An adaptive strong order 1 method for SDEs with discontinuous drift coefficient
aa r X i v : . [ m a t h . P R ] F e b AN ADAPTIVE STRONG ORDER 1 METHOD FOR SDES WITHDISCONTINUOUS DRIFT COEFFICIENT
LARISA YAROSLAVTSEVA
Abstract.
In recent years, an intensive study of strong approximation of stochastic differen-tial equations (SDEs) with a drift coefficient that may have discontinuities in space has begun.In many of these results it is assumed that the drift coefficient satisfies piecewise regularityconditions and the diffusion coefficient is Lipschitz continuous and non-degenerate at the dis-continuity points of the drift coefficient. For scalar SDEs of that type the best L p -error rateknown so far for approximation of the solution at the final time point is 3 / L p -error rate 3 / L p -error rate of at least 1 in terms of the averagenumber of evaluations of the driving Brownian motion for such SDEs. Introduction
In this article we consider a scalar autonomous stochastic differential equation (SDE)(1) dX t = µ ( X t ) dt + σ ( X t ) dW t , t ≥ ,X = x , where x ∈ R is the initial value, µ : R → R is the drift coefficient, σ : R → R is the diffusioncoefficient, W = ( W t ) t ≥ is a 1-dimensional Brownian motion and we assume that the SDE (1)has a unique strong solution X . Our computational task is L p -approximation of X by numericalmethods that are based on finitely many evaluations of the driving Brownian motion W at pointsin [0 ,
1] in the case when the drift coefficient µ may have finitely many discontinuity points.Strong approximation of SDEs with a discontinuous drift coefficient has gained a lot of interestin the literature in recent years. See [4, 5] for results on convergence in probability and almost sureconvergence of the Euler-Maruyama scheme and [1, 3, 6, 13, 14, 15, 18, 20, 22, 23, 24, 25, 26, 27]for results on L p -approximation. In many of these articles it is assumed that the drift coefficientsatisfies piecewise regularity conditions and the diffusion coefficient is Lipschitz continuous andnon-degenerate at the discontinuity points of the drift coefficient. For SDEs of that type the best L p -error rate known up to now for approximation of X is 3 /
4, see [18]. In the present articlewe construct for the first time in the literature a numerical method, which achieves an L p -errorrate of at least 1 for such SDEs.To be more precise, let us consider the following assumptions on the coefficients µ and σ . ( µ
1) 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 } ,( σ σ is Lipschitz continuous on R and σ ( ξ i ) = 0 for all i ∈ { , . . . , k } ,If ( µ
1) and ( σ
1) hold then the SDE (1) has a unique strong solution, see [13]. In [13, 14, 15, 18,20, 23] the L p -approximation of X under the assumptions ( µ
1) and ( σ
1) has been analyzed. Inparticular, in [13, 14] the transformed equidistant Euler-Maruyama scheme has been constructed,which achieves an L -error rate of at least 1 / W . After that, in [23] an adaptive Euler-Maruyama scheme has beenconstructed, which achieves up to a logarithmic factor an L -error rate of at least 1 / W used by the scheme. Finally, in [20] it has been proventhat the classical equidistant Euler-Maruyama scheme achieves for all p ∈ [1 , ∞ ) an L p -errorrate of at least 1 / W as in the case of SDEs withglobally Lipschitz continuous coefficients.In [18] the first higher-order method has been constructed for such SDEs. This method isbased on equidistant evaluations of W and achieves for all p ∈ [1 , ∞ ) an L p -error rate of at least3 / W if µ and σ satisfy ( µ
1) and ( σ
1) and additionallythe following piecewise regularity assumptions( µ µ has a Lipschitz continuous derivative on ( ξ i − , ξ i ) for every i ∈ { , . . . , k + 1 } ,( σ σ has a Lipschitz continuous derivative on ( ξ i − , ξ i ) for every i ∈ { , . . . , k + 1 } .Furthermore, in [22] it has been shown that for SDEs (1) with additive noise and a boundedand piecewise C b drift coefficient µ the equidistant Euler-Maruyama scheme in fact achieves an L -error rate of at least 3 / − in terms of the number of evaluations of W . Note that in this casethe Euler-Maruayama scheme coincides with the Milstein scheme.Recently in [21] it has been shown that an L p -error rate better than 3 / µ µ σ
1) and ( σ
2) by no numerical method based onevaluations of W at fixed time points in [0 , σ = 1 and if µ satisfies ( µ
1) and ( µ µ is bounded, increasing and there exists i ∈ { , . . . , k } such that µ ( ξ i +) = µ ( ξ i − ), then there exists c ∈ (0 , ∞ ) such that for all p ∈ [1 , ∞ ) and all n ∈ N ,(2) inf t ,...,t n ∈ [0 , g : R n → R measurable E (cid:2) | X − g ( W t , . . . , W t n ) | p (cid:3) /p ≥ cn / . Note that the lower bound (2) does not cover adaptive methods, i.e. methods that may choosethe number as well as the location of the evaluations of the Brownian motion W in a sequentialway dependent on the values of W observed so far. See e.g. [2, 9, 10, 12, 16, 17, 23, 29] forexamples of such methods. It is well-known that for a large class of SDEs (1) with globallyLipschitz continuous coefficients the best possible L p -error rate that can be achieved by non-adaptive methods coincides with the best possible L p -error rate that can be achieved by adaptivemethods and is equal to 1, see [16, 17]. Moreover, up to now there is no example of an SDE withglobally Lipschitz continuous coefficients known in the literature, for which adaptive methodsare superior to non-adaptive ones with respect to the L p -error rate. However, the superiorityof adaptive methods to non-adaptive ones with respect to the L p -error rate has recently been N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 3 demonstrated in [7, 19] for some examples of SDEs with non-globally Lipschitz continuous driftor diffusion coefficients.In view of the latter results it is natural to ask whether there exists an adaptive method thatachieves under the assumptions ( µ µ
2) and ( σ σ
2) a better L p -error rate than the rate3 /
4. To the best of our knowledge the answer to this question was not known in the literature upto now. In the present article we answer this question in the positive. More precisely, we constructa family of approximations b X δ with δ ∈ (0 , δ ] for some δ > b X δ is based on at most c · δ − adaptively chosen evaluations of W in the interval [0 ,
1] on averageand such that for all p ∈ [1 , ∞ ) and all δ ∈ (0 , δ ], E (cid:2) | X − b X δ | p (cid:3) /p ≤ c ( p ) · δ, where the constants c, c ( p ) ∈ (0 , ∞ ) do not depend on δ , see Theorem 2. Thus, the approxima-tions b X δ achieve an L p -error rate of at least 1 in terms of the average number of evaluations of W . The methods b X δ are obtained by applying a suitable transformation G : R → R to the strongsolution X of the SDE (1) such that the transformed solution Z = ( G ( X t )) t ≥ is a strong solu-tion of a new SDE with coefficients e µ and e σ which satisfy ( µ µ
2) and ( σ σ e µ is continuous, which implies that e µ is Lipschitz continuous. An adaptive quasi-Milstein scheme b Z δ = ( b Z δt ) t ≥ is used to approximate Z and the approximation b X δ is then givenby G − ( b Z δ ). The adaptive time stepping strategy used for the adaptive quasi-Milstein scheme b Z δ is an appropriate modification of the adaptive time stepping strategy used for the adaptiveEuler-Maruyama scheme in [23]. We add that an L p -error rate better than 1 can not be achievedin general under the assumptions ( µ µ
