Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations

Transcription

1 Error bounds for monotone approxmaton schemes for parabolc Hamlton-Jacob-Bellman equatons Guy Barles, Espen R. Jakobsen To cte ths verson: Guy Barles, Espen R. Jakobsen. Error bounds for monotone approxmaton schemes for parabolc Hamlton-Jacob-Bellman equatons. Math. Comp., 2007, 76 (260), pp <hal > HAL Id: hal Submtted on 26 Jan 2006 HAL s a mult-dscplnary open access archve for the depost and dssemnaton of scentfc research documents, whether they are publshed or not. The documents may come from teachng and research nsttutons n France or abroad, or from publc or prvate research centers. L archve ouverte plurdscplnare HAL, est destnée au dépôt et à la dffuson de documents scentfques de nveau recherche, publés ou non, émanant des établssements d ensegnement et de recherche franças ou étrangers, des laboratores publcs ou prvés.

2 ERROR BOUNDS FOR MONOTONE APPROXIMATION SCHEMES FOR PARABOLIC HAMILTON-JACOBI-BELLMAN EQUATIONS GUY BARLES AND ESPEN R. JAKOBSEN Abstract. We obtan non-symmetrc upper and lower bounds on the rate of convergence of general monotone approxmaton/numercal schemes for parabolc Hamlton Jacob Bellman Equatons by ntroducng a new noton of consstency. We apply our general results to varous schemes ncludng fnte dfference schemes, splttng methods and the classcal approxmaton by pecewse constant controls. ccsd , verson 1-26 Jan Introducton In ths artcle, we are nterested n the rate of convergence of general monotone approxmaton/numercal schemes for tme-dependent Hamlton Jacob Bellman (HJB) Equatons. In order to be more specfc, the HJB Equatons we consder are wrtten n the followng form (1.1) (1.2) where wth u t + F(t, x, u, Du, D 2 u) = 0 n Q T := (0, T] R N, u(0, x) = u 0 (x) n R N, F(t, x, r, p, X) = sup {L α (t, x, r, p, X)}, α A L α (t, x, r, p, X) := tr[a α (t, x)x] b α (t, x)p c α (t, x)r f α (t, x). The coeffcents a α, b α, c α, f α and the ntal data u 0 take values respectvely n S N, the space of N N symmetrc matrces, R N, R, R, and R. Under sutable assumptons (see (A1) n Secton 2), the ntal value problem (1.1)-(1.2) has a unque, bounded, Hölder contnuous, vscosty soluton u whch s the value functon of a fnte horzon, optmal stochastc control problem. We consder approxmaton/numercal schemes for (1.1)-(1.2) wrtten n the followng abstract way (1.3) S(h, t, x, u h (t, x), [u h ] t,x ) = 0 n G + h := G h \ {t = 0}, u h (0, x) = u h,0 (x) n G 0 h := G h {t = 0}, Date: January 26, Key words and phrases. Hamlton-Jacob-Bellman Equatons, swtchng system, vscosty soluton, approxmaton schemes, fnte dfference methods, splttng methods, convergence rate, error bound. Jakobsen was supported by the Research Councl of Norway, grant no /432. 1

3 2 BARLES AND JAKOBSEN where S s, loosely speakng, a consstent, monotone and unformly contnuous approxmaton of the equaton (1.1) defned on a grd/mesh G h Q T. The approxmaton parameter h can be mult-dmensonal, e.g. h could be ( t, x), t, x denotng tme and space dscretzaton parameters, x can be tself multdmensonal. The approxmate soluton s u h : G h R, [u h ] t,x s a functon defned from u h representng, typcally, the value of u h at other ponts than (t, x). We assume that the total scheme ncludng the ntal value s well-defned on some approprate subset of the space of bounded contnuous functons on G h. The abstract notaton was ntroduced by Barles and Sougands [3] to dsplay clearly the monotoncty of the scheme. One of the man assumptons s that S s non-decreasng n u h and non-ncreasng n [u h ] t,x wth the classcal orderng of functons. The typcal approxmaton schemes we have n mnd are varous fnte dfferences numercal scheme (see e.g. Kushner and Dupus [13] and Bonnans and Zdan [5]) and control schemes based on the dynamc programmng prncple (see e.g. Camll and Falcone [6]). However, for reasons explaned below, we wll not dscuss control schemes n ths paper. The am of ths paper s to obtan estmates on the rate of the convergence of u h to u. To obtan such results, one faces the double dffculty of havng to deal wth both fully nonlnear equatons and non-smooth solutons. Snce these equatons may be also degenerate, the (vscosty) solutons are expected to be no more than Hölder contnuous n general. Despte of these dffcultes, n the 80 s, Crandall & Lons [10] provded the frst optmal rates of convergence for frst-order equatons. We refer to Sougands [27] for more general results n ths drecton. For techncal reasons, the problem turns out to be more dffcult for second-order equatons, and the queston remaned open for a long tme. The breakthrough came n 1997 and 2000 wth Krylov s papers [20, 21], and by now there exsts several papers based on and extendng hs deas, e.g. [1, 2, 11, 18, 22, 23]. The man dea of Krylov s a method named by hmself shakng the coeffcents. Combned wth a standard mollfcaton argument, t allows one to get smooth subsolutons of the equaton whch approxmate the soluton. Then classcal arguments nvolvng consstency and monotoncty of the scheme yeld a one-sded bound on the error. Ths method uses n a crucal way the convexty of the equaton n u, Du, and D 2 u. It s much more dffcult to obtan the other bound and essentally there are two man approaches. The frst one conssts of nterchangng the role of the scheme and the equaton. By applyng the above explaned deas, one gets a sequence of approprate smooth subsolutons of the scheme and concludes by consstency and the comparson prncple for the equaton. Ths dea was used n dfferent artcles, see [1, 11, 18, 20, 23]. Here, the key dffculty s to obtan a contnuous dependence result for the scheme. Even though t s now standard to prove that the solutons of the HJB Equaton wth shaken coeffcents reman close to the soluton of the orgnal equaton, such type of results are not known for numercal schemes n general. We menton here the nce paper of Krylov [23] where such knd of results are obtaned by a trcky Bernsten type of argument. However, these results along wth the correspondng error bounds, only hold for equatons and schemes wth specal structures.

