and all ε ∈ (0 , ∞ ),(35) E (cid:20)Z [ ξ − ε,ξ + ε ] ( b X xn,t ) · (Σ xn,t ) dt (cid:21) = Z R [ ξ − ε,ξ + ε ] ( a ) · E (cid:2) L at ( b X xn ) (cid:3) da ≤ c · (1 + | x | ) · ε. By (24), (25) and the Lipschitz continuity of σ we obtain that there exist c , c ∈ (0 , ∞ ) suchthat for all x ∈ R , all n ∈ N and all t ∈ [0 , (cid:12)(cid:12) σ ( b X xn,t ) − (Σ xn,t ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) σ ( b X xn,t ) − Σ xn,t (cid:12)(cid:12) · (cid:0) | σ ( b X xn,t ) | + | Σ xn,t | (cid:1) ≤ c · (cid:0) | σ ( b X xn,t ) − σ ( b X xn,t n ) | + | σδ σ ( b X xn,t n ) | · | W t − W t n | (cid:1) · (cid:0) | b X xn,t | + (1 + | b X xn,t n | ) · (1 + | W t − W t n | ) (cid:1) ≤ c · (cid:0) | b X xn,t − b X xn,t n | + (1 + | b X xn,t n | ) · | W t − W t n | (cid:1) · (1 + k b X xn k ∞ ) · (1 + | W t − W t n | ) . Thus, using the H¨older inequality and Lemma 4 we conclude that there exist c ∈ (0 , ∞ ) suchthat for all x ∈ R , all n ∈ N and all t ∈ [0 , E (cid:2) | σ ( b X xn,t ) − (Σ xn,t ) | (cid:3) ≤ c · (1 + x ) · √ n . Since σ is continuous and σ ( ξ ) = 0 there exist κ, ε ∈ (0 , ∞ ) such that(37) inf z ∈ R : | z − ξ | <ε σ ( z ) ≥ κ. Using (35), (36) and (37) we obtain that there exists c ∈ (0 , ∞ ) such that for all x ∈ R , all n ∈ N and all ε ∈ (0 , ε ], Z P ( {| b X xn,t − ξ | ≤ ε } ) dt = 1 κ · E hZ κ · [ ξ − ε,ξ + ε ] ( b X xn,t ) dt i ≤ κ · E hZ [ ξ − ε,ξ + ε ] ( b X xn,t ) · σ ( b X xn,t ) dt i ≤ κ · E (cid:20)Z (cid:16) [ ξ − ε,ξ + ε ] ( b X xn,t ) · (Σ xn,t ) + (cid:12)(cid:12) σ ( b X xn,t ) − (Σ xn,t ) (cid:12)(cid:12)(cid:17) dt (cid:21) ≤ cκ · (1 + | x | + x ) · (cid:16) ε + 1 √ n (cid:17) , which completes the proof of the lemma. (cid:3) The following result provides moment estimates subject to the condition of a sign change ofthe process b X n − ξ at time t relative to its sign at the grid point t n . Lemma 7.
Let q ∈ [1 , ∞ ) , ξ ∈ R , and let A n,t = { ( b X n,t − ξ ) · ( b X n,t n − ξ ) ≤ } METHOD OF ORDER 3 / for all n ∈ N and t ∈ [0 , . Then there exists c ∈ (0 , ∞ ) such that for all n ∈ N , all ≤ s ≤ t ≤ with t n − s ≥ /n and all real-valued, non-negative, F s -measurable random variables Y , (38) E (cid:2) Y · | W t − W t n | q · A n,t (cid:3) ≤ cn q/ · E [ Y ] + cn q/ · Z R | z | q · E (cid:2) Y · {| b X n,tn − ( t − tn ) − ξ |≤ c √ n (1+ | z | ) } (cid:3) · e − z dz. Proof.
Choose K ∈ (0 , ∞ ) according to (24), put κ = K · (1 + | ξ | ) · (1 + k δ σ k ∞ )and choose n ∈ N \ { , } such that for all n ≥ n ,(39) 16 ln( n ) √ n ≤ κ · p ln( n ) √ n ≤ . Without loss of generality we may assume that n ≥ n . Let 0 ≤ s ≤ t ≤ t n − s ≥ /n and let Y be a real-valued, non-negative, F s -measurable random variable. If t = t n then (38)trivially holds for any c ∈ (0 , ∞ ).Now assume that t > t n and put Z = W t − W t n √ t − t n , Z = W t n − W t n − ( t − t n ) √ t − t n , Z = W t n − ( t − t n ) − W t n − /n p /n − ( t − t n ) . Below we show that(40) A n,t ∩ (cid:8) max i ∈{ , , } | Z i | ≤ p ln( n ) (cid:9) ⊂ (cid:8) | b X n,t n − ( t − t n ) − ξ | ≤ κ · (1 + | Z | + | Z | ) / √ n (cid:9) . Note that Z , Z , Z are independent and identically distributed standard normal random vari-ables. Moreover, ( Z , Z , Z ) is independent of F s since s ≤ t n − /n , ( Z , Z ) is independent of F t n − ( t − t n ) and b X n,t n − ( t − t n ) is F t n − ( t − t n ) -measurable. Using the latter three facts jointly with (40) and a standard estimate of standard normal tail probabilities we obtain that E (cid:2) Y · | W t − W t n | q · A n,t (cid:3) = ( t − t n ) q/ · E (cid:2) Y · | Z | q · A n,t (cid:3) ≤ n q/ · E (cid:2) Y · | Z | q · {| b X n,tn − ( t − tn ) − ξ |≤ κ · (1+ | Z | + | Z | ) / √ n } (cid:3) + 1 n q/ · E (cid:2) Y · | Z | q · { max i ∈{ , , } | Z i | > √ ln( n ) } (cid:3) = 2 πn q/ Z [0 , ∞ ) E (cid:2) Y · z q · {| b X n,tn − ( t − tn ) − ξ |≤ κ · (1+ z + z ) / √ n } (cid:3) · e − z z d ( z , z )+ 1 n q/ · E [ Y ] · E (cid:2) | Z | q · { max i ∈{ , , } | Z i | > √ ln( n ) } (cid:3) ≤ q/ πn q/ Z R E (cid:2) Y · (cid:0) | z + z |√ (cid:1) q · {| b X n,tn − ( t − tn ) − ξ |≤ √ κ · (1+ | z + z | / √ / √ n } (cid:3) · e − z z d ( z , z )+ 1 n q/ · E [ Y ] · E (cid:2) Z q (cid:3) / · (cid:0) P (cid:0)(cid:8) max i ∈{ , , } | Z i | > p ln( n ) (cid:9)(cid:1)(cid:1) / ≤ q/ √ πn q/ Z R E (cid:2) Y · | z | q · {| b X n,tn − ( t − tn ) − ξ |≤ √ κ · (1+ | z | ) / √ n } (cid:3) · e − z dz + p · · · · (2 q − n q/ · E [ Y ] · (cid:16) p π ln( n ) · n (cid:17) / , which yields (38).It remains to prove the inclusion (40). To this end let ω ∈ Ω and assume that(41) ω ∈ A n,t and max i ∈{ , , } | Z i ( ω ) | ≤ p ln( n ) . By (24) and (41),(42) | b X n,t n ( ω ) − ξ | ≤ | b X n,t ( ω ) − b X n,t n ( ω ) | = (cid:12)(cid:12)(cid:12) µ ( b X n,t n ( ω )) · ( t − t n ) + σ ( b X n,t n ( ω )) · p t − t n · Z ( ω )+ 12 σδ σ ( b X n,t n ( ω )) · ( t − t n ) · (cid:0) Z ( ω ) − (cid:12)(cid:12)(cid:12) ≤ K · (1 + | b X n,t n ( ω ) | ) · (cid:16) n + 1 √ n · | Z ( ω ) | + 1 √ n · k δ σ k ∞ · Z ( ω ) + 12 √ n (cid:17) . Observe that for all a, b ∈ R ,(43) 1 + | a | ≤ (1 + | a − b | ) · (1 + | b | ) . Moreover, (39) and (41) yield(44) Z ( ω ) + 12 √ n ≤ n ) + 12 √ n ≤ n )2 √ n ≤ . METHOD OF ORDER 3 / Combining (42) with (44) and employing (43) with a = b X n,t n ( ω ) and b = ξ we get(45) | b X n,t n ( ω ) − ξ | ≤ κ √ n · (1 + | b X n,t n ( ω ) − ξ | ) · (1 + | Z ( ω ) | ) . Similarly one can show that(46) | b X n,t n − ( t − t n ) ( ω ) − b X n,t n − /n ( ω ) | ≤ κ √ n · (1 + | b X n,t n − /n ( ω ) − ξ | ) · (1 + | Z ( ω ) | ) . Furthermore, by (24),(47) | b X n,t n ( ω ) − b X n,t n − ( t − t n ) ( ω ) | = (cid:12)(cid:12)(cid:12) µ ( b X n,t n − /n ( ω )) · ( t − t n ) + σ ( b X n,t n − /n ( ω )) · p t − t n · Z ( ω )+ 12 σ · δ σ ( b X n,t n − /n ( ω )) · (cid:0) u − ( t − t n ) (cid:1)(cid:12)(cid:12)(cid:12) ≤ K · (1 + | b X n,t n − /n ( ω ) | ) · (cid:16) n + 1 √ n · | Z ( ω ) | + 12 · k δ σ k ∞ · (cid:16) | u | + 1 n (cid:17)(cid:17) , where u = ( W t n ( ω ) − W t n − /n ( ω )) − ( W t n − ( t − t n ) ( ω ) − W t n − /n ( ω )) . Observing that for all a, b ∈ R , | ( a + b ) − b | ≤ ( | a | + | b | ) , and using (39) as well as (41) we obtain(48) | u | ≤ (cid:0)p t − t n · | Z ( ω ) | + p /n − ( t − t n ) · | Z ( ω ) | (cid:1) ≤
16 ln( n ) n ≤ √ n . Combining (47) with (48) and employing (43) with a = b X n,t n − /n ( ω ) and b = ξ we concludethat(49) | b X n,t n ( ω ) − b X n,t n − ( t − t n ) ( ω ) | ≤ κ √ n · (1 + | b X n,t n − /n ( ω ) − ξ | ) · (1 + | Z ( ω ) | ) . Clearly, we have(50) | b X n,t n − ( t − t n ) ( ω ) − ξ | ≤ | b X n,t n ( ω ) − b X n,t n − ( t − t n ) ( ω ) | + | b X n,t n ( ω ) − ξ | . By (39) and (41) we have for all i ∈ { , , } ,(51) κ √ n · (1 + | Z i ( ω ) | ) ≤ κ · p ln( n ) √ n ≤ . Using (45) and (51) we obtain(52) | b X n,t n ( ω ) − ξ | ≤ κ √ n · (1 + | Z ( ω ) | )1 − κ √ n · (1 + | Z ( ω ) | ) ≤ κ √ n · (1 + | Z ( ω ) | ) . Furthermore, by (46) and (51),1 + | b X n,t n − ( t − t n ) ( ω ) − ξ | ≥ | b X n,t n − /n ( ω ) − ξ | − | b X n,t n − ( t − t n ) ( ω ) − b X n,t n − /n ( ω ) |≥ (1 + | b X n,t n − /n ( ω ) − ξ | ) / , which jointly with (49) yields(53) | b X n,t n ( ω ) − b X n,t n − ( t − t n ) ( ω ) | ≤ κ √ n · (1 + | b X n,t n − ( t − t n ) ( ω ) − ξ | ) · (1 + | Z ( ω ) | ) . Combining (50), (52) and (53) we conclude that(54) | b X n,t n − ( t − t n ) ( ω ) − ξ | ≤ κ √ n · (1 + | b X n,t n − ( t − t n ) ( ω ) − ξ | ) · (1 + | Z ( ω ) | + | Z ( ω ) | ) . By (39) and (41),(55) 4 κ √ n · (1 + | Z ( ω ) | + | Z ( ω ) | ) ≤ κ · p ln( n ) √ n ≤ . Observing (55) we obtain from (54) that | b X n,t n − ( t − t n ) ( ω ) − ξ | ≤ κ √ n · (1 + | Z ( ω ) | + | Z ( ω ) | ) . This finishes the proof of (40). (cid:3)
Using Lemmas 5, 6 and 7 we can now establish the following two estimates on time averages ofmoments subject to the condition of sign changes of b X n − ξ relative to its sign at the gridpoints0 , /n, . . . , Lemma 8.
Let q ∈ [1 , ∞ ) , let ξ ∈ R satisfy σ ( ξ ) = 0 and let A n,t = { ( b X n,t − ξ ) · ( b X n,t n − ξ ) ≤ } for all n ∈ N and t ∈ [0 , . Then there exists c ∈ (0 , ∞ ) such that for all n ∈ N , all s ∈ [0 , − /n ) and all real-valued, non-negative, F s -measurable random variables Y , (56) Z s n +2 /n E (cid:2) Y · | W t − W t n | q · A n,t (cid:3) dt ≤ cn ( q +1) / · (cid:0) E [ Y ] + E (cid:2) Y · ( b X n,s n +1 /n − ξ ) (cid:3)(cid:1) and (57) Z s n +1 /n E (cid:2) Y · | W t − W t n | q · A n,t · ( b X n,t n +1 /n − ξ ) (cid:3) dt ≤ cn q/ · (cid:0) E [ Y ] + E (cid:2) Y · ( b X n,s n +1 /n − ξ ) (cid:3)(cid:1) . Proof.