2) and ( σ σ
2) by no adaptive method based onfinitely many evaluations of W , see [8, 16, 17] for corresponding lower error bounds.The implementation of our method requires the ability to evaluate the functions G and G − at each step of the adaptive quasi-Milstein scheme b Z δ . While the transformation G is knownexplicitly, this is so far not the case for G − , and therefore a numerical inverse of G has tobe used to approximate G − . This makes our method rather slow in practice. We conjecturehowever that the transformation of the SDE (1) is actually not needed and that an adaptivequasi-Milstein scheme for the SDE (1) itself achieves under the assumptions ( µ µ
2) and ( σ σ
2) an L p -error rate of at least 1 in terms of the average number of evaluations of W . The proofof this conjecture will be the subject of future work.We briefly describe the content of the paper. In Section 2 we introduce some notation. Section 3contains the construction and the error and cost analysis of the adaptive quasi-Milstein schemein the case when the coefficients of the SDE (1) satisfy the assumptions ( µ µ
2) and ( σ σ G that is then used to construct a method of order 1 under the assumptions( µ µ
2) and ( σ σ YAROSLAVTSEVA Notation
For A ⊂ R and x ∈ R we put d ( x, A ) = inf {| x − y | : y ∈ A } . For a function f : R → R wedefine d f : R → R by d f ( x ) = ( f ′ ( x ) , if f is differentiable in x, , otherwise.3. An adaptive quasi-Milstein scheme for SDEs with Lipschitz continuouscoefficients
Let (Ω , F , P ) be a complete probability space, let W : [0 , ∞ ) × Ω → R be a Brownian motion on(Ω , F , P ), let x ∈ R and let µ : R → R and σ : R → R be functions that satisfy the assumptions( µ µ
2) and ( σ σ µ is continuous. We consider the SDE(3) dX t = µ ( X t ) dt + σ ( X t ) dW t , t ≥ ,X = x . Observe that in this case both µ and σ are Lipschitz continuous on R , and therefore the SDE(3) has a unique strong solution and for every p ∈ [1 , ∞ ) it holds(4) E (cid:2) sup t ∈ [0 , | X t | p (cid:3) < ∞ . Put Θ = { ξ , . . . , ξ k } and for ε > ε = { x ∈ R : d ( x, Θ) < ε } . Let ε ∈ (0 ,
1] and assume that ε ≤
12 min { ξ i − ξ i − : i = 2 , . . . , k } if k ≥
2. For δ > ε δ = √ δ · log (1 /δ ) , ε δ = δ · log (1 /δ ) . Let δ ∈ (0 ,
1) be small enought such that for all δ ∈ (0 , δ ] it holds(5) ε δ ≤ ε δ ≤ ε / . For δ ∈ (0 , δ ] we define a time-continuous adaptive quasi-Milstein scheme b X δ = ( b X δt ) t ≥ recur-sively by(6) τ δ = 0 , b X δτ δ = x and(7) τ δi +1 = τ δi + h δ ( b X δτ δi ) , b X δt = b X δτ δi + µ ( b X δτ δi ) · ( t − τ δi ) + σ ( b X δτ δi ) · ( W t − W τ δi )+ 12 σd σ ( b X δτ δi ) · (cid:0) ( W t − W τ δi ) − ( t − τ δi ) (cid:1) , t ∈ ( τ δi , τ δi +1 ] , N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 5 for i ∈ N , where the step size function h δ : R → (0 ,
1) is defined by h δ ( x ) = δ, x Θ ε δ , (cid:16) d ( x, Θ)log (1 /δ ) (cid:17) , x ∈ Θ ε δ \ Θ ε δ ,δ · log (1 /δ ) , x ∈ Θ ε δ . (8)Note that the assumption (5) implies that Θ ε δ ⊆ Θ ε δ for all δ ∈ (0 , δ ] and hence h δ is well-defined for all δ ∈ (0 , δ ]. Moreover, h δ is continuous and it holds(9) δ · log (1 /δ ) ≤ h δ ≤ δ for all δ ∈ (0 , δ ]. We add that the step size function h δ we use for the adaptive quasi-Milsteinscheme b X δ is an appropriate modification of the step size function used for the adaptive Euler-Maruyama scheme in [23].For δ ∈ (0 , δ ] let N ( b X δ ) denote the number of evaluations of W used to compute b X δ , i.e. N ( b X δ ) = min { i ∈ N : τ δi ≥ } . Clearly, for all δ ∈ (0 , δ ], N ( b X δ ) ≤ ⌈ δ − log − (1 /δ ) ⌉ . We have the following upper bounds for the p -th root of the p -th mean of the maximum errorof b X δ on the time interval [0 ,
1] and for the average number of evaluations of W used to compute b X δ . Theorem 1.
Assume ( µ µ
2) and ( σ σ
2) and assume that µ is continuous. Let p ∈ [1 , ∞ ) .Then there exists c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] , (10) E (cid:2) sup t ∈ [0 , | X t − b X δt | p (cid:3) /p ≤ c · δ and (11) E [ N ( b X δ )] ≤ c · δ − . The proof of Theorem 1 is postponed to Section 5.4.
An adaptive strong order 1 method for SDEs with discontinuous driftcoefficient
As in Section 3 we consider a complete probability space (Ω , F , P ) and we assume that W : [0 , ∞ ) × Ω → R is a Brownian motion on (Ω , F , P ). In contrast to Section 3 we now turn toSDEs with a drift coefficient µ that may have discontinuity points.Let x ∈ R and let µ : R → R and σ : R → R be functions that satisfy the assumptions ( µ µ
2) and ( σ σ µ
1) implies the existence ofthe 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 ≥ ,X = x , which has a unique strong solution, see [13, Theorem 2.2]. YAROSLAVTSEVA
We now constuct an adaptive method for approximating the strong solution of the SDE (12)at the time 1. To this end we employ the transformation strategy from [18]. We use that X can be obtained by applying a Lipschitz continuous transformation to the strong solution of anSDE with coefficients e µ, e σ satisfying the assumptions ( µ µ
2) and ( σ σ e µ is continuous, and then we employ Theorem 1.We start by introducing the transformation procedure from [18]. 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. For the proofs of both lemmas see [18]. 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 inf x ∈ R G ′ z,α,ν ( x ) > . In particular, G z,α,ν has an inverse G − z,α,ν : R → R that is Lipschitzcontinuous. (ii) For every i ∈ { , . . . , k + 1 } , the function G ′ z,α,ν is differentiable on ( z i − , z i ) with Lips-chitz continuous derivatives G ′′ z,α,ν . (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 ( µ µ
2) and ( σ σ 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 N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 7 satisfy the assumptions ( µ µ
2) and ( σ σ e µ is continuous. 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 ≥ . Then the process Z is the unique strong solution of the SDE(17) dZ t = e µ ( Z t ) dt + e σ ( Z t ) dW t , t ≥ ,Z = G ξ,α,ν ( x )with e µ and e σ given by (15), see [18]. For every δ ∈ (0 , δ ] we use b Z δ = ( b Z δt ) t ≥ to denote thetime-continuous adaptive quasi-Milstein scheme (6), (7) associated to the SDE (17), i.e. b Z δ isdefined recursively by(18) τ δ = 0 , b Z δτ δ = G ξ,α,ν ( x )and(19) τ δi +1 = τ δi + h δ ( b Z δτ δi ) , b Z δt = b Z δτ δi + e µ ( b Z δτ δi ) · ( t − τ δi ) + e σ ( b Z δτ δi ) · ( W t − W τ δi )+ 12 e σd e σ ( b Z δτ δi ) · (cid:0) ( W t − W τ δi ) − ( t − τ δi ) (cid:1) , t ∈ ( τ δi , τ δi +1 ] , for i ∈ N , where the step size function h δ is given by (8).We approximate X by the stochastic process b X δ = ( b X δt ) t ≥ with b X δt = G − ξ,α,ν ( b Z δt ), t ≥
0. For δ ∈ (0 , δ ] let N ( b X δ ) denote the number of evaluations of W used to compute b X δ . We have thefollowing upper bounds for the p -th root of the p -th mean of the maximum error of b X δ on thetime interval [0 ,
1] and for the average number of evaluations of W used to compute b X δ . Theorem 2.
Assume ( µ µ
2) and ( σ σ p ∈ [1 , ∞ ) . Then there exists c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] , (20) E (cid:2) sup t ∈ [0 , | X t − b X δt | p (cid:3) /p ≤ c · δ and (21) E [ N ( b X δ )] ≤ c · δ − . Proof.