4 ERROR BOUNDS 3 The second approach conssts of consderng some approxmaton of the equaton or the assocated control problem and to obtan the other bound ether by probablstc arguments (as Krylov frst dd usng pecewse constant controls, [22, 21]) or by buldng a sequence of approprate smooth supersoluton of the equaton (see [2] where, as n the present paper, approxmatons by swtchng are consdered). The frst approach leads to better error bounds than the second one but t seems to work only for very specfc schemes and wth restrctons on the equatons. The second approach yelds error bounds n the general case but at the expense of lower rates. In ths paper we use the second approach by extendng the methods ntroduced n [2]. Compared wth the varous results of Krylov, we obtan better rates n most cases, our results apply to more general schemes, and we use a smpler, purely analytcal approach. In fact our method s robust n the sense that t apples to general schemes wthout any partcular form and under rather natural assumptons. However, we menton agan that n certan stuatons the frst approach can be used to get better rates, see n partcular [23]. The results n [2] apply to statonary HJB equatons set n whole space R N. In ths paper we extend these results to ntal value problems for tme-dependent HJB equatons. The latter case s much more nterestng n vew of applcatons, and from a mathematcal pont of vew, slghtly more dffcult. However, n our opnon the most mportant dfference between the two papers lays n the formulaton of the consstency requrements and the man (abstract) results. Here we ntroduce a new (and more general) formulaton that emphaszes more the non-symmetrcal feature of the upper and lower bounds and ther proofs. It s a knd of a recpe on how to obtan error bounds n dfferent stuatons, one whch we feel s easer to apply to new problems and gves better nsght nto how the error bounds are produced. We also present several techncal mprovements and smplfcatons n the proofs and, fnally, several new applcatons, some for whch error bounds have not appeared before: Fnte dfference methods (FDMs) usng the θ-method for tme dscretzaton, semdscrete splttng methods, and approxmaton by pecewse constant controls. The results for fnte dfference approxmatons can be compared wth the ones obtaned by Krylov n [21, 22]. As n [2], we get the rate 1/5 for monotone FDMs whle the correspondng result n [22] s 1/21. Of course, n specal stuatons the rate can be mproved to 1/2 whch s the optmal rate under our assumptons. We refer to [23] for the most general results n that drecton, and to [12] for the optmalty of the rate 1/2. The results for semdscrete splttng methods are new, whle the ones for the control approxmaton we get 1/10 whch s worse than 1/6 obtaned by Krylov n [22]. It would be nterestng to understand why Krylov s dong better than us here but not n the other cases. We conclude ths ntroducton by explanng the notatons we wll use throughout ths paper. By we mean the standard Eucldean norm n any R p type space (ncludng the space of N P matrces). In partcular, f X S N, then X 2 = tr(xx T ) where X T denotes the transpose of X. If w s a bounded functon from some set Q Q nto ether R, R M, or the space of N P matrces, we set w 0 = sup w(t, y). (t,y) Q

5 4 BARLES AND JAKOBSEN Furthermore, for δ (0, 1], we set [w] δ = sup (t,x) (s,y) w(t, x) w(s, y) ( x y + t s 1/2 ) δ and w δ = w 0 + [w] δ. Let C b (Q ) and C 0,δ (Q ), δ (0, 1], denote respectvely the space of bounded contnuous functons on Q and the subset of C b (Q ) n whch the norm δ s fnte. Note n partcular the choces Q = Q T and Q = R N. In the followng we always suppress the doman Q when wrtng norms. We denote by the component by component orderng n R M and the orderng n the sense of postve sem-defnte matrces n S N. For the rest of ths paper we let ρ denotes the same, fxed, postve smooth functon wth support n {0 < t < 1} { x < 1} and mass 1. From ths functon ρ, we defne the sequence of mollfers {ρ ε } ε>0 as follows, ρ ε (t, x) = 1 ( t ε N+2ρ ε 2, x ) n Q ε. The rest of ths paper s organzed as follows: In the next secton we present results on the so-called swtchng approxmaton for the problem (1.1)-(1.2). As n [2], these results are crucal to obtan the general results on the rate of convergence of approxmaton/numercal schemes and are of an ndependent nterest. Secton 3 s devoted to state and prove the man result on the rate of convergence. Fnally we present varous applcatons to classcal fnte dfference schemes, splttng method and on the classcal approxmaton by pecewse constant controls. 2. Convergence Rate for a Swtchng System In ths secton, we obtan the rate of convergence for a certan swtchng system approxmatons to the HJB equaton (1.1). Such approxmatons have be studed n [14, 7], and a vscosty solutons theory of swtchng systems can be found n [28, 17, 16]. We consder the followng type of swtchng systems, (2.1) F (t, x, v, t v, Dv, D 2 v ) = 0 n Q T, I := {1,...,M}, v(0, x) = v 0 (x) n R N, where the soluton v = (v 1,, v M ) s n R M, and for I, (t, x) Q T, r = (r 1,, r M ) R M, p t R, p x R N, and X S N, F s gven by { } F (t, x, r, p t, p x, X) = max p t + sup L α (t, x, r, p x, X); r M r, α A where the A s are subsets of A, L α s defned below (1.1), and for k > 0, M r = mn j {r j + k}. Fnally for the ntal data, we are nterested here n the case when v 0 = (u 0,...,u 0 ). Under sutable assumptons on the data (See (A1) below), we have exstence and unqueness of a soluton v of ths system. Moreover, t s not so dffcult to see that, as k 0, every component of v converge locally unformly to the soluton of the followng HJB equaton (2.2) u t + sup L α (x, u, Du, D 2 u) = 0 n Q T, α Ã u(0, x) = u 0 (x) n R N,

6 ERROR BOUNDS 5 where à = A. The objectve of ths secton s to obtan an error bound for ths convergence. For the sake of smplcty, we restrct ourselves to the stuaton where the solutons are n C 0,1 (Q T ),.e. when they are bounded, Lpschtz contnuous n x, and Hölder 1/2 n t. Such type of regularty s natural n ths context. However, t s not dffcult to adapt our approach to more general stuatons, and we gve results n ths drecton n Secton 6. We wll use the followng assumpton (A1) For any α A, a α = 1 2 σα σ αt for some N P matrx σ α. Moreover, there s a constant K ndependent of α such that u σ α 1 + b α 1 + c α 1 + f α 1 K. Assumpton (A1) ensures the well-posedness of all the equatons and systems of equatons we consder n ths paper; we refer the reader to the Appendx for a (partal) proof of ths clam. In the present stuaton, we have the followng well-posedness and regularty result. Proposton 2.1. Assume (A1). Then there exst unque solutons v and u of (2.1) and (2.2) respectvely, satsfyng v 1 + u 1 C, where the constant C only depends on T and K appearng n (A1). Furthermore, f w 1 and w 2 are sub- and supersolutons of (2.1) or (2.2) satsfyng w 1 (0, ) w 2 (0, ), then w 1 w 2. Remark 2.1. The functons σ α, b α, c α, f α are a pror only defned for tmes t [0, T]. But they can easly be extended to tmes [ r, T + r] for any r R + n such a way that (A1) stll holds. In vew of Proposton 2.1 we can then solve our ntal value problems (2.1) and (2.2) ether up to tme T + r and even, by usng a translaton n tme, on tme ntervals of the form [ r, T + r]. We wll use ths fact several tmes below. In order to obtan the rate of convergence for the swtchng approxmaton, we use a regularzaton procedure ntroduced by Krylov [21, 1]. Ths procedure requres the followng auxlary system (2.3) F ε (t, x, v ε, t v ε, Dv ε, D 2 v ε ) = 0 n Q T+ε 2, I, where v ε = (v ε 1,, vε M ), F ε (t, x, r, p t, p x, M) = { p t + max sup α A 0 s ε 2, e ε v ε (0, x) = v 0 (x) n R N, } L α (t + s, x + e, r, p x, X); r M r, and L and M are defned below (1.1) and (2.1) respectvely. Note that we use here the extenson mentoned n Remark 2.1. By Theorems A.1 and A.3 n the Appendx, we have the followng result: Proposton 2.2. Assume (A1). Then there exst a unque soluton v ε : Q T+ε 2 R of (2.3) satsfyng v ε ε vε v 0 C,