For s ∈ [0 ,
1] we use Y s to denote the set of all real-valued, non-negative, F s -measurablerandom variables.We first prove (56). Note that if t ≥ s n + 2 /n then t n − /n ≥ s n + 1 /n ≥ s . By Lemma 7 wethus obtain that there exists c ∈ (0 , ∞ ) such that for all n ∈ N , s ∈ [0 , − /n ) and Y ∈ Y s ,(58) Z s n +2 /n E (cid:2) Y · | W t − W t n | q · A n,t (cid:3) dt ≤ c · E [ Y ] n q/ + cn q/ Z R | z | q · e − z · Z s n +2 /n E (cid:2) Y · {| b X n,tn − ( t − tn ) − ξ |≤ c √ n (1+ | z | ) } (cid:3) dt dz = c · E [ Y ] n q/ + cn q/ Z R | z | q · e − z · Z − /ns n +1 /n E (cid:2) Y · {| b X n,t − ξ |≤ c √ n (1+ | z | ) } (cid:3) dt dz. METHOD OF ORDER 3 / Using the fact that for all n ∈ N and s ∈ [0 , − /n ) every Y ∈ Y s is F s n +1 /n -measurable andemploying the first part of Lemma 5 we obtain that for all n ∈ N , s ∈ [0 , − /n ), Y ∈ Y s and z ∈ R ,(59) Z − /ns n +1 /n E (cid:2) Y · {| b X n,t − ξ |≤ c √ n (1+ | z | ) } (cid:3) dt = E h Y · E hZ − /ns n +1 /n {| b X n,t − ξ |≤ c √ n (1+ | z | ) } dt (cid:12)(cid:12)(cid:12) b X n,s n +1 /n ii . Moreover, by the second part of Lemma 5 and by Lemma 6, there exists c ∈ (0 , ∞ ) such thatfor all n ∈ N , s ∈ [0 , − /n ), Y ∈ Y s , z ∈ R and P b X n,sn +1 /n -almost all x ∈ R ,(60) E hZ − /ns n +1 /n {| b X n,t − ξ |≤ c √ n (1+ | z | ) } dt (cid:12)(cid:12)(cid:12) b X n,s n +1 /n = x i = E hZ − /n − s n {| b X xn,t − ξ |≤ c √ n (1+ | z | ) } dt i ≤ c · (1 + x ) · (cid:16) c √ n · (1 + | z | ) + 1 √ n (cid:17) . Combining (59) and (60) and using the fact that for all a, b ∈ R ,1 + a ≤ a − b ) ) · (1 + b ) , we conclude that for all n ∈ N , s ∈ [0 , − /n ), Y ∈ Y s and z ∈ R ,(61) Z − /ns n +1 /n E (cid:2) Y · {| b X n,t − ξ |≤ c √ n (1+ | z | ) } (cid:3) dt ≤ c ( c +1) √ n · (1 + | z | ) · E (cid:2) Y · (1 + b X n,s n +1 /n ) (cid:3) ≤ c ( c +1) √ n · (1 + ξ ) · (1 + | z | ) · (cid:0) E [ Y ] + E (cid:2) Y · ( b X n,s n +1 /n − ξ ) (cid:3)(cid:1) . Inserting (61) into (58) and observing that R R (1 + | z | ) · | z | q · e − z / dz < ∞ completes the proofof (56).We next prove (57). Clearly, for all n ∈ N , s ∈ [0 , − /n ), t ∈ [ s n + 1 /n,
1] and all ω ∈ A n,t we have | b X n,t n +1 /n ( ω ) − ξ | ≤ | b X n,t n +1 /n ( ω ) − b X n,t ( ω ) | + | b X n,t ( ω ) − ξ |≤ | b X n,t n +1 /n ( ω ) − b X n,t ( ω ) | + | b X n,t ( ω ) − b X n,t n ( ω ) | . Using the fact that for all n ∈ N and s ∈ [0 , − /n ) every Y ∈ Y s is F s n +1 /n -measurable andemploying the H¨older inequality we therefore obtain that for all n ∈ N , all s ∈ [0 , − /n ), all Y ∈ Y s and all t ∈ [ s n + 1 /n, E (cid:2) Y · | W t − W t n | q · A n,t · ( b X n,t n +1 /n − ξ ) (cid:3) ≤ E (cid:2) Y · | W t − W t n | q · ( | b X n,t n +1 /n − b X n,t | + | b X n,t − b X n,t n | ) (cid:3) = E (cid:2) Y · E (cid:2) | W t − W t n | q · ( | b X n,t n +1 /n − b X n,t | + | b X n,t − b X n,t n | ) |F s n +1 /n (cid:3)(cid:3) ≤ E (cid:2) Y · (cid:0) E (cid:2) ( W t − W t n ) q |F s n +1 /n (cid:3)(cid:1) / · (cid:0) E (cid:2) ( | b X n,t n +1 /n − b X n,t | + | b X n,t − b X n,t n | ) |F s n +1 /n (cid:3)(cid:1) / (cid:3) . If t ≥ s n + 1 /n then t n ≥ s n + 1 /n . Hence, there exists c ∈ (0 , ∞ ) such that for all n ∈ N , all s ∈ [0 , − /n ) and all t ∈ [ s n + 1 /n, E (cid:2) ( W t − W t n ) q |F s n +1 /n (cid:3) = E (cid:2) ( W t − W t n ) q (cid:3) ≤ c/n q . Moreover, the first part of Lemma 5 implies that for all n ∈ N , all s ∈ [0 , − /n ) and all t ∈ [ s n + 1 /n,
1] it holds P -a.s. that(64) E (cid:2) ( | b X n,t n +1 /n − b X n,t | + | b X n,t − b X n,t n | ) |F s n +1 /n (cid:3) = E (cid:2) ( | b X n,t n +1 /n − b X n,t | + | b X n,t − b X n,t n | ) | b X n,s n +1 /n (cid:3) . By the second part of Lemma 5 and by Lemma 4 we obtain that there exist c , c ∈ (0 , ∞ ) suchthat for all n ∈ N , all s ∈ [0 , − /n ), all t ∈ [ s n + 1 /n,
1] and P b X n,sn +1 /n -almost all x ∈ R ,(65) E (cid:2) ( | b X n,t n +1 /n − b X n,t | + | b X n,t − b X n,t n | ) (cid:12)(cid:12) b X n,s n +1 /n = x (cid:3) = E [( | b X xn,t n − s n − b X xn,t − s n − /n | + | b X xn,t − s n − /n − b X xt n − s n − /n | ) (cid:3) ≤ c · (1 + x ) · /n ≤ c · (1 + ( x − ξ ) ) · /n . It follows from (62), (63), (64) and (65) that there exist c ∈ (0 , ∞ ) such that for all n ∈ N , all s ∈ [0 , − /n ) and all Y ∈ Y s ,(66) Z s n +1 /n E (cid:2) Y · | W t − W t n | q · A n,t · ( b X n,t n +1 /n − ξ ) (cid:3) dt ≤ cn q/ · Z s n +1 /n E (cid:2) Y · (1 + ( b X n,s n +1 /n − ξ ) ) (cid:3) dt ≤ cn q/ · (cid:0) E [ Y ] + E (cid:2) Y · ( b X n,s n +1 /n − ξ ) (cid:3)(cid:1) , which finishes the proof of (57) and completes the proof of the lemma. (cid:3) We are ready to establish the main result in this section, which provides a p -th mean estimateof the time average of | b X n,t − b X n,t n | q subject to a sign change of b X n,t − ξ relative to the sign of b X n,t n − ξ . Proposition 1.