Using the Lipschitz continuity of G − ξ,α,ν , see Lemma 1(i), the fact that e µ and e σ satisfythe assumptions ( µ µ
2) and ( σ σ e µ is continuous as well as theestimate (10) in Theorem 1 we obtain that there exist c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ], E (cid:2) sup t ∈ [0 , | X t − b X δt | p (cid:3) /p = E (cid:2) sup t ∈ [0 , | X t − G − ξ,α,ν ( b Z δt ) | p (cid:3) /p ≤ c · E (cid:2) sup t ∈ [0 , | Z t − b Z δt | p (cid:3) /p ≤ c · δ. Thus, (20) holds. The estimate (21) follows from the fact that N ( b X δ ) = N ( b Z δ ) and the estimate(11) in Theorem 1. (cid:3) YAROSLAVTSEVA Proof of Theorem 1
Throughout this section we assume that µ and σ satisfy ( µ µ
2) and ( σ σ µ is continuous. Moreover, for δ ∈ (0 , δ ] and t ∈ [0 ,
1] we put t δ = max { τ δi : i ∈ N , τ δi ≤ t } . We first briefly describe the structure of the proof of the error estimate (10) in Theorem 1and the relation of our analysis and the error analysis of the equidistant quasi-Milstein schemein [18]. Let b X δ,eq = ( b X δ,eqt ) t ≥ denote the equidistant quasi-Milstein scheme with step size δ , i.e. b X δ,eq is defined in the same way as b X δ in (7), but with h δ = δ in place of (8). For simplicitylet us restrict to the case p = 2. In [18] it is shown that there exists c ∈ (0 , ∞ ) such that for all δ ∈ { /n : n ∈ N } ,(22) E (cid:2) sup t ∈ [0 , | X t − b X δ,eqt | (cid:3) / ≤ c · δ + c · (cid:16)Z E (cid:2) | b X δ,eqt − b X δ,eqt δ | · S ( b X δ,eqt , b X δ,eqt δ ) (cid:3) dt (cid:17) / , where S = (cid:16) k +1 [ i =1 ( ξ i − , ξ i ) (cid:17) c is the set of pairs ( x, y ) in R , which do not allow for a joint Lipschitz estimate of | d µ ( x ) − d µ ( y ) | or of | d σ ( x ) − d σ ( y ) | if µ or σ is not differentiable at one of the points ξ , . . . , ξ k . Transformingthe condition ( b X δ,eqt , b X δ,eqt δ ) ∈ S into a condition solely on the sizes of the random variables | b X δt δ − ( t − t δ ) − ξ i | , | b X δt δ − ( t − t δ ) − b X δt δ | and | b X δt δ − b X δt | , where ξ i lies between b X δ,eqt and b X δ,eqt δ , andemploying a Markov-type property of b X δ,eq and occupation time estimates for b X δ,eq it is shownin [18] that there exists c ∈ (0 , ∞ ) such that for all δ ∈ { /n : n ∈ N } ,(23) Z E (cid:2) | b X δ,eqt − b X δ,eqt δ | · S ( b X δ,eqt , b X δ,eqt δ ) (cid:3) dt ≤ c · δ / . Combining (22) and (23) yields the rate of convergence 3 / b X δ,eq on the time interval [0 , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] the adaptive quasi-Milsteinscheme b X δ satisfies(24) E (cid:2) sup t ∈ [0 , | X t − b X δt | (cid:3) / ≤ c · δ + c · (cid:16)Z E (cid:2) | b X δt − b X δt δ | · S ( b X δt , b X δt δ ) (cid:3) dt (cid:17) / . However, we obtain a much better upper bound for the integral on the right hand side of (24)than the upper bound c · δ / in (23) in the case of the equidistant quasi-Milstein scheme b X δ,eq .More precisely, we show that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(25) Z E (cid:2) | b X δt − b X δt δ | · S ( b X δt , b X δt δ ) (cid:3) dt ≤ c · δ , N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 9 which jointly with (24) yields the error estimate (10). For the proof of (25) we split the integralon the left hand side of (25) into four terms using the identities1 = 1 (Θ ε ) c ( b X δt δ ) + 1 Θ ε \ Θ εδ ( b X δt δ ) + 1 Θ εδ \ Θ εδ ( b X δt δ ) + 1 Θ εδ ( b X δt δ ) , t ∈ [0 , , and prove the upper bound c · δ for each of the resulting terms employing uniform L p -estimatesof b X δ , appropriate upper bounds for the probabilities that the increments | b X δt − b X δt δ | are largecompared to the distance of b X δt δ from the set Θ as well as estimates for the expected value ofcertain occupation time functionals of b X δ . We add that for the proof of (25) it is crucial that theadaptive quasi-Milstein scheme b X δ uses smaller step sizes when it is close to the discontinuitypoints of µ .For the proof of the estimate (11) we proceed similarly to the cost analysis of the adaptiveEuler-Maruyama scheme in [22, Section 5].We briefly describe the structure of this section. In Section 5.1 we provide properties of therandom times τ δi and t δ that are crucial for our proofs. In Section 5.2 we prove L p -estimates ofthe adaptive quasi-Milstein scheme b X δ . Section 5.3 contains estimates for the expected value ofoccupation time functionals of b X δ as well as estimates for the probabilities that the increments | b X δt − b X δt δ | of the adaptive quasi-Milstein scheme are large compared to the distance of the actualvalue of the scheme b X δt δ from the set Θ, which finally lead to the proof of the estimate (25),see Proposition 1. The results in Sections 5.2 and 5.3 are then used in Section 5.4 to derive theerror estimate (10) in Theorem 1. Section 5.5 is devoted to the proof of the estimate (11) inTheorem 1.Throughout the following we will employ the following facts, which are an immediate conse-quence of the assumptions ( µ µ
2) and ( σ σ
2) and the assumption that µ is continuous.Namely, the function µ is Lipschitz continuous on R , the functions µ and σ satisfy a lineargrowth condition, i.e.(26) ∃ K ∈ (0 , ∞ ) ∀ x ∈ R : | µ ( x ) | + | σ ( x ) | ≤ K · (1 + | x | ) , the functions d µ and d σ are bounded, i.e.(27) k d µ k ∞ + k d σ k ∞ < ∞ , and it holds(28) ∃ c ∈ (0 , ∞ ) ∀ f ∈ { µ, σ } ∀ i ∈ { , . . . , k + 1 } ∀ x, y ∈ ( ξ i − , ξ i ) : | f ( y ) − f ( x ) − f ′ ( x )( y − x ) | ≤ c · | y − x | . Properties of the random times τ δi and t δ . Let ( F t ) t ≥ denote the augmentation ofthe filtration generated by W , i.e. for all t ≥ F t = σ (cid:0) σ ( { W s : s ∈ [0 , t ] } ) ∪ N (cid:1) , where N = { N ∈ F : P ( N ) = 0 } . For a stopping time τ : Ω → [0 , ∞ ) let F τ denote the σ -algebraof τ -past, i.e. F τ = { A ∈ F : A ∩ { τ ≤ t } ∈ F t for all t ≥ } . Moreover, for a random time τ : Ω → [0 , ∞ ) define a stochastic process W τ : [0 , ∞ ) × Ω → R by W τt = W τ + t − W τ , t ≥ . The following two lemmas provide the properties of the random times τ δi and t δ that are crucialfor our proofs. Lemma 3.
Let δ ∈ (0 , δ ] . Then for all i ∈ N , (i) τ δi is a stopping time and b X δτ δi is F τ δi / B ( R ) -measurable, (ii) τ δi +1 is F τ δi / B ([0 , ∞ )) -measurable, (iii) W τ δi is a Brownian motion and independent of F τ δi and (iv) τ δi ∧ is a stopping time and b X δτ δi ∧ is F τ δi ∧ / B ( R ) -measurable, (v) τ δi +1 ∧ is F τ δi ∧ / B ([0 , ∞ )) -measurable, (vi) W τ δi ∧ is a Brownian motion and independent of F τ δi ∧ .Proof. We prove (i) by induction on i ∈ N . Clearly, (i) holds for i = 0. Next, assume that(i) holds for some i ∈ N . Then using the definition (7) of τ δi +1 we conclude that τ δi +1 is F τ δi / B ([0 , ∞ ))-measurable and τ δi +1 ≥ τ δi . Applying [11, Exercise 1.2.14] we thus obtain that τ δi +1 is a stopping time. This in particular yields that W τ δi +1 is F τ δi +1 / B ( R )-measurable and W τ δi is F τ δi / B ( R )-measurable. Thus, using the fact that F τ δi ⊂ F τ δi +1 as well as the induction as-sumption we obtain from the definition (7) of b X δτ δi +1 that b X δτ δi +1 is F τ δi +1 / B ( R )-measurable. Thedefinition (7) of τ δi +1 and (i) imply (ii). The strong Markov property of W yields (iii).For the proof of (iv)-(vi) put s δi = τ δi ∧ , i ∈ N , observe that s δ = 0 , b X δs δ = x and s δi +1 = ( s δi + h δ ( b X δs δi )) ∧ , b X δs δi +1 = b X δs δi + µ ( b X δs δi ) · ( s δi +1 − s δi ) + σ ( b X δs δi ) · ( W s δi +1 − W s δi )+ 12 σd σ ( b X δs δi ) · (cid:0) ( W s δi +1 − W s δi ) − ( s δi +1 − s δi ) (cid:1) for i ∈ N and proceed similarly to the proof of (i)-(iii). (cid:3) Lemma 4.
Let δ ∈ (0 , δ ] and t ∈ [0 , ∞ ) . Then W t δ is a Brownian motion and independent of b X δt δ .Proof. Clearly, W t δ is continuous. Employing Lemma 3(i),(ii),(iii) we obtain that for all A ∈B ( C ([0 , ∞ ); R )), P ( W t δ ∈ A ) = ∞ X i =0 P ( W τ δi ∈ A, τ δi ≤ t < τ δi +1 ) = ∞ X i =0 P ( W τ δi ∈ A ) · P ( τ δi ≤ t < τ δi +1 ) = P ( W ∈ A ) . N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 11
Thus, W t δ is a Brownian motion. Applying the latter fact as well as Lemma 3(i),(ii),(iii) weconclude that for all A ∈ B ( C ([0 , ∞ ); R )) and all B ∈ B ( R ), P ( W t δ ∈ A, b X δt δ ∈ B ) = ∞ X i =0 P ( W τ δi ∈ A, b X δτ δi ∈ B, τ δi ≤ t < τ δi +1 )= ∞ X i =0 P ( W τ δi ∈ A ) · P ( b X δτ δi ∈ B, τ δi ≤ t < τ δi +1 )= P ( W t δ ∈ A ) · ∞ X i =0 P ( b X δτ δi ∈ B, τ δi ≤ t < τ δi +1 )= P ( W t δ ∈ A ) · P ( b X δt δ ∈ B ) , which shows that W t δ and b X δt δ are independent and completes the proof of the lemma. (cid:3) L p estimates of the adaptive quasi-Milstein scheme. Using Lemma 3(i) one can showin a straightforward way that for all δ ∈ (0 , δ ] and all t ∈ [0 , ∞ ),(29) b X δt = x + Z t µ ( b X δs δ ) ds + Z t (cid:0) σ ( b X δs δ ) + σd σ ( b X δs δ ) · ( W s − W s δ ) (cid:1) dW s P -a.s.Employing (29) we obtain the following uniform L p -estimates for b X δ , δ ∈ (0 , δ ]. Lemma 5.