7 6 BARLES AND JAKOBSEN where v solves (2.1) and the constant C only depends on T and K from (A1). Furthermore, f w 1 and w 2 are sub- and supersolutons of (2.3) satsfyng w 1 (0, ) w 2 (0, ), then w 1 w 2. We are now n a poston to state and prove the man result of ths secton. Theorem 2.3. Assume (A1) and v 0 = (u 0,...,u 0 ). If u and v are the solutons of (2.2) and (2.1) respectvely, then for k small enough, 0 v u Ck 1/3 n Q T, I, where C only depends on T and K from (A1). Proof. Snce w = (u,...,u) s a subsoluton of (2.1), comparson for (2.1) (Proposton 2.1) yelds u v for I. To get the other bound, we use an argument suggested by P.-L. Lons [24] together wth the regularzaton procedure of Krylov [21]. Consder frst system (2.3). It follows that, for every 0 s ε 2, e ε, t v ε + sup L α (t + s, x + e, v ε (t, x), Dvε, D2 v ε ) 0 n Q T+ε2, I. α A After a change of varables, we see that for every 0 s ε 2, e ε, v ε (t s, x e) s a subsoluton of the followng system of uncoupled equatons (2.4) t w + sup α A L α (t, x, w, Dw, D 2 w ) = 0 n Q ε T, I, where Q ε T := (ε2, T) R N. Defne v ε := v ε ρ ε where {ρ ε } ε s the sequence of mollfers defned at the end of the ntroducton. A Remann-sum approxmaton shows that v ε (t, x) can be vewed as the lmt of convex combnatons of v ε (t s, x e) s for 0 < s < ε 2 and e < ε. Snce the v ε (t s, x e) s are subsolutons of the convex equaton (2.4), so are the convex combnatons. By the stablty result for vscosty subsolutons we can now conclude that v ε s tself a subsoluton of (2.4). We refer to the Appendx n [1] for more detals. On the other hand, snce v ε s a contnuous subsoluton of (2.3), we have v ε mn j vε j + k n Q T+ε 2, I. It follows that max v ε(t, x) mn v ε(t, x) k n Q T+ε2, and hence v ε v ε j 0 k,, j I. Then, by the defnton and propertes of v ε, we have t v ε t v εj 0 C k ε 2, Dn v ε D n v εj 0 C k, n N,, j I, εn where C depends only on ρ and the unform bounds on v ε and Dv ε,.e. on T and K gven n (A1). Furthermore, from these bounds, we see that for ε < 1, tv εj + sup L α [v εj ] t v ε sup L α [v ε ] C k α A α A ε 2 n Q ε T,, j I. Here, as above, C only depends on ρ, T and K. Snce v ε s a subsoluton of (2.4), ths means that, t v ε + sup L α (x, v ε, Dv ε, D 2 v ε ) C k α A ε 2 n Q ε T, I.

8 ERROR BOUNDS 7 From assumpton (A1) and the structure of the equaton, we see that v ε te Kt C k ε 2 s a subsoluton of equaton (2.2) restrcted to Q ε T. Comparson for (2.2) restrcted to Q ε T (Proposton 2.1) yelds ( v ε u e Kt v ε (ε 2, ) u(ε 2, ) 0 + Ct k ) ε 2 n Q ε T, I. Regularty of u and v (Proposton 2.1) mples that u(t, ) v (t, ) 0 ([u] 1 + [v ] 1 )ε n [0, ε 2 ]. Hence by Proposton 2.2, regularty of u and v ε, and propertes of mollfers, we have v u v v ε + v ε u C(ε + k ε 2 ) n Qε T, I. Mnmzng w.r.t. ε now yelds the result. 3. Convergence rate for the HJB equaton In ths secton we derve our man result, an error bound for the convergence of the soluton of the scheme (1.3) to the soluton of the HJB Equaton (1.1)-(1.2). As n [2], ths result s general and derved usng only PDE methods, and t extends and mproves earler results by Krylov [20, 21], Barles and Jakobsen [1, 18]. Compared to [2], we consder here the tme-dependent case and ntroduce a new, mproved, formulaton of the consstency requrement. Throughout ths secton, we assume that (A1) holds and we recall that, by Proposton 2.1, there exsts a unque C 0,1 -soluton u of (1.1) satsfyng u 1 C, where the constant C only depends on T and K from (A1). In Secton 6, we wll weaken assumpton (A1) and gve results for C 0,β solutons, β (0, 1). In order to get a lower bound bound on the error, we have to requre a techncal assumpton: If {α } I A s a suffcently refned grd for A, the soluton assocated to the control set {α } I s close to u. In fact for ths to be true we need to assume that the coeffcents σ α, b α, c α, f α can be approxmated unformly n (t, x) by σ α, b α, c α, f α. The precse assumpton s: (A2) For every δ > 0, there are M N and {α } M =1 A, such that for any α A, nf 1 M ( σα σ α 0 + b α b α 0 + c α c α 0 + f α f α 0 ) < δ. We pont out that ths assumptons s automatcally satsfed f ether A s a fnte set or f A s compact and σ α, b α, c α, f α are unformly contnuous functons of t, x, and α. Next we ntroduce the followng assumptons for the scheme (1.3). (S1) (Monotoncty) There exsts λ, µ 0, h 0 > 0 such that f h h 0, u v are functons n C b (G h ), and φ(t) = e µt (a + bt) + c for a, b, c 0, then S(h, t, x, r + φ(t), [u + φ] t,x ) S(h, t, x, r, [v] t,x ) + b/2 λc n G + h. (S2) (Regularty) For every h and φ C b (G h ), the functon (t, x) S(h, t, x, φ(t, x), [φ] t,x ) s bounded and contnuous n G + h and the functon r S(h, t, x, r, [φ] t,x ) s unformly contnuous for bounded r, unformly n (t, x) G + h.

9 8 BARLES AND JAKOBSEN Remark 3.1. In (S1) and (S2) we may replace C b (G h ) by any relevant subset of ths space. The pont s that (1.3) has to make sense for the class of functons used. In Secton 4, C b (R N ) s tself the relevant class of functons, whle, n Secton 5, t s C({0, 1,..., n T }; C 0,1 (R N )) (snce G h = {0, 1,..., n T } R N ). Assumptons (S1) and (S2) mply a comparson result for the scheme (1.3), see Lemma 3.2 below. Let us now state the key consstency condtons. (S3)() (Sub-consstency) There exsts a functon E 1 ( K, h, ε) such that for any sequence {φ ε } ε>0 of smooth functons satsfyng β0 t D β φ ε (x, t) Kε 1 2β0 β where β = N =1 β, the followng nequalty holds: n Q T, for any β 0 N, β = (β ) N N, S(h, t, x, φ ε (t, x), [φ ε ] t,x ) φ εt + F(t, x, φ, Dφ ε, D 2 φ ε ) + E 1 ( K, h, ε) n G + h. (S3)() (Super-consstency) There exsts a functon E 2 ( K, h, ε) such that for any sequence {φ ε } ε of smooth functons satsfyng β0 t D β φ ε (x, t) Kε 1 2β0 β the followng nequalty holds: n Q T, for any β 0 N, β N N, S(h, t, x, φ ε (t, x), [φ ε ] t,x ) φ εt + F(t, x, φ, Dφ ε, D 2 φ ε ) E 2 ( K, h, ε) n G + h. Typcally the φ ε we have n mnd n (S3) are of the form χ ε ρ ε where (χ ε ) ε s a sequence of unformly bounded functons n C 0,1 and ρ ε s the mollfer defned at the end of the ntroducton. The man result n ths paper s the followng: Theorem 3.1. Assume (A1), (S1), (S2) and that (1.3) has a unque soluton u h n C b (G h ). Let u denote the soluton of (1.1)-(1.2), and let h be suffcently small. (a) (Upper bound) If (S3)() holds, then there exsts a constant C dependng only µ, K n (S1), (A1) such that u u h e µt (u 0 u 0,h ) C mn ε>0 ( ε + E 1 ( K, ) h, ε) n G h, where K = u 1. (b) (Lower bound) If (S3)() and (A3) holds, then there exsts a constant C dependng only µ, K n (S1), (A1) such that u u h e µt (u 0 u 0,h ) 0 C mn ε>0 where K = u 1. ( ε 1/3 + E 2 ( K, ) h, ε) n G h, The motvaton for ths new formulaton of the upper and lower bounds s threefold: () n some applcatons, E 1 E 2 and therefore t s natural to have such dsymmetry n the consstency requrement (see Secton 5), () from the proof t can be seen that the upper bound (a) s proven ndependently of the lower bound (b), and most mportantly, () the new formulaton descrbes completely how the bounds are obtaned from the consstency requrements. The good h-dependence and the bad ε dependence of E 1 and E 2 are combned n the mnmzaton process to gve the fnal bounds, see Remark 3.2 below.