Let ξ ∈ R satisfy σ ( ξ ) = 0 and let A n,t = { ( b X n,t − ξ ) · ( b X n,t n − ξ ) ≤ } METHOD OF ORDER 3 / for all n ∈ N and t ∈ [0 , . Then for all p, q ∈ [1 , ∞ ) there exists c ∈ (0 , ∞ ) such that for all n ∈ N , (67) E h(cid:12)(cid:12)(cid:12)Z | b X n,t − b X n,t n | q · A n,t dt (cid:12)(cid:12)(cid:12) p i /p ≤ cn ( q +1) / . Proof.
Clearly, it suffices to consider only the case p ∈ N . Fix q ∈ [1 , ∞ ). For n, p ∈ N put a n,p = E h(cid:12)(cid:12)(cid:12)Z | W t − W t n | q · A n,t dt (cid:12)(cid:12)(cid:12) p i . We prove by induction on p that for every p ∈ N there exists c ∈ (0 , ∞ ) such that for all n ∈ N ,(68) a n,p ≤ cn ( q +1) p/ . First, consider the case p = 1. Using (56) in Lemma 8 with s = 0 and Y = 1 we obtain thatthere exists c ∈ (0 , ∞ ) such that for all n ≥ a n, ≤ Z /n E (cid:2) | W t − W t n | q · A n,t (cid:3) dt + Z /n E (cid:2) | W t − W t n | q (cid:3) dt ≤ cn ( q +1) / · (cid:0) E (cid:2)(cid:0) b X n, /n − ξ (cid:1) (cid:3)(cid:1) + cn q/ ≤ cn ( q +1) / · (cid:0) ξ + E (cid:2) b X n, /n (cid:3)(cid:1) . Employing Lemma 4 we thus conclude that (68) holds for p = 1.Next, let r ∈ N and assume that (68) holds for all p ∈ { , . . . , r } . For n ∈ N and t ∈ [0 ,
1] put Y n,t = | W t − W t n | q . We then have for all n ∈ N ,(69) a n,r +1 = ( r + 1)! · Z Z t . . . Z t r E h r +1 Y i =1 Y n,t i · A n,ti i dt r +1 . . . dt . For n ∈ N and 0 ≤ t ≤ . . . ≤ t r ≤ J ,n ( t , . . . , t r ) = Z ( t rn +2 /n ) ∧ t r E h r +1 Y i =1 Y n,t i · A n,ti i dt r +1 ,J ,n ( t , . . . , t r ) = Z t rn +2 /n ) ∧ E h r +1 Y i =1 Y n,t i · A n,ti i dt r +1 . By the H¨older inequality there exists c ∈ (0 , ∞ ) such that for all n ∈ N and all 0 ≤ t ≤ · · · ≤ t r +1 ≤ E h r +1 Y i =1 Y n,t i · A n,ti i ≤ E h(cid:16) Y n,t r +1 · r Y i =1 Y /rn,t i (cid:17) · (cid:16) r Y i =1 Y ( r − /rn,t i · A n,ti (cid:17)i ≤ E h Y rn,t r +1 · r Y i =1 Y n,t i i /r · E h r Y i =1 Y n,t i · A n,ti i ( r − /r ≤ cn q · E h r Y i =1 Y n,t i · A n,ti i ( r − /r . Hence there exists c ∈ (0 , ∞ ) such that for all n ∈ N and all 0 ≤ t ≤ · · · ≤ t r ≤ J ,n ( t , . . . , t r ) ≤ cn q +1 · E h r Y i =1 Y n,t i · A n,ti i ( r − /r . Clearly, for all n ∈ N and all 0 ≤ t ≤ . . . ≤ t r ≤ t r ≥ − /n we have(72) J ,n ( t , . . . , t r ) = 0 . Furthermore, if t r ∈ [0 , − /n ) then ( t rn + 2 /n ) ∧ t rn + 2 /n , and by applying (56) inLemma 8 with s = t r and Y = Q ri =1 Y n,t i · A n,ti we obtain that there exists c ∈ (0 , ∞ ) such thatfor all n ∈ N and all 0 ≤ t ≤ . . . ≤ t r ≤ J ,n ( t , . . . , t r ) ≤ cn ( q +1) / · (cid:16) E h r Y i =1 Y n,t i · A n,ti i + E h r Y i =1 Y n,t i · A n,ti · ( b X n,t rn +1 /n − ξ ) i(cid:17) . Combining (71) to (73) with (69) and employing the induction hypothesis we conclude thatthere exists c , c ∈ (0 , ∞ ) such that for all n ∈ N ,(74) a n,r +1 ≤ c · (cid:16) a n,r n ( q +1) / + b n,r n ( q +1) / + a ( r − /rn,r n q +1 (cid:17) ≤ c · (cid:16) n ( q +1)( r +1) / + b n,r n ( q +1) / (cid:17) , where b n,r = Z Z t . . . Z t r − E h(cid:16) r Y i =1 Y n,t i · A n,ti (cid:17) · ( b X n,t rn +1 /n − ξ ) i dt r . . . dt . We proceed with estimating the term b n,r . Using (57) in Lemma 8 with s = 0 and Y = 1 aswell as Lemma 4 we obtain that there exist c , c ∈ (0 , ∞ ) such that for all n ∈ N , Z /n E (cid:2) Y n,t · A n,t · ( b X n,t n +1 /n − ξ ) (cid:3) dt ≤ c n q/ · (cid:0) E [( b X n,t n +1 /n − ξ ) ] (cid:1) ≤ c n q/ . Furthermore, by employing Lemma 4 again we see that there exist c ∈ (0 , ∞ ) such that for all n ∈ N , Z /n E (cid:2) Y n,t · A n,t · ( b X n,t n +1 /n − ξ ) (cid:3) dt ≤ Z /n E (cid:2) Y n,t ] / · E [( b X n,t n +1 /n − ξ ) (cid:3) / dt ≤ cn q/ . METHOD OF ORDER 3 / It follows that there exist c ∈ (0 , ∞ ) such that for all n ∈ N ,(75) b n, ≤ cn q/ . Next, we assume that r ≥