Let p ∈ [1 , ∞ ) . Then there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] , (30) E (cid:2) sup t ∈ [0 , | b X δt | p (cid:3) /p ≤ c. Moreover, there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] , all ∆ ∈ [0 , and all t ∈ [0 , − ∆] , (31) E (cid:2) sup s ∈ [ t,t +∆] | b X δs − b X δt | p (cid:3) /p ≤ c · √ ∆ . Proof.
We first show that for all δ ∈ (0 , δ ] and all i ∈ N ,(32) E (cid:2) | b X δτ δi | p (cid:3) < ∞ . Let δ ∈ (0 , δ ]. We prove (32) by induction on i ∈ N . Clearly, (32) holds for i = 0. Next, assumethat (32) holds for some i ∈ N . By (7), (26) and (27) there exist c , c ∈ (0 , ∞ ) such that | b X δτ δi +1 | p ≤ c · (cid:0) | b X δτ δi | p + | µ ( b X δτ δi ) | p · δ p + | σ ( b X δτ δi ) | p · | W τ δi +1 − W τ δi | p + 12 | σd σ ( b X δτ δi ) | p · ( | W τ δi +1 − W τ δi | p + δ p ) (cid:1) ≤ c · (1 + | b X δτ δi | p ) · (1 + sup t ∈ [0 ,δ ] | W τ δi t | p + sup t ∈ [0 ,δ ] | W τ δi t | p ) . Using the independence of b X δτ δi and W τ δi , the fact that W τ δi is a Brownian motion as well as theinduction assumption we therefore conclude that E (cid:2) | b X δτ δi +1 | p (cid:3) ≤ c · (1 + E (cid:2) | b X δτ δi | p (cid:3) ) · (1 + E (cid:2) sup t ∈ [0 ,δ ] | W τ δi t | p (cid:3) + E (cid:2) sup t ∈ [0 ,δ ] | W τ δi t | p (cid:3) ) < ∞ , which completes the proof of (32).For δ ∈ (0 , δ ] put(33) n δ = ⌈ δ − log − (1 /δ ) ⌉ . It follows from (32) that for all δ ∈ (0 , δ ],(34) sup t ∈ [0 , E (cid:2) | b X δt δ | p (cid:3) = sup t ∈ [0 , n δ X i =0 E (cid:2) | b X δτ δi | p · { t δ = τ δi } (cid:3) ≤ n δ X i =0 E (cid:2) | b X δτ δi | p (cid:3) < ∞ . We next prove (30). By (29), for all δ ∈ (0 , δ ] and all t ∈ [0 , E (cid:2) sup s ∈ [0 ,t ] | b X δs | p (cid:3) ≤ p · | x | p + 3 p · E h(cid:12)(cid:12)(cid:12)Z t | µ ( b X δu δ ) | du (cid:12)(cid:12)(cid:12) p i + 3 p · E h sup s ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)Z s (cid:0) σ ( b X δu δ ) + σ · d σ ( b X δu δ ) · ( W u − W u δ ) (cid:1) dW u (cid:12)(cid:12)(cid:12) p i . Using the H¨older inequality, the Burkholder-Davis-Gundy inequality, (26) and (27) we concludethat there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all t ∈ [0 , E (cid:2) sup s ∈ [0 ,t ] | b X δs | p (cid:3) ≤ c + c · Z t E (cid:2) | b X δu δ | p (cid:3) du + c · Z t E (cid:2) (1 + | b X δu δ | p ) · | W u − W u δ | p (cid:3) du. Lemma 4 implies that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all u ∈ [0 , E (cid:2) (1 + | b X δu δ | p ) · | W u − W u δ | p (cid:3) ≤ E (cid:2) (1 + | b X δu δ | p ) · sup s ∈ [0 ,δ ] | W u δ s | p (cid:3) = E (cid:2) (1 + | b X δu δ | p ) (cid:3) · E (cid:2) sup s ∈ [0 ,δ ] | W s | p (cid:3) ≤ c · E (cid:2) | b X δu δ | p (cid:3) . Combining (35) and (36) we conclude that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ]and all t ∈ [0 , E (cid:2) sup s ∈ [0 ,t ] | b X δs | p (cid:3) ≤ c + c · Z t E (cid:2) | b X δu δ | p (cid:3) du. Employing (34) we therefore obtain that for all δ ∈ (0 , δ ],(38) E (cid:2) sup s ∈ [0 , | b X δs | p (cid:3) < ∞ . Moreover, by (37), for all δ ∈ (0 , δ ] and all t ∈ [0 , E (cid:2) sup s ∈ [0 ,t ] | b X δs | p (cid:3) ≤ c + c · Z t E (cid:2) sup u ∈ [0 ,s ] | b X δu | p (cid:3) ds. Applying the Gronwall inequality completes the proof of (30).
N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 13
For the proof of (31) observe that for all δ ∈ (0 , δ ], all ∆ ∈ [0 ,
1] and all t ∈ [0 , − ∆], E (cid:2) sup s ∈ [ t,t +∆] | b X δs − b X δt | p (cid:3) ≤ p · E h(cid:12)(cid:12)(cid:12)Z t +∆ t | µ ( b X δu δ ) | du (cid:12)(cid:12)(cid:12) p i + 2 p · E h sup s ∈ [ t,t +∆] (cid:12)(cid:12)(cid:12)Z st (cid:0) σ ( b X δu δ ) + σ · d σ ( b X δu δ ) · ( W u − W u δ ) (cid:1) dW u (cid:12)(cid:12)(cid:12) p i and employ the H¨older inequality, the Burkholder-Davis-Gundy inequality, (26), (27), (36) and(i). (cid:3) Occupation time estimates for the adaptive quasi-Milstein scheme.
We first pro-vide an estimate for the expected value of occupation time functionals of b X δ . Lemma 6.
Let f : [0 , ∞ ) → [0 , ∞ ) be B ([0 , ∞ )) / B ([0 , ∞ )) -measurable and let γ > . Then thereexists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all ε ∈ (0 , ε ] , E hZ f ( d ( b X δt , Θ)) · Θ ε ( b X δt ) dt i ≤ c · Z ε f ( x ) dx + c · sup x ∈ [0 ,ε ] f ( x ) · ( ε − γ + δ − γ (cid:1) . Proof.