10 ERROR BOUNDS 9 Snce the mnmum s acheved for ε 1, the upper bound s n general much better than the lower bound (n partcular n cases where E 1 = E 2 ). Fnally note that the exstence of a u h n C b (G h ) must be proved for each partcular scheme S. We refer to [20, 21, 1, 18] for examples of such arguments. Remark 3.2. In the case of a fnte dfference method wth a tme step t and maxmal mesh sze n space x, a standard formulaton of the consstency requrement would be (S3 ) There exst fnte sets I N N N 0, Ī N 0 N N and constants K c 0, k β, k β for β = (β 0, β ) I, β = ( β 0, β ) Ī such that for every h = ( t, x) > 0, (t, x) G + h, and smooth functons φ: φt + F(t, x, φ, Dφ, D 2 φ) S(h, t, x, φ(t, x), [φ] t,x ) K c β0 t D β φ 0 t k β + K c β 0 t D β φ 0 x k β. β I The correspondng verson of (S3) s obtaned by pluggng φ ε nto (S3 ) and usng the estmates on ts dervatves. The result s β Ī E 1 ( K, h, ε) = E 2 ( K, h, ε) = KK c ε 1 2β0 β t k β + KK c β I β Ī ε 1 2 β 0 β x k β. From ths formula we see that the dependence n the small parameter ε s bad snce all the exponents of ε are negatve, whle the dependence on t, x s good snce ther exponents are postve. Remark 3.3. Assumpton (S1) contans two dfferent knds of nformaton. Frst, by takng φ 0 t mples that the scheme s nondecreasng wth respect to the [u] argument. Second, by takng u v t ndcates that a parabolc equaton an equaton wth a u t term s beng approxmated. Both these ponts play a crucal role n the proof of the comparson prncple for (1.3) (Lemma 3.2 below). To better understand that assumpton (S1) mples parabolcty of the scheme, consder the followng more restrctve assumpton: (S1 ) (Monotoncty) There exsts λ 0, K > 0 such that f u v, u, v C b (G h ), and φ : [0, T] R s smooth, then S(h, t, x, r + φ(t), [u + φ] t,x ) S(h, t, x, r, [v] t,x ) + φ (t) K t φ 0 λφ + (t) n G + h. Here h = ( t, h ) where h representng a small parameter related to e.g. the space dscretzaton. It s easy to see that (S1 ) mples (S1), e.g. wth the same value for λ and the followng values of µ and h 0 : µ = λ + 1 and h 1 0 = 2 Ke (λ+1)t (λ + 1)(2 + (λ + 1)T). Assumpton (S1 ) s satsfed for all monotone fnte dfference n tme approxmatons of (1.1), e.g. monotone Runge-Kutta methods and monotone mult-step methods, both explct and mplct methods. We have emphaszed the word monotone because whereas many Runge Kutta methods actually lead to monotone schemes for (1.1) (possbly under a CFL condton), t seems that the most commonly used

11 10 BARLES AND JAKOBSEN multstep methods (Adams-Bashforth, BDS) do not. We refer to [26] for an example of a multstep method that yelds a monotone approxmaton of (1.1). Proof of Theorem 3.1. We start by provng that condtons (S1) and (S2) mply a comparson result for bounded contnuous sub and supersolutons of (1.3). Lemma 3.2. Assume (S1), (S2), and that u, v C b (G h ) satsfy where g 1, g 2 C b (G h ). Then S(h, t, x, u(t, x), [u] t,x ) g 1 n G + h, S(h, t, x, v(t, x), [v] t,x ) g 2 n G + h, u v e µt (u(0, ) v(0, )) te µt (g 1 g 2 ) + 0, where λ and µ are gven by (S1). Proof. 1. Frst, we notce that t suffces to prove the lemma n the case (3.1) (3.2) u(0, x) v(0, x) 0 n G 0 h, g 1 (t, x) g 2 (t, x) 0 n G h. The general case follows from ths result after notng that, by (S1), w = v + e µt ( (u(0, ) v(0, )) t (g 1 g 2 ) + 0 ), satsfes S(h, t, x, w(t, x), [w] t,x ) g 1 n G + h and u(0, x) w(0, x) 0 n G0 h. 2. We assume that (3.1) and (3.2) hold and, for b 0, we set ψ b (t) = e µt 2bt where µ s gven by (S1) and M(b) = sup G h {u v ψ b }. We have to prove that M(0) 0 and we argue by contradcton assumng that M(0) > Frst we consder some b 0 for whch M(b) > 0 and take a sequence {(t n, x n )} n G h such that δ n := M(b) (u v ψ b )(t n, x n ) 0 as n. Snce M(b) > 0 and (3.1) holds, t n > 0 for all suffcently large n and for such n, we have g 1 S(h, t n, x n, u, [u] tn,x n ) (u subsoluton) S(h, t n, x n, v + ψ b + M(b) δ n, [v + ψ b + M(b)] tn,x n ) (S1), φ 0 ω(δ n ) + S(h, t n, x n, v + ψ b + M(b), [v + ψ b + M(b)] tn,x n ) (S2) ω(δ n ) + b λm(b) + S(h, t n, x n, v, [v] tn,x n ) ω(δ n ) + b λm(b) + g 2, (S1), φ = ψ + M (v supersoluton) where we have dropped the dependence n t n, x n of u, v and ψ b for the sake of smplcty of notaton. Recallng (3.2) and sendng n lead to b λm(b) Snce M(b) M(0), the above nequalty yelds a contradcton for b large, so for such b, M(b) 0. On the other hand, snce M(b) s a contnuous functon of b and M(0) > 0, there exsts a mnmal soluton b > 0 of M( b) = 0. For δ > 0

12 ERROR BOUNDS 11 satsfyng b δ > 0, we have M( b δ) > 0 and M( b δ) 0 as δ 0. But, by 3 we have b δ λm( b δ), whch s a contradcton for δ small enough snce b > 0. Now we turn to the proof of the upper bound,.e. of (a). We just sketch t snce t reles on the regularzaton procedure of Krylov whch s used n Secton 2. We also refer to Krylov [20, 21], Barles and Jakobsen [1, 18] for more detals. The man steps are: 1. Introduce the soluton u ε of u ε t + sup 0 s ε 2, e ε F(t + s, x + e, u ε (t, x), Du ε, D 2 u ε ) = 0 n Q T+ε 2, u ε (x, 0) = u 0 (x) n R N. Essentally as a consequence of Proposton 2.1, t follows that u ε belongs to C 0,1 (Q T ) wth a unform C 0,1 (Q T )-bound K. 2. By analogous arguments to the ones used n Secton 2, t s easy to see that u ε := u ε ρ ε s a subsoluton of (1.1). By combnng regularty and contnuous dependence results (Theorem A.3 n the Appendx), we also have u ε u 0 Cε where C only depends T and K n (A1). 3. Pluggng u ε nto the scheme and usng (S3)() and the unform estmates on u ε we get S(h, t, x, u ε (t, x), [u ε ] t,x ) E 1 ( K, h, ε) n G + h, where K s the above mentoned C 0,1 unform estmate on u ε whch depends only on the data and s essentally the same as for u. 4. Use Lemma 3.2 to compare u ε and u h and conclude by usng the control we have on u u ε and by takng the mnmum n ε. We now provde the proof of the lower bound,.e. of (b). Unfortunately, contrarly to the proof of (a), we do not know how to obtan a sequence of approxmate, global, smooth supersolutons. As n [2], we are gong to obtan approxmate almost smooth supersolutons whch are n fact supersolutons whch are smooth at the rght ponts. We buld them by consderng the followng swtchng system approxmaton of (1.1): (3.3) F ε (t, x, v ε, t v ε, Dv ε, D 2 v ε ) = 0 v ε (0, x) = v 0 (x) n R N, where v ε = (v ε 1,, vε M ), v 0 = (u 0,...,u 0 ), (3.4) F ε n Q T+2ε 2, I := {1,...,M}, (t, x, r, p t, p x, X) = { } max p t + mn (t + s ε 2, x + e, r 0 s ε 2, e ε Lα, p x, X); r M r, and L and M are defned below (1.1) and (2.1) respectvely. The soluton of ths system s expected to be close to the soluton of (1.1) f k and ε are small and {α } I A s a suffcently refned grd for A. Ths s where the assumpton (A2) plays a role. For equaton (3.3), we have the followng result.