2, and for n ∈ N and 0 ≤ t ≤ . . . ≤ t r − ≤ K ,n ( t , . . . , t r − ) = Z ( t r − n +1 /n ) ∧ t r − E h(cid:16) r Y i =1 Y n,t i · A n,ti (cid:17) · ( b X n,t rn +1 /n − ξ ) i dt r ,K ,n ( t , . . . , t r − ) = Z t r − n +1 /n ) ∧ E h(cid:16) r Y i =1 Y n,t i · A n,ti (cid:17) · ( b X n,t rn +1 /n − ξ ) i dt r . Proceeding similarly to (70) and employing Lemma 4 we conclude that there exists c ∈ (0 , ∞ )such that for all n ∈ N and all 0 ≤ t ≤ . . . ≤ t r ≤ E h(cid:16) r Y i =1 Y n,t i · A n,ti (cid:17) · ( b X n,t rn +1 /n − ξ ) i ≤ E h Y r − n,t r · (cid:16) r − Y i =1 Y n,t i (cid:17) · ( b X n,t rn +1 /n − ξ ) r − i r − · E h r − Y i =1 Y n,t i · A n,ti i r − r − ≤ cn q · E h r − Y i =1 Y n,t i · A n,ti i r − r − . Hence there exists c ∈ (0 , ∞ ) such that for all n ∈ N and all 0 ≤ t ≤ · · · ≤ t r − ≤ K ,n ( t , . . . , t r − ) ≤ cn q +1 · E h r − Y i =1 Y n,t i · A n,ti i r − r − . Clearly, for all n ∈ N and all 0 ≤ t ≤ . . . ≤ t r − ≤ t r − ≥ − /n we have(78) K ,n ( t , . . . , t r − ) = 0 . Furthermore, if t r − ∈ [0 , − /n ) then ( t r − n + 1 /n ) ∧ t r − n + 1 /n , and by applying (57) inLemma 8 with s = t r − and Y = Q r − i =1 Y n,t i · A n,ti we obtain that there exists c ∈ (0 , ∞ ) suchthat for all n ∈ N and all 0 ≤ t ≤ · · · ≤ t r − ≤ K ,n ( t , . . . , t r − ) ≤ cn q/ · (cid:16) E h r − Y i =1 Y n,t i · A n,ti i + E h(cid:16) r − Y i =1 Y n,t i · A n,ti (cid:17) · ( b X n,t r − n +1 /n − ξ ) i(cid:17) . Using (77) to (79) and employing the induction hypothesis we thus conclude that there exist c , c ∈ (0 , ∞ ) such that for all n ∈ N ,(80) b n,r ≤ c · (cid:16) a ( r − / ( r − n,r − n q +1 + a n,r − n q/ + b n,r − n q/ (cid:17) ≤ c n ( q +1) r/ + c n q/ · b n,r − ≤ c n ( q +1) r/ + c n ( q +1) / · b n,r − . Using (75) and (80) we obtain by induction that there exist c , c ∈ (0 , ∞ ) such that for all n ∈ N ,(81) b n,r ≤ c n ( q +1) r/ + c n ( q +1)( r − / · b n, ≤ c n ( q +1) r/ . Inserting the estimate (81) into (74) yields that there exists c ∈ (0 , ∞ ) such that for all n ∈ N ,(82) a n,r +1 ≤ cn ( q +1)( r +1) / , which completes the proof of (68).We turn to the proof of (67). By the definition of b X n and by (24) and (25) we see that thereexists c ∈ (0 , ∞ ) such that for all n ∈ N and all t ∈ [0 , | b X n,t − b X n,t n | ≤ c · (1 + | b X n,t n | ) · (1 /n + | W t − W t n | + | W t − W t n | ) . Using the fact that for all n ∈ N , all t ∈ [0 ,
1] and all ω ∈ A n,t we have | b X n,t n ( ω ) | ≤ | ξ | + | b X n,t n ( ω ) − ξ | ≤ | ξ | + | b X n,t ( ω ) − b X n,t n ( ω ) | we therefore conclude that there exists c ∈ (0 , ∞ ) such that for all n ∈ N and all t ∈ [0 , | b X n,t − b X n,t n | · A n,t ≤ c · ( | W t − W t n | · A n,t + R n,t ) , where R n,t = (1 + | b X n,t − b X n,t n | ) · (1 /n + | W t − W t n | ) + | b X n,t − b X n,t n | · | W t − W t n | . Employing Lemma 4 we obtain that for every r ∈ N there exists c ∈ (0 , ∞ ) such that for all n ∈ N and all t ∈ [0 , E (cid:2) | R n,t | r (cid:3) ≤ c/n r , which yields that there exists c ∈ (0 , ∞ ) such that for all n ∈ N ,(86) E h(cid:12)(cid:12)(cid:12)Z | R n,t | q dt (cid:12)(cid:12)(cid:12) p i /p ≤ c/n q . Using (84) and (86) as well as (68) we conclude that there exist c , c ∈ (0 , ∞ ) such that for all n ∈ N , E h(cid:12)(cid:12)(cid:12)Z | b X n,t − b X n,t n | q · A n,t dt (cid:12)(cid:12)(cid:12) p i /p ≤ c · E h(cid:12)(cid:12)(cid:12)Z | W t − W t n | q · A n,t dt (cid:12)(cid:12)(cid:12) p i /p + c · E h(cid:12)(cid:12)(cid:12)Z | R n,t | q dt (cid:12)(cid:12)(cid:12) p i /p ≤ c · (1 /n ( q +1) / + 1 /n q ) ≤ c /n ( q +1) / , which finishes the proof of the proposition. (cid:3) METHOD OF ORDER 3 / Proof of the estimates (10) and (11) . For n ∈ N and t ∈ [0 ,
1] we put A t = Z t µ ( X s ) ds, b A n,t = Z t µ ( b X n,s n ) ds and B t = Z t σ ( X s ) dW s , b B n,t = Z t (cid:0) σ ( b X n,s n ) + σδ σ ( b X n,s n ) · ( W s − W s n ) (cid:1) dW s as well as U n,t = Z t σδ µ ( b X n,s n ) · ( W s − W s n ) ds and we use the decomposition(87) X t − b X n,t = ( A t − b A n,t − U n,t ) + ( B t − b B n,t ) + U n,t . Furthermore, we put S µ = (cid:16) k µ +1 [ ℓ =1 ( ξ ℓ − , ξ ℓ ) (cid:17) c , S σ = (cid:16) k σ +1 [ ℓ =1 ( η ℓ − , η ℓ ) (cid:17) c and we note that S µ = ∪ k µ ℓ =1 { ( x, y ) ∈ R : ( x − ξ ℓ ) · ( y − ξ ℓ ) ≤ } and S σ = ∪ k σ ℓ =1 { ( x, y ) ∈ R : ( x − η ℓ ) · ( y − η ℓ ) ≤ } . Observing the assumption (B3) we thus obtain by Proposition 1that there exists c ∈ (0 , ∞ ) such that for all n ∈ N and q ∈ { , } ,(88) E h(cid:12)(cid:12)(cid:12)Z | b X n,t − b X n,t n | q · S µ ∪ S σ ( b X n,t , b X n,t n ) dt (cid:12)(cid:12)(cid:12) p i ≤ c/n p ( q +1) / . For all n ∈ N and t ∈ [0 ,