Clearly, it is enought to show that for all i ∈ { , . . . , k } there exists c ∈ (0 , ∞ ) such thatfor all δ ∈ (0 , δ ] and all ε ∈ (0 , ε ],(39) E hZ f ( | b X δt − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( b X δt ) dt i ≤ c · Z ε f ( x ) dx + c · sup x ∈ [0 ,ε ] f ( x ) · ( ε − γ + δ − γ (cid:1) . In the following fix i ∈ { , . . . , k } .Let δ ∈ (0 , δ ]. For t ∈ [0 ,
1] putΣ δt = σ ( b X δt δ ) + σd σ ( b X δt δ ) · ( W t − W t δ ) . Using (26), (27), (29) and Lemma 5 we conclude that b X δ is a continuous semi-martingale withquadratic variation(40) h b X δ i t = Z t (Σ δs ) ds, t ∈ [0 , . For a ∈ R let L a ( b X δ ) = ( L at ( b X δ )) t ∈ [0 , denote the local time of b X δ at the point a . Thus, for all a ∈ R and all t ∈ [0 , | b X δt − a | = | x − a | + Z t sgn( b X δs − a ) · µ ( b X δs δ ) ds + Z t sgn( b X δs − a ) · Σ δs dW s + L at ( b X δ ) , where sgn( y ) = 1 (0 , ∞ ) ( y ) − ( −∞ , ( y ) for y ∈ R , see, e.g. [28, Chap. VI]. Hence, for all a ∈ R and all t ∈ [0 , L at ( b X δ ) ≤ | b X δt − x | + Z t | µ ( b X δs δ ) | ds + (cid:12)(cid:12)(cid:12)Z t sgn( b X δs − a ) · Σ δs dW s (cid:12)(cid:12)(cid:12) ≤ Z t | µ ( b X δs δ ) | ds + (cid:12)(cid:12)(cid:12)Z t Σ δs dW s (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)Z t sgn( b X δs − a ) · Σ δs dW s (cid:12)(cid:12)(cid:12) . Using (26), (41), the H¨older inequality, the Burkholder-Davis-Gundy inequality and Lemma 5we obtain that there exist c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ], all a ∈ R and all t ∈ [0 , E (cid:2) L at ( b X δ ) (cid:3) ≤ c · Z (cid:0) E (cid:2) | b X δs δ | (cid:3)(cid:1) ds + c (cid:16)Z E (cid:2) (Σ δs ) (cid:3) ds (cid:17) / ≤ c + c (cid:16)Z E (cid:2) (Σ δs ) (cid:3) ds (cid:17) / . Moreover, by (26), (27), Lemma 4 and Lemma 5 there exist c , c ∈ (0 , ∞ ) such that for all s ∈ [0 ,
1] and all δ ∈ (0 , δ ],(43) E (cid:2) (Σ δs ) (cid:3) ≤ c · E (cid:2) (1 + | b X δs δ | ) · (1 + | W s − W s δ | ) (cid:3) ≤ c · E (cid:2) (1 + | b X δs δ | ) ] · E [(1 + sup u ∈ [0 ,δ ] | W s δ u | ) (cid:3) ≤ c . Combining (42) and (43) we obtain that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ], all a ∈ R and all t ∈ [0 , E (cid:2) L at ( b X δ ) (cid:3) ≤ c. Using (40), (44) and the occupation time formula it follows that there exists c ∈ (0 , ∞ ) suchthat for all δ ∈ (0 , δ ] and all ε ∈ (0 , ε ],(45) E (cid:20)Z f ( | b X δt − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( b X δt ) · (Σ δt ) dt (cid:21) = Z R f ( | a − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( a ) · E (cid:2) L at ( b X δ ) (cid:3) da ≤ c · Z ε f ( x ) dx. By (26), (27) and the Lipschitz continuity of σ we obtain that there exist c , c ∈ (0 , ∞ ) suchthat for all δ ∈ (0 , δ ] and all t ∈ [0 , (cid:12)(cid:12) σ ( b X δt ) − (Σ δt ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) σ ( b X δt ) − Σ δt (cid:12)(cid:12) · (cid:0) | σ ( b X δt ) | + | Σ δt | (cid:1) ≤ c · (cid:0) | σ ( b X δt ) − σ ( b X δt δ ) | + | σδ σ ( b X δt δ ) | · | W t − W t δ | (cid:1) · (cid:0) | b X δt | + (1 + | b X δt δ | ) · (1 + | W t − W t δ | ) (cid:1) ≤ c · (cid:0) | b X δt − b X δt δ | + (1 + | b X δt δ | ) · sup u ∈ [0 ,δ ] | W t δ u | (cid:1) · (1 + sup s ∈ [0 , | b X δs | ) · (1 + sup u ∈ [0 ,δ ] | W t δ u | ) . Thus, using the H¨older inequality, Lemma 5 and Lemma 4 we conclude that for all q ∈ [1 , ∞ )there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all t ∈ [0 , E (cid:2) | σ ( b X δt ) − (Σ δt ) | q (cid:3) /q ≤ c · √ δ. Since σ is continuous and σ ( ξ i ) = 0 there exist κ i , ρ i ∈ (0 , ∞ ) such that(47) inf x ∈ R : | x − ξ i |≤ ρ i σ ( x ) ≥ κ i . N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 15
Using (45), (46), (47) and the H¨older inequality we obtain that for all q ∈ (1 , ∞ ) there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all ε ∈ (0 , ρ i ∧ ε ],(48) E hZ f ( | b X δt − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( b X δt ) dt i ≤ κ · E hZ f ( | b X δt − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( b X δt ) · σ ( b X δt ) dt i ≤ κ · E hZ f ( | b X δt − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( b X δt ) · (Σ δt ) dt i + 1 κ · E hZ f ( | b X δt − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( b X δt ) · (cid:12)(cid:12) σ ( b X δt ) − (Σ δt ) (cid:12)(cid:12) dt i ≤ c · Z ε f ( x ) dx + c · sup x ∈ [0 ,ε ] f ( x ) · √ δ · Z (cid:0) P ( | b X δt − ξ i | ≤ ε ) (cid:1) /q dt ≤ c · Z ε f ( x ) dx + c · sup x ∈ [0 ,ε ] f ( x ) · √ δ · (cid:16)Z P ( | b X δt − ξ i | ≤ ε ) dt (cid:17) /q . Note that in the case of f = 1 the estimate (48) yields that for all q ∈ (1 , ∞ ) there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all ε ∈ (0 , ρ i ∧ ε ], Z P ( | b X δt − ξ i | ≤ ε ) dt ≤ c · ε + c · √ δ · (cid:16)Z P ( | b X δt − ξ i | ≤ ε ) dt (cid:17) /q . Thus, observing that ε ∈ (0 ,
1] and δ ∈ (0 ,
1) and using the Young inequality we obtain thatfor all q ∈ (1 ,
2] there exist c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all ε ∈ (0 , ρ i ∧ ε ],(49) Z P ( | b X δt − ξ i | ≤ ε ) dt ≤ c · ε + c · √ δ · (cid:16) c · ε + c · √ δ (cid:17) /q ≤ c · ε + c · δ + q . It follows from (48) and (49) that for all q ∈ (1 ,
2] there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all ε ∈ (0 , ρ i ∧ ε ], E hZ f ( | b X δt − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( b X δt ) dt i ≤ c · Z ε f ( x ) dx + c · sup x ∈ [0 ,ε ] f ( x ) · √ δ · (cid:0) ε q + δ q + q (cid:1) . By the Young inequality, for all q ∈ (1 , δ ∈ (0 , δ ] and all ε ∈ (0 , ρ i ∧ ε ], √ δ · ε q ≤ δ / + 23 ε q . Combining the latter two estimates we conclude that for all q ∈ (1 ,
2] there exists c ∈ (0 , ∞ )such that for all δ ∈ (0 , δ ] and all ε ∈ (0 , ρ i ∧ ε ], E hZ f ( | b X δt − ξ i | ) · [ ξ i − ε,ξ i + ε ] ( b X δt ) dt i ≤ c · Z ε f ( x ) dx + c · sup x ∈ [0 ,ε ] f ( x ) · (cid:0) ε q + δ + q + q (cid:1) . This yields (39) and completes the proof of the lemma. (cid:3)
The following lemma provides upper bounds for the probabilities that increments of theadaptive quasi-Milstein scheme are large compared to the actual distance of the scheme fromthe set Θ.
Lemma 7.
Let α ∈ (0 , ∞ ) and q ∈ [1 , ∞ ) . Then there exist c , c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all t ∈ [0 , , (i) P ( | b X δt δ − b X δt | ≥ α · ε δ , b X δt δ ∈ Θ ε δ ) ≤ c · δ q , (ii) P ( | b X δt − b X δt δ | ≥ α · d ( b X δt δ , Θ) , b X δt δ ∈ Θ ε δ \ Θ ε δ ) ≤ c · δ q , (iii) P ( | b X δt − b X δt δ | ≥ α · ε δ , b X δt δ ∈ Θ ε \ Θ ε δ ) ≤ c · δ q .Proof. Define Φ : R × C ([0 , ∞ ); R ) → C ([0 , ∞ ); R ) byΦ( y, u )( t ) = y + µ ( y ) · t + σ ( y ) · u ( t ) + 12 σd σ ( y ) · ( u ( t ) − t )for y ∈ R , u ∈ C ([0 , ∞ ); R ) and t ∈ [0 , ∞ ) and observe that there exists κ ∈ (0 , ∞ ) such thatfor all y ∈ Θ ε , all u ∈ C ([0 , ∞ ); R ) and all t ∈ [0 , ∞ ),(50) | Φ( y, u )( t ) − y | ≤ κ · ( t + | u ( t ) | + u ( t )) . We first proof (i). Using Lemma 4 we obtain that for all δ ∈ (0 , δ ] and all t ∈ [0 , P ( | b X δt δ − b X δt | ≥ α · ε δ , b X δt δ ∈ Θ ε δ )= P ( | Φ( b X δt δ , W t δ )( t − t δ ) − b X δt δ | ≥ α · ε δ , b X δt δ ∈ Θ ε δ ) ≤ P ( sup s ∈ [0 ,h δ ( b X δtδ )] | Φ( b X δt δ , W t δ )( s ) − b X δt δ | ≥ α · ε δ , b X δt δ ∈ Θ ε δ )= Z Θ εδ P ( sup s ∈ [0 ,h δ ( y )] | Φ( y, W )( s ) − y | ≥ α · ε δ ) P b X δtδ ( dy ) . By (50), for all δ ∈ (0 , δ ] and all y ∈ Θ ε δ ,(52) P ( sup s ∈ [0 ,h δ ( y )] | Φ( y, W )( s ) − y | ≥ α · ε δ ) ≤ P (cid:0) h δ ( y ) + sup s ∈ [0 ,h δ ( y )] | W s | + sup s ∈ [0 ,h δ ( y )] W s ≥ αε δ κ (cid:1) ≤ P (cid:0) h δ ( y ) + sup s ∈ [0 ,h δ ( y )] | W s | ≥ αε δ κ (cid:1) + P (cid:0) sup s ∈ [0 ,h δ ( y )] W s ≥ αε δ κ (cid:1) = P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ αε δ κ − h δ ( y ) (cid:1) + P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ √ αε δ √ κ (cid:1) . Recall that for all δ ∈ (0 , δ ] and all y ∈ Θ ε δ we have h δ ( y ) = δ log (1 /δ ). Moreover, by [23,Lemma 3.4], there exists c ∈ (0 , ∞ ) such that for all u ∈ (0 , ∞ ) and all x ∈ R ,(53) P ( sup s ∈ [0 ,u ] | W s | ≥ x ) ≤ c · e − x √ u . N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 17