13 12 BARLES AND JAKOBSEN Lemma 3.3. Assume (A1). (a) There exsts a unque soluton v ε of (3.3) satsfyng v ε 1 K, where K only depends on T and K from (A1). (b) Assume n addton (A2) and let u denote the soluton of (1.1). For I, we ntroduce the functons v ε : [ ε2, T + ε 2 ] R N R defned by v ε (t, x) := vε (t ε2, x). Then, for any δ > 0, there are M N and {α } M =1 A such that max u v ε 0 C(ε + k 1/3 + δ), where C only depends on T and K from (A1). In order to smplfy the arguments of the proof of the lower bound (to have the smplest possble formulaton of Lemma 3.5 below), we need the solutons of the equaton wth shaken coeffcents to be defned n a slghtly larger doman than Q T. More precsely on Q ε T := ( ε2, T + ε 2 ] R N. Ths s the role of the v ε s. In fact they solve the same system of equatons as the v ε s but on Qε T and wth (t + s ε 2, x + e, r Lα, p x, X) beng replaced by L α (t + s, x + e, r, p x, X) n (3.4). The (almost) smooth supersolutons of (1.1) we are lookng for are bult out of the v ε s by mollfcaton. Before gvng the next lemma, we remnd the reader that the sequence of mollfers {ρ ε } ε s defned at the end of the ntroducton. Lemma 3.4. Assume (A1) and defne v ε := ρ ε v ε : Q T+ε2 R for I. (a) There s a constant C dependng only on T and K from (A1), such that v εj v ε C(k + ε) n Q T+ε2,, j I. (b) Assume n addton that ε (8 sup [v ε ] 1) 1 k. For every (t, x) Q T, f j := argmn I v ε (t, x), then t v εj (t, x) + L αj (t, x, v εj (t, x), Dv εj (t, x), D 2 v εj (t, x)) 0. The proofs of these two lemmas wll be gven at the end of ths secton. The key consequence s the followng result whch s the corner-stone of the proof of the lower bound. Lemma 3.5. Assume (A1) and that ε (8 sup [v ε ] 1) 1 k. Then the functon w := mn I v ε s an approxmate supersoluton of the scheme (1.3) n the sense that S(h, t, x, w(t, x), [w] t,x ) E 2 ( K, h, ε) n G + h, where K comes from Lemma 3.3. Proof. Let (t, x) Q T and j be as n Lemma 3.4 (b). We see that w(t, x) = v εj (t, x) and w v εj n G h, and hence the monotoncty of the scheme (cf. (S1)) mples that S(h, t, x, w(t, x), [w] t,x ) S(h, t, x, v εj (t, x), [v εj ] t,x ). But then, by (S3)(), S(h, t, x, w(t, x), [w] t,x ) t v εj (t, x) + L αj (t, x, v εj (t, x), Dv εj (t, x), D 2 v εj (t, x)) E 2 ( K, h, ε), and the proof complete by applyng Lemma 3.4 (b).

14 ERROR BOUNDS 13 It s now straghtforward to conclude the proof of the lower bound, we smply choose k = 8 sup [v ε ] 1ε and use Lemma 3.2 to compare u h and w. Ths yelds u h w e µt (u h,0 w(0, )) te µt E 2 ( K, h, ε) n G h. But, by Lemmas 3.3 (b) and 3.4 (a), we have and therefore w u 0 C(ε + k + k 1/3 + δ), u h u e µt (u h,0 u 0 ) te µt E 2 ( K, h, ε) + C(ε + k + k 1/3 + δ) n G h, for some constant C. In vew of our choce of k, we conclude the proof by mnmzng w.r.t ε. Now we gve the proofs of Lemmas 3.3 and 3.4. Proof of Lemma We frst approxmate (1.1) by v t + sup L α (t, x, v, Dv, D 2 v) = 0 n Q T, I v(0, x) = u 0 (x) n R N. From assumpton (A2) and Lemmas A.1 and A.3 n the Appendx, t follows that there exsts a unque soluton v of the above equaton satsfyng v u 0 Cδ, where C only depends on T and K from (A1). 2. We contnue by approxmatng the above equaton by the followng swtchng system { } max t v + L α (t, x, v, Dv, D 2 v ); v M v = 0 n Q T, I, v(0, x) = v 0 (x) n R N, where v 0 = (u 0,...,u 0 ) and M s defned below (2.1). From Proposton 2.1 and Theorem 2.3 n Secton 2 we have exstence and unqueness of a soluton v = ( v 1,, v M ) of the above system satsfyng v v 0 Ck 1/3, I, where C only depends on the mollfer ρ, T, and K from (A1). 3. The swtchng system defned n the prevous step s nothng but (3.3) wth ε = 0 or (2.3) wth the A s beng sngletons. Theorems A.1 and A.3 n the Appendx yeld the exstence and unqueness of a soluton v ε : Q T+2ε 2 R of (3.3) satsfyng v ε ε vε v 0 C, where C only depends on T and K from (A1). 4. The proof s complete by combnng the estmates n steps 1 3, and notng that v ε v ε [v ε ] 1 ε n Q T+ε 2 and (A2) s only needed n step 1. Proof of Lemma 3.4. We start by (a). From the propertes of mollfers and the Hölder contnuty of v ε, t s mmedate that (3.5) v ε v ε Cε n Q T+ε 2, I,