1] we have(89) | µ ( X t ) − µ ( b X n,t n ) − σδ µ ( b X n,t n ) · ( W t − W t n ) |≤ | µ ( X t ) − µ ( b X n,t ) | + | µ ( b X n,t ) − µ ( b X n,t n ) − δ µ ( b X n,t n ) · ( b X n,t − b X n,t n ) | + | δ µ ( b X n,t n ) · ( b X n,t − b X n,t n − σ ( b X n,t n ) · ( W t − W t n )) | = | µ ( X t ) − µ ( b X n,t ) | + | µ ( b X n,t ) − µ ( b X n,t n ) − δ µ ( b X n,t n ) · ( b X n,t − b X n,t n ) | · S cµ ( b X n,t , b X n,t n )+ | µ ( b X n,t ) − µ ( b X n,t n ) − δ µ ( b X n,t n ) · ( b X n,t − b X n,t n ) | · S µ ( b X n,t , b X n,t n )+ (cid:12)(cid:12) δ µ ( b X n,t n ) · ( µ ( b X n,t n )( t − t n ) + σδ σ ( b X n,t n ) · (( W t − W t n ) − ( t − t n ))) (cid:12)(cid:12) . Using the assumption (B1) as well as (24), (25) and (26) we thus obtain that there exists c ∈ (0 , ∞ ) such that for all n ∈ N and all t ∈ [0 , | µ ( X t ) − µ ( b X n,t n ) − σδ µ ( b X n,t n ) · ( W t − W t n ) |≤ c · | X t − b X n,t | + c · | b X n,t − b X n,t n | + c · | b X n,t − b X n,t n | · S µ ( b X n,t , b X n,t n )+ c · (1 + | b X n,t n | ) · (1 /n + | W t − W t n | ) . Using (90) as well as Lemma 4 and (88) with q = 1 we conclude that there exist c , c ∈ (0 , ∞ )such that for all n ∈ N and all t ∈ [0 , E h sup ≤ s ≤ t | A s − b A n,s − U n,s | p i ≤ E hZ t | µ ( X s ) − µ ( b X n,s n ) − σδ µ ( b X n,s n ) · ( W s − W s n ) | p ds i ≤ c · Z t E (cid:2) | X s − b X n,s | p (cid:3) ds + c · Z t E (cid:2) | b X n,s − b X n,s n | p (cid:3) ds + c · E h(cid:12)(cid:12)(cid:12)Z t | b X n,s − b X n,s n | · S µ ( b X n,s , b X n,s n ) ds (cid:12)(cid:12)(cid:12) p i + c · Z t E (cid:2) (1 + | b X n,s n | p ) · (1 /n p + | W s − W s n | p (cid:3) ds ≤ c · Z t E [ | X s − b X n,s | p ] ds + c /n p . Proceeding similarly to (89) and (90) one obtains that there exists c ∈ (0 , ∞ ) such that forall n ∈ N and all t ∈ [0 , | σ ( X t ) − σ ( b X n,t n ) − σδ σ ( b X n,t n ) · ( W t − W t n ) |≤ c · | X t − b X n,t | + c · | b X n,t − b X n,t n | + c · | b X n,t − b X n,t n | · S σ ( b X n,t , b X n,t n )+ c · (1 + | b X n,t n | ) · (1 /n + | W t − W t n | ) . Employing the Burkholder-Davis-Gundy inequality, Lemma 4, (88) with q = 2 and (92) we thenconclude analogously to the derivation of (91) that there exist c , c , c ∈ (0 , ∞ ) such that forall n ∈ N and all t ∈ [0 , E h sup ≤ s ≤ t | B t − b B n,t | p i ≤ c · E h(cid:12)(cid:12)(cid:12)Z t | σ ( X s ) − σ ( b X n,s n ) − σδ σ ( b X n,s n ) · ( W s − W s n ) | ds (cid:12)(cid:12)(cid:12) p/ i ≤ c · Z t E (cid:2) | X s − b X n,s | p (cid:3) ds + c · Z t E (cid:2) | b X n,s − b X n,s n | p (cid:3) ds + c · E h(cid:12)(cid:12)(cid:12)Z t | b X n,s − b X n,s n | · S σ ( b X n,s , b X n,s n ) ds (cid:12)(cid:12)(cid:12) p/ i + c · Z t E (cid:2) (1 + | b X n,s n | p ) · (1 /n p + | W s − W s n | p (cid:3) ds ≤ c · Z t E [ | X s − b X n,s | p ] ds + c /n p/ . METHOD OF ORDER 3 / Combining (87) with (91) and (93) we see that there exists c ∈ (0 , ∞ ) such that for all n ∈ N and all t ∈ [0 , E (cid:2) sup ≤ s ≤ t | X t − b X n,t | p (cid:3) ≤ c · Z t E (cid:2) sup ≤ u ≤ s | X u − b X n,u | p (cid:3) ds + c/n p/ + E (cid:2) sup ≤ s ≤ t | U n,s | p (cid:3) . Note that E (cid:2) k X − b X n k p ∞ (cid:3) < ∞ due to (9) and Lemma 4. Below we show that there exists c ∈ (0 , ∞ ) such that for all n ∈ N ,(95) E (cid:2) sup ≤ s ≤ | U n,s | p (cid:3) ≤ c/n p . Inserting (95) into (94) and applying the Gronwall inequality then yields the error estimate inTheorem 3.We turn to the proof of (95). Clearly, for all n ∈ N , all ℓ ∈ { , . . . , n − } and all s ∈ [ ℓ/n, ( ℓ + 1) /n ] we have(96) U n,s = U n,ℓ/n + σδ µ ( b X n,ℓ/n ) · Z sℓ/n ( W u − W ℓ/n ) du, which jointly with Lemma 4 shows that the sequence ( U n,ℓ/n ) ℓ =0 ,...,n is a martingale. Further-more, using (24) and (25) we obtain from (96) that there exists c ∈ (0 , ∞ ) such that for all n ∈ N ,(97) sup ≤ s ≤ | U n,s | ≤ max ℓ =0 ,...,n − | U n,ℓ/n | + max ℓ =0 ,...,n − | σδ µ ( b X n,ℓ/n ) | · Z ( ℓ +1) /nℓ/n | W u − W ℓ/n | du ≤ max ℓ =0 ,...,n | U n,ℓ/n | + c · (1 + k b X n k ∞ ) · max ℓ =0 ,...,n − Z ( ℓ +1) /nℓ/n | W u − W ℓ/n | du. Clearly, for all q ∈ [1 , ∞ ) there exists c ∈ (0 , ∞ ) such that for all n ∈ N ,(98) E h(cid:12)(cid:12)(cid:12)Z ( ℓ +1) /nℓ/n | W u − W ℓ/n | du (cid:12)(cid:12)(cid:12) q i ≤ E h n q − · Z ( ℓ +1) /nℓ/n | W u − W ℓ/n | q du i ≤ cn q/ . Employing the Burkholder-Davis-Gundy inequality as well as (24), (25), Lemma 4 and (98) weobtain that there exist c , c , c ∈ (0 , ∞ ) such that for all n ∈ N ,(99) E (cid:2) max ℓ =0 ,...,n | U n,ℓ/n | p i ≤ E h(cid:16) n − X ℓ =0 (cid:16) σδ µ ( b X n,ℓ/n ) · Z ( ℓ +1) /nℓ/n ( W u − W ℓ/n ) du (cid:17) (cid:17) p/ i ≤ c · E (cid:2) (1 + k b X n k p ∞ ) (cid:3) / · E h(cid:16) n − X ℓ =0 (cid:16)Z ( ℓ +1) /nℓ/n ( W u − W ℓ/n ) du (cid:17) (cid:17) p i / ≤ c · (cid:16) n − X ℓ =0 E h(cid:16)Z ( ℓ +1) /nℓ/n | W u − W ℓ/n | du (cid:17) p i /p (cid:17) p/ ≤ c n p . Furthermore, by (98) and Lemma 4 we see that there exists c , c ∈ (0 , ∞ ) such that for all n ∈ N ,(100) E h(cid:16) (1 + k b X n k ∞ ) · max ℓ =0 ,...,k − Z ( ℓ +1) /nℓ/n | W u − W ℓ/n | du (cid:17) p i ≤ c · E (cid:2) (1 + k b X n k p ∞ ) (cid:3) / · E h n − X ℓ =0 (cid:16)Z ( ℓ +1) /nℓ/n ( W u − W ℓ/n ) du (cid:17) p i / ≤ c n p . Combining (97) with (99) and (100) yields (95) and completes the proof of the estimate (10) inTheorem 3.It remains to prove (11). In the case k σ = 0 we have S σ = ∅ . Then the estimates (91) and (95)still hold true but instead of the estimate (93) we obtain(101) E (cid:2) sup ≤ s ≤ t | B t − b B n,t | p (cid:3) ≤ c · Z t E (cid:2) | X s − b X n,s | p (cid:3) ds + c /n p , where c ∈ (0 , ∞ ) neither depends on n nor on t . Combining (91), (95) and (101) we obtain thatthere exists c ∈ (0 , ∞ ) such that for all n ∈ N and all t ∈ [0 , E (cid:2) sup ≤ s ≤ t | X t − b X n,t | p (cid:3) ≤ c · Z t E (cid:2) sup ≤ u ≤ s | X u − b X n,u | p (cid:3) ds + c/n p . Applying the Gronwall inequality we now obtain the estimate (11) from (102)7.
Proof of Lemmas 1, 2
We make use of the following result, which is straightforward to check.
Lemma 9.
Let −∞ ≤ a < b ≤ ∞ and let f, g : R → R be Lipschitz continuous on ( a, b ) . Assumefurther that there exists c ∈ (0 , ∞ ) such that g is constant on the set ( −∞ , c ) ∪ ( c, ∞ ) . Then f · g is Lipschitz continuous on ( a, b ) . Proof of Lemma 1.
We first show that G z,α,ν satisfies (i). It is straightforward to checkthat G z,α,ν is differentiable on R with G ′ z,α,ν ( x ) = 1 + k X i =1 α i ν · | x − z i | ν · (cid:16) − (cid:16) x − z i ν (cid:17) (cid:17) · (cid:16) − (cid:16) x − z i ν (cid:17) (cid:17) · [ z i − ν,z i + ν ] ( x )for all x ∈ R . Note that for every i ∈ { , . . . , k } the mapping x | x − z i | ν · (1 − ( x − z i ν ) ) · (1 − x − z i ν ) ) · [ z i − ν,z i + ν ] ( x ) is Lipschitz continuous on R . Thus, as a finite linear combinationof Lipschitz continuous functions, G ′ z,α,ν is Lipschitz continuous on R as well. Clearly, for all x ∈ { z , . . . , z k } ∪ R \ S ki =1 ( z i − ν, z i + ν ) we have(103) G ′ z,α,ν ( x ) = 1and for all i ∈ { , . . . , k } and all x ∈ [ ξ i − ν, ξ i + ν ] we have G ′ z,α,ν ( x ) ≥ − | α i | ν > , which finishes the proof of part (i) of the lemma. METHOD OF ORDER 3 / Next we show that G z,α,ν satisfies (ii) and (iii). Note that the intervals [ z i − ν, z i + ν ], i =1 , . . . , k , are pairwise disjoint. Observing (103) it is easy to check that G ′ z,α,ν is two timesdifferentiable on ∪ k +1 i =1 ( z i − , z i ) with(104) G ′′ z,α,ν ( x ) = − α i · ψ i ( x ) , if x ∈ ( z i − ν, z i ) , α i · ψ i ( x ) , if x ∈ ( z i , z i + ν ) , , if x ∈ R \ S kj =1 ( z j − ν, z j + ν ) ,G ′′′ z,α,ν ( x ) = − α i /ν · η i ( x ) , if x ∈ ( z i − ν, z i ) , α i /ν · η i ( x ) , if x ∈ ( z i , z i + ν ) , , if x ∈ R \ S kj =1 ( z j − ν, z j + ν ) , where(105) ψ i ( x ) = (cid:16) − (cid:16) x − z i ν (cid:17) (cid:17) (cid:16) − (cid:16) x − z i ν (cid:17) + 45 (cid:16) x − z i ν (cid:17) (cid:17) ,η i ( x ) = (cid:16) − (cid:16) x − z i ν (cid:17) (cid:17)(cid:16) − x − z i ν + 312 (cid:16) x − z i ν (cid:17) − (cid:16) x − z i ν (cid:17) (cid:17) . Obviously, on each interval ( z i − , z i ), G ′′ z,α,ν and G ′′′ z,α,ν are Lipschitz continuous, and we have G ′′ z,α,ν ( z i − ) = − α i = − G ′′ z,α,ν ( z i +). This finishes the proof of Lemma 1. (cid:3) Proof of Lemma 2.