Hence, there exist c , c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all y ∈ Θ ε δ , P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ αε δ κ − h δ ( y ) (cid:1) ≤ c · e − α κ log (1 /δ )+ δ log (1 /δ ) ≤ c · δ q as well as P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ √ αε δ √ κ (cid:1) ≤ c · e − √ α √ κδ ≤ c · δ q . The latter two estimates together with (51) and (52) imply (i).We next proof (ii). Proceeding similarly to (51) and (52) we obtain that for all δ ∈ (0 , δ ] andall t ∈ [0 , P ( | b X δt − b X δt δ | ≥ α · d ( b X δt δ , Θ) , b X δt δ ∈ Θ ε δ \ Θ ε δ ) ≤ Z Θ εδ \ Θ εδ P ( sup s ∈ [0 ,h δ ( y )] | Φ( y, W )( s ) − y | ≥ α · d ( y, Θ)) P b X δtδ ( dy )and for all δ ∈ (0 , δ ] and all y ∈ Θ ε δ \ Θ ε δ ,(55) P ( sup s ∈ [0 ,h δ ( y )] | Φ( y, W )( s ) − y | ≥ α · d ( y, Θ)) ≤ P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ α κ d ( y, Θ) − h δ ( y ) (cid:1) + P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ √ α √ κ p d ( y, Θ) (cid:1) . Recall that for all δ ∈ (0 , δ ] and all y ∈ Θ ε δ \ Θ ε δ we have h δ ( y ) = (cid:16) d ( y, Θ)log (1 /δ ) (cid:17) . Hence, applying(53) we obtain that there exist c , c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all y ∈ Θ ε δ \ Θ ε δ , P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ α κ d ( y, Θ) − h δ ( y ) (cid:1) ≤ c · e − α κ log (1 /δ )+ √ h δ ( y ) ≤ c · δ q as well as P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ √ α √ κ p d ( y, Θ) (cid:1) ≤ c · e − √ α √ κ · log (1 /δ ) √ d ( y, Θ) ≤ c · e − √ α √ κ · log(1 /δ ) δ / ≤ c · δ q . The latter two estimates together with (54) and (55) yield (ii).We finally prove (iii). Proceeding similarly to (51) and (52) we obtain that for all δ ∈ (0 , δ ]and all t ∈ [0 , P ( | b X δt − b X δt δ | ≥ α · ε δ , b X δt δ ∈ Θ ε \ Θ ε δ ) ≤ Z Θ ε \ Θ εδ P ( sup s ∈ [0 ,h δ ( y )] | Φ( y, W )( s ) − y | ≥ α · ε δ ) P b X δtδ ( dy )and for all δ ∈ (0 , δ ] and all y ∈ Θ ε \ Θ ε δ ,(57) P ( sup s ∈ [0 ,h δ ( y )] | Φ( y, W )( s ) − y | ≥ α · ε δ ) ≤ P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ αε δ κ − h δ ( y ) (cid:1) + P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ √ αε δ √ κ (cid:1) . Recall that for all δ ∈ (0 , δ ] and all y ∈ Θ ε \ Θ ε δ we have h δ ( y ) = δ . Applying (53) we thereforeobtain that there exist c , c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all y ∈ Θ ε \ Θ ε δ , P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ αε δ κ − h δ ( y ) (cid:1) ≤ c · e − α κ log (1 /δ )+ √ δ ≤ c · δ q as well as P (cid:0) sup s ∈ [0 ,h δ ( y )] | W s | ≥ √ αε δ √ κ (cid:1) ≤ c · e − √ α √ κ · log(1 /δ ) δ / ≤ c · δ q . The latter two estimates together with (56) and (57) imply (iii) and complete the proof of thelemma. (cid:3)
Next, put(58) S = (cid:16) k +1 [ ℓ =1 ( ξ ℓ − , ξ ℓ ) (cid:17) c and note that S = ∪ kℓ =1 { ( x, y ) ∈ R : ( x − ξ ℓ ) · ( y − ξ ℓ ) ≤ } . We are ready to establisch the mainresult in this section, which provides a p -th mean estimate of the time average of | b X δt − b X δt δ | subject to the condition that the pair ( b X δt , b X δt δ ) lies in the set S . Proposition 1.
Let p ∈ [1 , ∞ ) . Then there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] , (59) E h(cid:12)(cid:12)(cid:12)Z | b X δt − b X δt δ | · S ( b X δt , b X δt δ ) dt (cid:12)(cid:12)(cid:12) p i /p ≤ c · δ . Proof.
For δ ∈ (0 , δ ] and i ∈ { , , , } let E δi = E hZ | b X δt − b X δt δ | p · S ( b X δt , b X δt δ ) · O δi ( b X δt δ ) dt i , where O δ = (Θ ε ) c , O δ = Θ ε \ Θ ε δ , O δ = Θ ε δ \ Θ ε δ , O δ = Θ ε δ . Then for all δ ∈ (0 , δ ],(60) E h(cid:12)(cid:12)(cid:12)Z | b X δt − b X δt δ | · S ( b X δt , b X δt δ ) dt (cid:12)(cid:12)(cid:12) p i ≤ E hZ | b X δt − b X δt δ | p · S ( b X δt , b X δt δ ) dt i ≤ X i =1 E δi . Below we show that for all i ∈ { , , , } there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(61) E δi ≤ c · δ p . Clearly, (60) and (61) imply (59).It remains to prove (61). We start with the analysis of E δ . For all δ ∈ (0 , δ ] and all t ∈ [0 , { ( b X δt , b X δt δ ) ∈ S } ∩ { b X δt δ ∈ O δ } ⊆ {| b X δt − b X δt δ | ≥ ε } . N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 19
Thus, using the Markov inequality and Lemma 5 we obtain that there exist c , c ∈ (0 , ∞ ) suchthat for all δ ∈ (0 , δ ], E δ ≤ Z E (cid:2) | b X δt − b X δt δ | p · {| b X δt − b X δtδ |≥ ε } (cid:3) dt ≤ Z E (cid:2) | b X δt − b X δt δ | p (cid:3) / · ( P ( | b X δt − b X δt δ | ≥ ε )) / dt ≤ ε p Z E (cid:2) | b X δt − b X δt δ | p (cid:3) dt ≤ c ε p Z (cid:0) E (cid:2) | b X δt − b X δ ∨ ( t − δ ) | p (cid:3) + E (cid:2) | b X δt δ − b X δ ∨ ( t − δ ) | p (cid:3)(cid:1) dt ≤ c ε p Z E (cid:2) sup s ∈ [0 ∨ ( t − δ ) ,t ] | b X δs − b X δ ∨ ( t − δ ) | p (cid:3) dt ≤ c · δ p , which shows that (61) holds for i = 1.We next estimate E δ . Using Lemma 5 we obtain that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(62) E δ ≤ Z E (cid:2) | b X δt − b X δt δ | p (cid:3) / · ( P (( b X δt , b X δt δ ) ∈ S, b X δt δ ∈ O δ )) / dt ≤ c · δ p · Z ( P (( b X δt , b X δt δ ) ∈ S, b X δt δ ∈ Θ ε \ Θ ε δ )) / dt. Moreover, using Lemma 7(iii) with α = 1 and q = 2 p we conclude that there exists c ∈ (0 , ∞ )such that for all δ ∈ (0 , δ ] and all t ∈ [0 , P (( b X δt , b X δt δ ) ∈ S, b X δt δ ∈ Θ ε \ Θ ε δ ) ≤ P ( | b X δt − b X δt δ | ≥ ε δ , b X δt δ ∈ Θ ε \ Θ ε δ ) ≤ c · δ p . The latter estimate together with (62) yields (61) for i = 2.We next estimate E δ . Similarly to (62) we obtain that there exists c ∈ (0 , ∞ ) such that forall δ ∈ (0 , δ ],(63) E δ ≤ c · δ p · Z ( P (( b X δt , b X δt δ ) ∈ S, b X δt δ ∈ Θ ε δ \ Θ ε δ )) / dt. Moreover, using Lemma 7(ii) with α = 1 and q = 2 p we conclude that there exists c ∈ (0 , ∞ )such that for all δ ∈ (0 , δ ] and all t ∈ [0 , P (( b X δt , b X δt δ ) ∈ S, b X δt δ ∈ Θ ε δ \ Θ ε δ ) ≤ P ( | b X δt − b X δt δ | ≥ d ( b X δt δ , Θ) , b X δt δ ∈ Θ ε δ \ Θ ε δ ) ≤ c · δ p . The latter estimate together with (63) yields (61) for i = 3.We finally extimate E δ . Note that for all δ ∈ (0 , δ ], all t ∈ [0 ,
1] and all ω ∈ { b X δt δ ∈ Θ ε δ } wehave t − t δ ( ω ) ≤ δ · log (1 /δ ). Thus, using Lemma 5 we obtain that there exist c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(64) E δ ≤ Z E (cid:2) | b X δt − b X δt δ | p · Θ εδ ( b X δt δ ) (cid:3) dt ≤ Z E (cid:2) | b X δt − b X δt δ | p · Θ εδ ( b X δt δ ) (cid:3) / · ( P ( b X δt δ ∈ Θ ε δ )) / dt ≤ c Z E (cid:2) sup s ∈ [0 ∨ ( t − δ · log (1 /δ )) ,t ] | b X δs − b X δ ∨ ( t − δ · log (1 /δ )) | p (cid:3) / · ( P ( b X δt δ ∈ Θ ε δ )) / dt ≤ c · δ p · log p (1 /δ ) · Z ( P ( b X δt δ ∈ Θ ε δ )) / dt ≤ c · δ p · log p (1 /δ ) · (cid:16)Z P ( b X δt δ ∈ Θ ε δ ) dt (cid:17) / . Employing Lemma 6 with f = 1 and γ = 1 / α = 1 and q = 1 we obtainthat there exist c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(65) Z P ( b X δt δ ∈ Θ ε δ ) dt = Z P ( | b X δt − b X δt δ | < ε δ , b X δt δ ∈ Θ ε δ ) dt + Z P ( | b X δt − b X δt δ | ≥ ε δ , b X δt δ ∈ Θ ε δ ) dt ≤ Z P ( b X δt ∈ Θ ε δ ) dt + Z P ( | b X δt − b X δt δ | ≥ ε δ , b X δt δ ∈ Θ ε δ ) dt ≤ c · ε δ + c · δ ≤ c · δ · log (1 /δ ) . The latter estimate together with (64) implies that there exist c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ], E δ ≤ c · δ p +1 / · log p +2 (1 /δ ) ≤ c · δ p , which shows that (61) holds for i = 4 and completes the proof of the proposition. (cid:3) Convergence analysis.