15 14 BARLES AND JAKOBSEN where C = 2 max [ v ε] 1 = 2 max [v ε] 1 depends only on T and K from (A1). Furthermore as we ponted out after the statement of Lemma 3.3, v ε solves a swtchng system n Q ε T, so argung as n the proof of Theorem 2.3 n Secton 2 leads to 0 max v ε mn v ε k n Q ε T. From these two estmates, (a) follows. Now consder (b). Fx an arbtrary pont (t, x) Q T and set Then, by defnton of M and j, we have j = argmn I v ε (t, x). v εj (t, x) M j v ε (t, x) = max j {v εj(t, x) v ε (t, x) k} k, and the bound (3.5) leads to v j(t, ε x) M j v ε (t, x) k + 2 max[v ε ] 1 2ε. Next, by usng the Hölder contnuty of v ε (Lemma 3.3), for any ( t, x) Q ε T, we have v j ε ( t, x) M j v ε ( t, x) k + 2 max[v ε ] 1(2ε + x x + t t 1/2 ). From ths we conclude that f x x < ε, t t < ε 2, and ε (8 max [v ε ] 1) 1 k, then v ε j( t, x) M j v ε ( t, x) < 0, and by equaton (3.3) and the defnton of v ε, v ε (t, x) = v ε (t ε 2, x), t v j ε ( t, x) + nf ( t + s, x + e, v ε 0 s ε 2, e ε Lαj j ( t, x), D v j ε ( t, x), D 2 v j ε ( t, x)) = 0. After a change of varables, we see that, for every 0 s < ε 2, e < ε, (3.6) t v j ε (t s, x e)(t, x) + L αj (t, x, v j ε (t s, x e), D vε j (t s, x e), D2 v j ε (t s, x e)) 0. In other words, for every 0 s < ε 2, e < ε, v j ε (t s, x e) s a (vscosty) supersoluton at (t, x) of (3.7) χ t + L αj (t, x, χ, Dχ, D 2 χ) = 0. By mollfyng (3.6) (w.r.t. the (s, e)-argument) we see that v εj s also a smooth supersoluton of (3.7) at (t, x) and hence a (vscosty) supersoluton of the HJB equaton (1.1) at (t, x). Ths s correct snce v εj can be vewed as the lmt of convex combnatons of supersolutons v j ε (t s, x e) of the lnear and hence concave equaton (3.7), we refer to the proof of Theorem 2.3 and to the Appendx n [1] for the detals. We conclude the proof by notng that snce v εj s smooth, t s n fact a classcal supersoluton of (1.1) at x.

16 ERROR BOUNDS Monotone Fnte Dfference Methods In ths secton, we apply our man result to fnte dfference approxmatons of (1.1) based on the ϑ-method approxmaton n tme and two dfferent approxmatons n space: One proposed by Kushner [13] whch s monotone when a s dagonal domnant and a (more) general approach based on drectonal second dervatves proposed by Bonnans and Zdan [5], but see also Dong and Krylov [11]. For smplcty we take h = ( t, x) and consder the unform grd G h = t{0, 1,...,n T } xz N Dscretzaton n space. To explan the methods we frst wrte equaton (1.1) lke { } u t + sup α A L α u c α (t, x)u f α (t, x) = 0 n Q T, where L α φ(t, x) = tr[a α (t, x)d 2 φ(t, x)] + b α (t, x)dφ(t, x). To obtan a dscretzaton n space we approxmate L by a fnte dfference operator L h, whch we wll take to be of the form (4.1) L α hφ(t, x) = β S C α h (t, x, β)(φ(t, x + β x) φ(t, x)), for (t, x) G h, where the stencl S s a fnte subset of Z N \ {0}, and where (4.2) Ch α (t, x, β) 0 for all β S, (t, x) G+ h, h = ( x, t) > 0, α A. The last assumpton says that the dfference approxmaton s of postve type. Ths s a suffcent assumpton for monotoncty n the statonary case. () The approxmaton of Kushner. We denote by {e } N =1 the standard bass n RN and defne (4.3) L α h φ = N =1 [ a α 2 + ( a α+ j 2 + j aα j j 2 j ) + b α+ δ + b α δ ] φ, where b + = max{b, 0}, b = ( b) + (b = b + b ), and δ ± w(x) = ± 1 x {w(x ± e x) w(x)}, w(x) = 1 x 2 {w(x + e x) 2w(x) + w(x e x)}, + j w(x) = 1 2 x 2 {2w(x) + w(x + e x + e j x) + w(x e x e j x)} 1 2 x 2 {w(x + e x) + w(x e x) + w(x + e j x) + w(x e j x)}, j w(x) = 1 2 x 2 {2w(x) + w(x + e x e j x) + w(x e x + e j x)} x 2 {w(x + e x) + w(x e x) + w(x + e j x) + w(x e j x)}.

17 16 BARLES AND JAKOBSEN The stencl s S = {±e, ±(e ± e j ) :, j = 1,...,N}, and t s easy to see that the coeffcents n (4.1) are C α h (t, x, ±e ) C α h (t, x, e h ± e j h) = aα (x) 2 x 2 a α j (x) 4 x 2 + bα± (x) x, j = aα± j (x) 2 x 2, j, Ch α (t, x, e h ± e j h) = aα j (x) 2 x 2, j. The approxmaton s of postve type (4.2) f and only f a s dagonal domnant,.e. (4.4) a α (t, x) j a α j(t, x) 0 n Q T, α A, = 1,...,N. () The approxmaton of Bonnans and Zdan. We assume that there s a (fnte) stencl S Z N \ {0} and a set of postve coeffcents {ā β : β S} R + such that (4.5) a α (t, x) = β S ā α β (t, x)βt β n Q T, α A. Under assumpton (4.5) we may rewrte the operator L usng second order drectonal dervatves D 2 β = tr[ββt D 2 ] = (β D) 2, L α φ(t, x) = β S ā α β(t, x)d 2 βφ(t, x) + b α (t, x)dφ(t, x). The approxmaton of Bonnans and Zdan s gven by L α h φ = N [ (4.6) ā α β βφ + δ + b α β S =1 where β s an approxmaton of Dβ 2 gven by β w(x) = b α+ δ ] φ, 1 β 2 {w(x + β x) 2w(x) + w(x β x)}. x2 In ths case, the stencl s S = ± S {±e : = 1,...,N} and the coeffcents correspondng to (4.1) are gven by Ch α (t, x, ±e ) = bα± (x) x, Ch α (t, x, ±β) = āα β (t, x) β 2 x 2, = 1,...,N, β S, and the sum of the two whenever β = e. Under assumpton (4.5), whch s more general than (4.4) (see below), ths approxmaton s always of postve type. For both approxmatons there s a constant C > 0, ndependent of x, such that, for every φ C 4 (R N ) and (t, x) G + h, (4.7) L α φ(t, x) L α hφ(t, x) C( b α 0 D 2 φ 0 x + a α 0 D 4 φ 0 x 2 ).

18 ERROR BOUNDS The fully dscrete scheme. To obtan a fully dscrete scheme, we apply the ϑ-method, ϑ [0, 1], to dscretze the tme dervatve. The result s the followng scheme, (4.8) u(t, x) = u(t t, x) (1 ϑ) t sup{ L α h u cα u f α }(t t, x) α A ϑ t sup{ L α h u cα u f α }(t, x) n G + h. α A The case ϑ = 0 and ϑ = 1 correspond to the forward and backward Euler tmedscretzatons respectvely, whle for ϑ = 1/2 the scheme s a generalzaton of the second order n tme Crank-Ncholson scheme. Note that the scheme s mplct except for the value ϑ = 0. We may wrte (4.8) n the form (1.3) by settng S(h, t, x, r, [u] t,x ) = sup α A [ 1 t + (1 ϑ)cα (1 ϑ) {[ 1 t ϑcα + ϑ Ch ]r α (t, x, β) β S β S C α h (t, x, β) ][u] t,x ( t, 0) [ ]} Ch α (t, x, β) ϑ[u] t,x (0, β x) + (1 ϑ)[u] t,x ( t, β x), β S where [u] t,x (s, y) = u(t + s, x + y). Under assumpton (4.2) the scheme (4.8) s monotone (.e. satsfes (S1) or (S1 )) provded the followng CFL condtons hold ( t (1 ϑ) c α (t, x) + ) (4.9) Ch α (t, x, β) 1, β S (4.10) ( t ϑ c α (t, x) ) Ch α (t, x, β) 1. β S Furthermore, n vew of (A1) and (4.7), Taylor expanson n (4.8) yelds the followng consstency result for smooth functons φ and (t, x) G + h, φ t + F(t, x, φ, Dφ, D 2 φ) S(h, t, x, φ, [φ] t,x ) C( t φ tt 0 + x D 2 φ 0 + x 2 D 4 φ 0 + (1 ϑ) t( Dφ t 0 + D 2 φ t 0 )). The (1 ϑ) t-term s a non-standard term comng from the fact that we need the equaton and the scheme to be satsfed n the same pont, see assumpton (S3). The necessty of ths assumpton follows from the proof of Theorem 3.1. We have seen that f (4.2) and (4.7) hold along wth the CFL condtons (4.9) and (4.10) then the scheme (4.8) satsfes assumptons (S1) (S3) n Secton 3. Theorem 3.1 therefore yelds the followng error bound: Theorem 4.1. Assume (A1), (A2), (4.2), (4.7), (4.9), (4.10) hold. If u h C b (G h ) s the soluton of (4.8) and u s the soluton of (1.1), then there s C > 0 such that n G h e µt (u 0 u 0,h) 0 C h 1 5 u uh e µt (u 0 u 0,h ) C h 1 2, where h := x 2 + t. Remark 4.1. Except when ϑ = 1, the CFL condton (4.9) essentally mples that t C x 2. Therefore t and x 2 play essentally the same role. Also note that the CFL condton (4.10) s satsfed f e.g. t (sup α (c α ) + 0 ) 1.