Due to Lemma 1 the functions e µ and e σ are well-defined. Recallfrom Lemma 1(i) that there exists c ∈ (0 , ∞ ) such that G ′ ξ,α,ν = 1 on ( −∞ , c ) ∪ ( c, ∞ ). Hence G ′′ ξ,α,ν = 0 on ( −∞ , c ) ∪ ( c, ∞ ). By means of Lemma 9 we can thus conclude that G ′ ξ,α,ν · µ and G ′′ ξ,α,ν · σ are Lipschitz continuous on each of the intervals ( ξ , ξ ) , . . . , ( ξ k , ξ k +1 ) and that G ′ ξ,α,ν · σ is Lipschitz continuous on R . Observing Lemma 1(i),(iii) we see that for each i ∈ { , . . . , k } ,( G ′ ξ,α,ν · µ + G ′′ ξ,α,ν · σ )( ξ i − ) = µ ( ξ i − ) − α i · σ ( ξ i )= ( µ ( ξ i − ) + µ ( ξ i +)) / G ′ ξ,α,ν · µ + G ′′ ξ,α,ν · σ )( ξ i )= µ ( ξ i +) + α i · σ ( ξ i ) = ( G ′ ξ,α,ν · µ + G ′′ ξ,α,ν · σ )( ξ i +) . Hence G ′ ξ,α,ν · µ + G ′′ ξ,α,ν · σ is continuous on R and Lipschitz continuous on each of the intervals( ξ , ξ ) , . . . , ( ξ k , ξ k +1 ), which yields Lipschitz continuity of the latter function on the whole realline. Finally, recall that by Lemma 1, G − ξ,α,ν is Lipschitz continous. This shows that e µ and e σ satisfy the assumption (B1).Using the assumption (A3), Lemma 1(i),(ii) and the fact that G − (( ξ i − , ξ i )) = ( ξ i − , ξ i ) forall i ∈ { , . . . , k + 1 } we immediately obtain that for each i ∈ { , . . . , k + 1 } the functions e µ and e σ are differentiable on ( ξ i − , ξ i ) with derivatives e µ ′ = ( µ ′ + G ′′ ξ,α,ν /G ′ ξ,α,ν · ( µ + σ · σ ′ ) + G ′′′ ξ,α,ν /G ′ ξ,α,ν · σ ) ◦ G − ξ,α,ν , e σ ′ = ( σ ′ + G ′′ ξ,α,ν /G ′ ξ,α,ν · σ ) ◦ G − ξ,α,ν . Using the assumption (A3) and Lemma 1(i),(ii) again we can now derive by iteratively applyingLemma 9 (with any extension of µ ′ and σ ′ to the whole real line) that for each i ∈ { , . . . , k + 1 } the functions e µ ′ and e σ ′ are Lipschitz continous on ( ξ i − , ξ i ). Hence e µ and e σ satisfy the assumption(B2) with k µ = k σ = k and η i = ξ i for i ∈ { , . . . , k } . Finally, note that G ξ,α,ν ( ξ i ) = ξ i for each i ∈ { , . . . , k } , which yields that e σ ( ξ i ) = σ ( ξ i ) = 0 for each i ∈ { , . . . , k } . Hence e σ satisfies theassumption (B3), which finishes the proof of Lemma 2. (cid:3) References [1]
G¨ottlich, S., Lux, K., and Neuenkirch, A.
The Euler scheme for stochastic differential equations withdiscontinuous drift coefficient: A numerical study of the convergence rate. arXiv:1705.04562 (2017), 18 pages.[2]
Gy¨ongy, I.
A note on Euler’s approximations.
Potential Anal. 8 , 3 (1998), 205–216.[3]
Gy¨ongy, I., and Krylov, N.
Existence of strong solutions for Itˆo’s stochastic equations via approximations.
Probab. Theory Related Fields 105 , 2 (1996), 143–158.[4]
Halidias, N., and Kloeden, P. E.
A note on the Euler-Maruyama scheme for stochastic differentialequations with a discontinuous monotone drift coefficient.
BIT 48 , 1 (2008), 51–59.[5]
Hefter, M., Herzwurm, A., and M¨uller-Gronbach, T.
Lower error bounds for strong approximationof scalar sdes with non-lipschitzian coefficients.
Ann. Appl. Probab. 29 , 1 (2019), 178–216.[6]
Hofmann, N., M¨uller-Gronbach, T., and Ritter, K.
The optimal discretization of stochastic differentialequations.
J. Complexity 17 (2001), 117–153.[7]
Kloeden, P. E., and Platen, E.
Numerical solution of stochastic differential equations , vol. 23 of
Appli-cations of Mathematics (New York) . Springer-Verlag, Berlin, 1992.[8]
Kruse, R., and Wu, Y.
A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients.
Discrete Contin. Dyn. Syst. Ser. B (online first) (2018).[9]
Leobacher, G., and Sz¨olgyenyi, M.
A numerical method for SDEs with discontinuous drift.
BIT 56 , 1(2016), 151–162.[10]
Leobacher, G., and Sz¨olgyenyi, M.
A strong order 1/2 method for multidimensional SDEs with discon-tinuous drift.
Ann. Appl. Probab. 27 (2017), 2383–2418.[11]
Leobacher, G., and Sz¨olgyenyi, M.
Convergence of the Euler-Maruyama method for multidimensionalSDEs with discontinuous drift and degenerate diffusion coefficient.
Numer. Math. 138 , 1 (2018), 219–239.[12]
Mao, X.
Stochastic differential equations and applications , second ed. Horwood Publishing Limited, Chich-ester, 2008.[13]
M¨uller-Gronbach, T.
Optimal pointwise approximation of SDEs based on Brownian motion at discretepoints.
Ann. Appl. Probab. 14 , 4 (2004), 1605–1642.[14]
M¨uller-Gronbach, T., and Yaroslavtseva, L.
On the performance of the Euler-Maruyama scheme forSDEs with discontinuous drift coefficient. arXiv:1809.08423 (2018).[15]
Neuenkirch, A., Sz¨olgyenyi, M., and Szpruch, L.
An adaptive Euler-Maruyama scheme for stochasticdifferential equations with discontinuous drift and its convergence analysis.
SIAM J. Numer. Anal. 57 (2019),378–403.[16]
Ngo, H.-L., and Taguchi, D.
Strong rate of convergence for the Euler-Maruyama approximation of sto-chastic differential equations with irregular coefficients.
Math. Comp. 85 , 300 (2016), 1793–1819.[17]
Ngo, H.-L., and Taguchi, D.
On the Euler-Maruyama approximation for one-dimensional stochastic dif-ferential equations with irregular coefficients.
IMA J. Numer. Anal. 37 , 4 (2017), 1864–1883.[18]
Ngo, H.-L., and Taguchi, D.
Strong convergence for the Euler-Maruyama approximation of stochasticdifferential equations with discontinuous coefficients.
Statist. Probab. Lett. 125 (2017), 55–63.[19]
Revuz, D., and Yor, M.
Continuous martingales and Brownian motion , third ed. Springer-Verlag, Berlin,1995.
Faculty of Computer Science and Mathematics, University of Passau, Innstrasse 33, 94032 Pas-sau, Germany
E-mail address : [email protected] Faculty of Computer Science and Mathematics, University of Passau, Innstrasse 33, 94032 Pas-sau, Germany
E-mail address ::