In this subsection we proof the estimate (10). Clearly, it is enoughto consider the case p ∈ N \ { } . For δ ∈ (0 , δ ] and t ∈ [0 ,
1] we put A t = Z t µ ( X s ) ds, b A δt = Z t µ ( b X δs δ ) ds and B t = Z t σ ( X s ) dW s , b B t = Z t (cid:0) σ ( b X δs δ ) + σd σ ( b X δs δ ) · ( W s − W s δ ) (cid:1) dW s as well as U δt = Z t σd µ ( b X δs δ ) · ( W s − W s δ ) ds and we use the decomposition(66) X t − b X δt = ( A t − b A δt − U δt ) + ( B t − b B δt ) + U δt . N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 21
Recall the definition (58) of the set S . For all δ ∈ (0 , δ ], all s ∈ [0 ,
1] and all f ∈ { µ, σ } wehave | f ( X s ) − f ( b X δs δ ) − σd f ( b X δs δ ) · ( W s − W s δ ) |≤ | f ( X s ) − f ( b X δs ) | + | f ( b X δs ) − f ( b X δs δ ) − d f ( b X δs δ ) · ( b X δs − b X δs δ ) | · S c ( b X δs , b X δs δ )+ | f ( b X δs ) − f ( b X δs δ ) − d f ( b X δs δ ) · ( b X δs − b X δs δ ) | · S ( b X δs , b X δs δ )+ (cid:12)(cid:12) d f ( b X δs δ ) · ( µ ( b X δs δ )( s − s δ ) + σd σ ( b X δs δ ) · (( W s − W s δ ) − ( s − s δ ))) (cid:12)(cid:12) . Using the Lipschitz continuity of µ and σ as well as (26), (27) and (28) we thus obtain thatthere exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ], all s ∈ [0 ,
1] and all f ∈ { µ, σ } ,(67) | f ( X s ) − f ( b X δs δ ) − σd f ( b X δs δ ) · ( W s − W s δ ) |≤ c · | X s − b X δs | + c · | b X δs − b X δs δ | + c · | b X δs − b X δs δ | · S ( b X δs , b X δs δ )+ c · (1 + | b X δs δ | ) · ( δ + | W s − W s δ | ) . Employing (67), Lemma 5 and Proposition 1 we conclude that there exist c , c , c ∈ (0 , ∞ )such that for all δ ∈ (0 , δ ] and all t ∈ [0 , E h sup ≤ s ≤ t | A s − b A δs − U δs | p i ≤ E h(cid:12)(cid:12)(cid:12)Z t | µ ( X s ) − µ ( b X δs δ ) − σd µ ( b X δs δ ) · ( W s − W s δ ) | ds (cid:12)(cid:12)(cid:12) p i ≤ c · Z t E (cid:2) | X s − b X δs | p (cid:3) ds + c · Z t E (cid:2) | b X δs − b X δs δ | p (cid:3) ds + c · E h(cid:12)(cid:12)(cid:12)Z t | b X δs − b X δs δ | · S ( b X δs , b X δs δ ) ds (cid:12)(cid:12)(cid:12) p i + c · Z t E (cid:2) u ∈ [0 , | b X δu | p (cid:3) / · E (cid:2) δ p + sup u ∈ [0 ∨ ( s − δ ) ,s ] | W s − W u | p (cid:3) / ds ≤ c · Z t E (cid:2) | X s − b X δs | p (cid:3) ds + c · E h(cid:12)(cid:12)(cid:12)Z t | b X δs − b X δs δ | · S ( b X δs , b X δs δ ) ds (cid:12)(cid:12)(cid:12) p/ i + c · Z t E (cid:2) sup u ∈ [0 ∨ ( s − δ ) ,s ] | b X δu − b X δ ∨ ( s − δ ) | p (cid:3) ds + c · δ p ≤ c · Z t E (cid:2) | X s − b X δs | p (cid:3) ds + c · δ p . Using the Burkholder-Davis-Gundy inequality, (67), Lemma 5 and Proposition 1 we obtainthat there exist c , c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all t ∈ [0 , E h sup ≤ s ≤ t | B s − b B δs | p i ≤ c · E h(cid:12)(cid:12)(cid:12)Z t | σ ( X s ) − σ ( b X δs δ ) − σd σ ( b X δs δ ) · ( W s − W s δ ) | ds (cid:12)(cid:12)(cid:12) p/ i ≤ c · Z t E (cid:2) | X s − b X δs | p (cid:3) ds + c · Z t E (cid:2) | b X δs − b X δs δ | p (cid:3) ds + c · E h(cid:12)(cid:12)(cid:12)Z t | b X δs − b X δs δ | · S ( b X δs , b X δs δ ) ds (cid:12)(cid:12)(cid:12) p/ i + c · Z t E (cid:2) u ∈ [0 , | b X δu | p (cid:3) / · E (cid:2) δ p + sup u ∈ [0 ∨ ( s − δ ) ,s ] | W s − W u | p (cid:3) / ds ≤ c · Z t E (cid:2) | X s − b X δs | p (cid:3) ds + c · δ p . Combining (66) with (68) and (69) we conclude that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ] and all t ∈ [0 , E (cid:2) sup ≤ s ≤ t | X t − b X δt | p (cid:3) ≤ c · Z t E (cid:2) sup ≤ u ≤ s | X u − b X δu | p (cid:3) ds + c · δ p + E (cid:2) sup ≤ s ≤ t | U δs | p (cid:3) . Note that E (cid:2) sup ≤ u ≤ | X u − b X δu | p (cid:3) < ∞ due to (4) and Lemma 5. Below we show that thereexists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(71) E (cid:2) sup ≤ s ≤ | U δs | p (cid:3) ≤ c · δ p . Inserting (71) into (70) and applying the Gronwall inequality then yields the error estimate (10)in Theorem 1.We turn to the proof of (71). Clearly, for all δ ∈ (0 , δ ], all i ∈ N and all s ∈ [ τ δi , τ δi +1 ],(72) U δs ∧ = U δτ δi ∧ + σd µ ( b X δτ δi ∧ ) · Z s ∧ τ δi ∧ ( W u − W τ δi ) du. For δ ∈ (0 , δ ] let n δ be given by (33). Using (26) and (27) we obtain from (72) that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(73) sup ≤ s ≤ | U δs | = max i =0 ,...,n δ − sup τ δi ≤ s ≤ τ δi +1 | U δs ∧ |≤ max i =0 ,...,n δ − | U δτ δi ∧ | + max i =0 ,...,n δ − | σd µ ( b X δτ δi ∧ ) | · Z τ δi +1 ∧ τ δi ∧ | W u − W τ δi | du ≤ max i =0 ,...,n δ − | U δτ δi ∧ | + c · (1 + sup ≤ s ≤ | b X δs | ) · max i =0 ,...,n δ − Z τ δi +1 ∧ τ δi ∧ | W u − W τ δi | du. N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 23