19 18 BARLES AND JAKOBSEN Remark 4.2. Even though the above consstency relatonshp s not qute the standard one, t gves the correct asymptotc behavor of our scheme. Frst of all note that the new term, the (1 ϑ)-term, behaves just lke the t and x 2 terms. To see ths, we note that accordng to (S3) we only need the above relaton when φ s replaced by φ ε defned n (S3). But for φ ε we have φ ε,tt 0 D 4 φ ε 0 D 2 φ ε,t 0 Kε 3. By the CFL condtons (4.9) and (4.10) we have essentally that x 2 t, so t φ ε,tt 0 x 2 D 4 φ ε 0 t D 2 φ ε,t 0 K x 2 ε 3. Next note that for ϑ = 1/2 (the Cranck-Ncholson case) the scheme s formally second order n tme. However ths s no longer the case for the monotone verson. It s only frst order n tme due to the CFL condton whch mples that x 2 D 4 φ = C t D 4 φ. Proof. In ths case E 1 ( K, h, ε) = E 2 ( K, h, ε) = C( tε 3 + xε 1 + x 2 ε 3 + (1 ϑ) t(ε 2 + ε 3 )). So we have to mnmze w.r.t. ε the followng functons ε + C( tε 3 + xε 1 + x 2 ε 3 ), ε 1/3 + C( tε 3 + xε 1 + x 2 ε 3 ). By mnmzng separately n t and x, one fnds that ε has to be lke t 1/4 and x 1/2 n the frst case, and that ε 1/3 has to be lke t 1/10 and x 1/5 n the second case. The result now follows by takng ε = max( t 1/4, x 1/2 ) n the frst case and ε 1/3 = max( t 1/10, x 1/5 ) n the second case Remarks. For approxmatons of nonlnear equatons monotoncty s a key property snce t ensures (along wth consstency) that the approxmate solutons converge to the correct generalzed soluton of the problem (the vscosty soluton n our case). Ths s not the case for nonmonotone methods, at least not n any generalty. However, the monotoncty requrement poses certan problems. Monotone schemes are low order schemes, and maybe more mportantly, t s not always possble to fnd consstent monotone approxmatons for a gven problem. To see the last pont we note that n general the second dervatve coeffcent matrx a s only postve semdefnte, whle the monotone schemes of Kushner and Bonnans/Zdan requre the stronger assumptons (4.4) and (4.5) respectvely. In fact, n Dong and Krylov [11] t was proved that f an operator L admts an approxmaton L h of the form (4.1) whch s of postve type, then a has to satsfy (4.5) (at least f a s bounded). Ths s a problem n real applcatons, e.g. n fnance, and t was ths problem was the motvaton behnd the approxmaton of Bonnans and Zdan. Frst of all we note that ther condton (4.5) s more general than (4.4) because any symmetrc N N matrx a can be decomposed as a = N =1 j (a a j )e e T + a+ j 2 (e + e j )(e + e j ) T + a j 2 (e e j )(e e j ) T, where the coeffcents are nonnegatve f and only f a s dagonal domnant. More mportantly, t turns out that any symmetrc postve semdefnte matrx can be approxmated by a sequence of matrces satsfyng (4.5). In Bonnans, Ottenwaelter,

20 ERROR BOUNDS 19 and Zdan [4], ths was proved n the case of symmetrc 2 2 matrces along wth an explct error bound and an algorthm for computng the approxmate matrces. Because of contnuous dependence results for the equatons, convergence of the coeffcents mmedately mply convergence of the solutons of the correspondng equatons. Hence the Bonnans/Zdan approxmaton yelds a way of approxmatng general problems where a s only postve semdefnte. 5. Semgroup Approxmatons and Splttng Methods In ths secton, we consder varous approxmatons of semgroups obtaned by a sem-dscretzaton n tme. In order to smplfy the presentaton we start by specalzng Theorem 3.1 to the semgroup settng. To be precse we consder onestep n tme approxmatons of (1.1) gven by (5.1) u h (t n, x) = S h (t n, t n 1 )u h (t n 1, x) n R N, u h (0, x) = u h,0 (x) n R N, where t 0 = 0 < t 1 < < t n < < t nt = T, h := max n (t n+1 t n ), and the approxmaton semgroup S h satsfes the followng sub and superconsstency requrements: There exst a constant K c, a subset I of N N N, and constants γ β, δ β for β I such that for any smooth functons φ, 1 [ ] (5.2) S h (t n, t n 1 ) 1 φ(t n 1, x) F(t, x, φ, Dφ, D 2 φ) t=tn t K c β0 t D β φ γ β 0 tδ β, β I where β = (β 0, β ) I for β 0 N and β N N, and n a smlar way 1 [ ] (5.3) S h (t n, t n 1 ) 1 φ(t n 1, x) F(t, x, φ, Dφ, D 2 φ) t=tn t K c β0 t D β φ γ β 0 t δ β, β Ī wth correspondng data K c, Ī, γ β, δ β. We say that the semgroup s monotone f φ ψ S h (t n, t n 1 )φ S h (t n, t n 1 )ψ, n = 1,...,n T, for all contnuous bounded functons φ, ψ for whch S h (t)φ and S h (t)ψ are well defned. We have the followng corollary to Theorem 3.1. Proposton 5.1. Assume (A1), (A2), and that S h s a monotone semgroup satsfyng (5.2) and (5.3) and whch s defned on a subset of C b (R N ). If u s the soluton of (1.1) and u h s the soluton of (5.1), then n R N, where C( u 0 u h,0 0 + t 1 10 r1 ) u u h C( u 0 u h,0 0 + t 1 4 r2 ) { r 1 := mn β I { r 2 := mn β Ī δ β 3(2β 0 + β 1)γ β + 1 δ β (2β 0 + β 1) γ β + 1 where β denotes the sum of the components of β. }, },

21 20 BARLES AND JAKOBSEN Proof. We defne where S(h, t n, x, u h, [u h ] tn,x) = 1 t ( ) u h (t n, x) [u h ] tn,x, [u h ] tn,x = S h (t n 1, t n )u h (t n 1, x). To apply Theorem 3.1, we just have to check that (S1) (S3) hold and ths s clear for (S1) and (S2) (see Remark 3.1). For (S3)(), note that by (5.2) we have whch leads to φ t + F(t n, x, φ, Dφ, D 2 φ) S(h, t n, x, φ, [φ] tn,x) t φ 0 t + K c β0 t D β φ γ β 0 tδ β, β I E 1 ( K, h, ε) = 1 2 Kε 1 4 t + K c ( Kε 1 2β0 β ) γ β t δ β. The upper bound now follows by optmzng wth respect to ε as n the proof of Theorem 4.1. In a smlar way we may use (5.3) to defne E 2 and then conclude the lower bound. Remark 5.1. In vew of the consstency requrements (5.2) and (5.3), for schemes lke (5.1) t s natural to thnk that only the x-varable s really playng a role and that one can get results on the rate of convergence by usng ths specal semgroup type structure. More specfcally, one mght thnk that a dfferent proof usng a mollfcaton of the soluton wth respect to the space varable only, can produce the estmates n an easer and maybe better way. We tred ths strategy but we could not avod usng the short tme expanson of the soluton of the HJB Equaton assocated wth smooth ntal data (the short tme expanson of the semgroup), and ths leads to worse rates, even n cases where F s smooth. One way of understandng ths wthout justfyng t completely conssts of lookng at our estmates for the φ tt -term (cf. (S3)() and ()). The present approach leads to an estmate of order ε 3, whle f we use the short tme expanson, we are lead to a worse estmate of order ε 4. We refer the reader to Subsecton 5.1 and n partcular to Lemma 5.6 below, where short tme expansons for sem-groups are obtaned and used to study the rate of convergence for splttng problems Semdscrete splttng. We consder an equaton of the form (5.4) where β I u t + F 1 (D 2 u) + F 2 (D 2 u) = 0 n Q T, F j (X) = sup{ tr[a α j X] fα j }, j = 1, 2, α A and a α j 0 are matrces and fj α real numbers. We assume that they are both unformly bounded n α and are ndependent of (t, x). It follows that F 1 and F 2 are Lpschtz contnuous and that (A1) s satsfed. Let S denote the semgroup of (5.4),.e. S( t)φ s the soluton at tme t = t of (5.4) wth ntal value φ. Smlarly, let S 1 and S 2 denote the semgroups assocated wth the equatons u t + F 1 (D 2 u) = 0 and u t + F 1 (D 2 u) = 0.

22 ERROR BOUNDS 21 We can defne a semdscrete splttng method by takng (5.1) wth t n := n t and (5.5) S h (t n 1, t n ) = S 1 ( t)s 2 ( t). Under the current assumptons all these semgroups map W 1, (R N ) nto tself, they are monotone, and they are nonexpansve, S(t)φ 0 φ 0, for φ W 1, (R N ) and where S denotes one of the semgroups above. As soon as we know the consstency relaton for ths scheme, we can fnd an error bound usng Theorem 5.1. However, contrarly to the case of fnte dfferent schemes n the prevous secton, here the precse form of the consstency requrement s not well known. We are gong to provde such results under dfferent assumptons on F 1, F 2. Our frst result s the followng: Lemma 5.2. Under the above assumptons, f n addton DF 1 W 1, (S N ) and DF 2 W 3, (S N ), then C( t D 2 φ t 0 + t 2 D 3 φ 4 0 ) h.o.t. 1 t [S h(t) 1]φ(t n 1, x) + F 1 (D 2 φ(t n, x)) + F 2 (D 2 φ(t n, x)) C( t D 2 φ t 0 + t D 3 φ 2 0 ) + h.o.t. for all smooth functons φ, where h.o.t. stands for hgher order terms. Remark 5.2. Due to the convexty of the equaton, n ths example the upper and lower bounds are dfferent. Remark 5.3. We have only stated the prncpal error terms, the terms decdng the rate. The other terms are put n the h.o.t. category. Snce the prncpal error terms need not be the lowest order terms (see the frst nequalty n Lemma 5.2), maybe a better name than h.o.t. would be the less mportant terms. A drect consequence of Proposton 5.1 s the followng result: Corollary 5.3. Let u h denote the soluton of (5.1) where S h s defned n (5.5) and u h,0 = u 0, and let u be the soluton of (5.4) wth ntal value u 0. Under the assumptons of Lemma 5.2 we have C t 1 13 u uh C t 2 9 n t{0, 1, 2,..., n T } R N. Next, we gve the result when F 1 and F 2 are assumed to be only Lpschtz contnuous (whch s the natural regularty assumpton here). In ths case the consstency relaton s: Lemma 5.4. Under the above assumptons, f F 1 and F 2 are only Lpschtz contnuous, we have 1 t [S h(t) 1]φ(t n 1, x) + F 1 (D 2 φ(t n, x)) + F 2 (D 2 φ(t n, x)) C t D 2 φ t 0 + C t 1 2 D 3 φ 0 + h.o.t. for all smooth functons φ. Agan as a drect consequence of Proposton 5.1 we have the followng error bound:

23 22 BARLES AND JAKOBSEN Corollary 5.5. Under the assumptons of Corollary 5.3 but where F 1 and F 2 are only assumed to be Lpschtz contnuous, we have C t 1 14 u uh C t 1 6 n t{0, 1, 2,..., n T } R N. Remark 5.4. We see a slght reducton of the rates n the Lpschtz case but not as mportant as one mght have guessed. For frst order equatons these methods lead to the same rates n the smooth and Lpschtz cases. Remark 5.5. If we change operators S 1, S 2 so that S 1 (t)φ and S 2 (t)φ denote the vscosty solutons of u(x) = φ(x) tf 1 (D 2 u(x)) n R N, u(x) = φ(x) tf 2 (D 2 u(x)) n R N, respectvely, then the statements of Corollary 5.3 and 5.5 stll hold. In the proofs of Lemmas 5.2 and 5.4 we wll use the followng lemma: Lemma 5.6. Let S be the semgroup assocated to the equaton u t + F(D 2 u) = 0, where F s Lpschtz, convex, and non-ncreasng. Defne F δ by F δ = F ρ δ, where ρ δ (X) = δ N2 ρ(x/δ) and ρ s a smooth functon on S(N) wth mass one and support n B(0, 1). Then for any smooth functon φ, and S(t)φ φ + t F δ (D 2 φ) tδ D F t2 D F 0 D F δ 0 D 4 φ 0, S(t)φ φ + t F δ (D 2 φ) 1 2 t2 D F 0 ( D 2 Fδ 0 D 3 φ D F δ 0 D 4 φ 0 ). The proof of ths result wll be gven after the proofs of Lemmas 5.2 and 5.4. Proofs of Lemmas 5.2 and 5.4. In order to treat the two results at the same tme, we mollfy F 1 and F 2 and consder F 1,δ and F 2,δ (see Lemma 5.6 for the defntons). By Lemma 5.6 we have the followng (small tme) expansons: (5.6) (5.7) S j (t)φ φ + tf j,δ (D 2 φ) tδ DF j t2 DF j 0 DF j,δ 0 D 4 φ 0, S j (t)φ φ + tf j,δ (D 2 φ) 1 2 t2 DF j 0 ( D 2 F j,δ 0 D 3 φ DF j,δ 0 D 4 φ 0 ), for smooth functons φ and j = 1, 2. Now we want to fnd an (small tme) expanson for S h. We wrte S h (t)φ φ + t(f 1 + F 2 )(D 2 φ) = [S 1 (t)s 2 (t)φ S 1 (t)(φ tf 2,δ (D 2 φ))] + [S 1 (t)(φ tf 2,δ (D 2 φ)) φ + tf 1,δ (D 2 φ) + tf 2,δ (D 2 φ)] + t[(f 1 + F 2 )(D 2 φ) (F 1,δ + F 2,δ )(D 2 φ)]. In vew of the Lpschtz regularty and convexty of F 1 and F 2, the last term on rght hand sde s between Ctδ (Lpschtz regularty) and 0 (convexty), whle the