Let δ ∈ (0 , δ ]. Employing (72), (26), (27) and Lemma 5 one can show by induction on i ∈ { , . . . , n δ − } that E (cid:2) | U δτ δi ∧ (cid:12)(cid:12)(cid:3) < ∞ for all i ∈ { , . . . , n δ − } . Moreover, using Lemma3(vi),(v) one can show by induction on i ∈ { , . . . , n δ − } that U δτ δi ∧ is F τ δi ∧ / B ( R )-measurablefor all i ∈ { , . . . , n δ − } . Finally, observe that for all i ∈ { , . . . , n δ − } , Z τ δi +1 ∧ τ δi ∧ ( W u − W τ δi ) du = Z ( τ δi +1 ∧ − ( τ δi ∧ W τ δi ∧ u du. Using Lemma 3(vi),(v),(vi) we therefore obtain that for all i ∈ { , . . . , n δ − } , E (cid:2) U δτ δi +1 ∧ |F τ δi ∧ (cid:3) = U δτ δi ∧ + σd µ ( b X δτ δi ∧ ) · Z ( τ δi +1 ∧ − ( τ δi ∧ E (cid:2) W τ δi ∧ u |F τ δi ∧ (cid:3) du = U δτ δi ∧ . Hence, the sequence ( U δτ δi ∧ , F τ δi ∧ ) i ∈{ ,...,n δ − } is a martingale.Employing the Burkholder-Davis-Gundy inequality as well as (26), (27), Lemma 5 and Lemma4 we conclude that there exist c , c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(74) E (cid:2) max i =0 ,...,n δ − | U δτ δi ∧ | p i ≤ E h(cid:16) n δ − X i =0 (cid:16) σd µ ( b X δτ δi ∧ ) · Z τ δi +1 ∧ τ δi ∧ ( W u − W τ δi ) du (cid:17) (cid:17) p/ i ≤ c · E (cid:2) (1 + sup ≤ s ≤ | b X δs | p ) (cid:3) / · E h(cid:16) n δ − X i =0 (cid:16)Z τ δi +1 ∧ τ δi ∧ ( W u − W τ δi ) du (cid:17) (cid:17) p i / ≤ c · E h(cid:16) n δ − X i =0 (( τ δi +1 ∧ − ( τ δi ∧ · Z τ δi +1 ∧ τ δi ∧ ( W u − W τ δi ) du (cid:17) p i / ≤ c · δ p/ · E h(cid:16)Z ( W u − W u δ ) du (cid:17) p i / ≤ c · δ p/ · (cid:16)Z E (cid:2) sup s ∈ [0 ,δ ] | W u δ s | p (cid:3) du (cid:17) / ≤ c · δ p . Furthermore, using Lemma 5 and Lemma 4 we obtain that there exists c ∈ (0 , ∞ ) such thatfor all δ ∈ (0 , δ ],(75) E h(cid:16) (1 + sup ≤ s ≤ | b X δs | ) · max i =0 ,...,n δ − Z τ δi +1 ∧ τ δi ∧ | W u − W τ δi | du (cid:17) p i ≤ E (cid:2) (1 + sup ≤ s ≤ | b X δs | p ) (cid:3) / · E h n δ − X i =0 (cid:16)Z τ δi +1 ∧ τ δi ∧ | W u − W τ δi | du (cid:17) p i / ≤ c · E h n δ − X i =0 (( τ δi +1 ∧ − ( τ δi ∧ p − · Z τ δi +1 ∧ τ δi ∧ | W u − W τ δi | p du i / ≤ c · δ p − · (cid:16)Z E (cid:2) sup s ∈ [0 ,δ ] | W u δ s | p (cid:3) du (cid:17) / ≤ c · δ p − ≤ c · δ p . Combining (73) with (74) and (75) yields (71) and completes the proof of the estimate (10) inTheorem 1.5.5.
Cost analysis.
In this subsection we proof the estimate (11). Clearly, for all δ ∈ (0 , δ ]and all i ∈ N we have 1 = Z τ δi τ δi − τ δi − τ δi − dt = Z τ δi τ δi − h δ ( b X δt δ ) dt. Thus, for all δ ∈ (0 , δ ], N ( b X δ ) = 1 + ∞ X i =1 { τ δi < } = 1 + ∞ X i =1 { τ δi < } · Z τ δi τ δi − h δ ( b X δt δ ) dt ≤ Z h δ ( b X δt δ ) dt. For δ ∈ (0 , δ ] and i ∈ { , , } put I δi = E hZ h δ ( b X δt δ ) · O δi ( b X δt δ ) dt i , where O δ = (Θ ε δ ) c , O δ = Θ ε δ \ Θ ε δ , O δ = Θ ε δ . Then for all δ ∈ (0 , δ ],(76) E [ N ( b X δ )] ≤ X i =1 I δi . Clearly,(77) I δ = δ − · Z P (cid:0) b X δt δ ∈ (Θ ε δ ) c (cid:1) dt ≤ δ − . Moreover, observing (65) we obtain that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(78) I δ = δ − · log − (1 /δ ) · Z P (cid:0) b X δt δ ∈ Θ ε δ (cid:1) dt ≤ c · δ − . N ADAPTIVE METHOD OF ORDER 1 FOR SDES WITH DISCONTINUOUS DRIFT COEFFICIENT 25
Below we show that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(79) I δ ≤ c · δ − . Combining (76) to (79) we obtain (11).It remains to prove (79). For δ ∈ (0 , δ ] and t ∈ [0 ,
1] put D δt = (cid:8) | b X δt − b X δt δ | ≤ d ( b X δt δ , Θ) (cid:9) . Clearly, for all δ ∈ (0 , δ ],(80) I δ = I δ , + I δ , , where I δ , = E hZ h δ ( b X δt δ ) · O δ ( b X δt δ ) · D δt dt i , I δ , = E hZ h δ ( b X δt δ ) · O δ ( b X δt δ ) · ( D δt ) c dt i . Observing the fact that the distance function d ( · , Θ) : R → [0 , ∞ ) is Lipschitz continuous withLipschitz seminorm 1, i.e. for all x, y ∈ R , | d ( x, Θ) − d ( y, Θ) | ≤ | x − y | , we obtain that for all δ ∈ (0 , δ ] and all t ∈ [0 , { b X δt δ ∈ O δ } ∩ D δt ⊆ { b X δt ∈ Θ ε δ \ Θ ε δ } ∩ (cid:8) d ( b X δt δ , Θ) ≤ d ( b X δt , Θ) ≤ d ( b X δt δ , Θ) (cid:9) . Thus, for all δ ∈ (0 , δ ],(81) I δ , = log (1 /δ ) · E hZ d ( b X δt δ , Θ) · O δ ( b X δt δ ) · D δt dt i ≤
94 log (1 /δ ) · E hZ d ( b X δt , Θ) · Θ εδ \ Θ εδ ( b X δt ) dt i . For δ ∈ (0 , δ ] put ε δ = δ / · log (1 /δ ) and observe that ε δ ≤ ε δ ≤ ε δ for all δ ∈ (0 , δ ]. Hence,(81) implies that for all δ ∈ (0 , δ ], I δ , ≤
94 log (1 /δ ) · E hZ d ( b X δt , Θ) · Θ εδ \ Θ εδ ( b X δt ) dt i ++ 94 log (1 /δ ) · E hZ d ( b X δt , Θ) · Θ εδ \ Θ εδ ( b X δt ) dt i ≤
94 log (1 /δ ) · E hZ ε δ , d ( b X δt , Θ)) · Θ εδ ( b X δt ) dt i + 94 log (1 /δ ) · E hZ ε δ , d ( b X δt , Θ)) · Θ εδ ( b X δt ) dt i . Applying Lemma 6 with f = 1 / max( ε δ , · ) and γ = 1 / f = 1 / max( ε δ , · ) and γ = 1 / c , c , c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(82) I δ , ≤ c · log (1 /δ ) · (cid:16)Z ε δ ε δ , x ) dx + ( ε δ ) − · ( ε δ + δ )+ Z ε δ ε δ , x ) dx + ( ε δ ) − · (cid:0) ( ε δ ) + δ (cid:1)(cid:17) ≤ c · log (1 /δ ) · (cid:0) ( ε δ ) − + ( ε δ ) − · ε δ + ( ε δ ) − + ( ε δ ) − · ( ε δ ) (cid:1) ≤ c · δ − . Moreover, employing (9) and Lemma 7(ii) with α = 1 / q = 2 we obtain that there exists c ∈ (0 , ∞ ) such that for all δ ∈ (0 , δ ],(83) I δ , ≤ δ − · log − (1 /δ ) · Z P (cid:0) | b X δt − b X δt δ | > d ( b X δt δ , Θ) , b X δt δ ∈ Θ ε δ \ Θ ε δ (cid:1) dt ≤ c · log − (1 /δ ) . Combining (80), (82) and (83) we obtain (79). This completes the proof of the estimate (11) inTheorem 1.
Acknowledgement
I am grateful to Thomas M¨uller-Gronbach for stimulating discussions on the topic of thisarticle.
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