ERROR BOUNDS FOR MONOTONE APPROXIMATION SCHEMES FOR PARABOLIC HAMILTON-JACOBI-BELLMAN EQUATIONS

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1 MATHEMATICS OF COMPUTATION Volume 76, Number 260, October 2007, Pages S (07) Artcle electroncally publshed on Aprl 20, 2007 ERROR BOUNDS FOR MONOTONE APPROXIMATION SCHEMES FOR PARABOLIC HAMILTON-JACOBI-BELLMAN EQUATIONS GUY BARLES AND ESPEN R. JAKOBSEN Abstract. We obtan nonsymmetrc 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. Our results are robust and general they mprove and extend earler results by Krylov, Barles, and Jakobsen. We apply our general results to varous schemes ncludng Crank Ncholson type fnte dfference schemes, splttng methods, and the classcal approxmaton by pecewse constant controls. In the frst two cases our results are new, and n the last two cases the results are obtaned by a new method whch we develop here. 1. 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. 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,thespaceofN N symmetrc matrces, R N, R, R, andr. 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. Receved by the edtor June 24, 2005 and, n revsed form, June 29, Mathematcs Subject Classfcaton. Prmary 65M15, 65M06, 35K60, 35K70, 49L25. 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 / c 2007 Amercan Mathematcal Socety Reverts to publc doman 28 years from publcaton

2 1862 GUY BARLES AND ESPEN R. JAKOBSEN 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 Gh 0 := G h {t =0}, 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, where x can tself be multdmensonal. The approxmate soluton s u h : G h R, and[u h ] t,x s a functon defned from u h representng, typcally, the value of u h at ponts other 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 grd G h wll vary from applcaton to applcaton, and when G h Q T s dscrete (as s not always the case!), any functon on G h s automatcally contnuous. The abstract notaton S was ntroduced by Barles and Sougands [3] to dsplay clearly the monotoncty of the scheme. One of the man assumptons s that S s nondecreasng n u h and nonncreasng n [u h ] t,x wth the classcal orderng of functons. The typcal approxmaton schemes we have n mnd are varous fnte dfferences schemes (see, e.g., Kushner and Dupus [12] and Bonnans and Zdan [5]), (operator) splttng methods (see, e.g., Tourn [27] and references theren), and control schemes based on the dynamc programmng prncple (see, e.g., Camll and Falcone [6]). However, we wll not dscuss control schemes n ths paper, snce better results can be obtaned usng the dfferent approach of [1]. 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 nonsmooth solutons. Snce these equatons may also be degenerate, the (vscosty) solutons are expected to be no more than Hölder contnuous n general. Despte these dffcultes, n the 1980s, Crandall and Lons [9] provded the frst optmal rates of convergence for frst-order equatons. We refer to Sougands [26] 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 [19, 20], and by now there exst several papers based on and extendng hs deas, e.g. [1, 2, 10, 17, 21, 22]. One of the man deas 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, andd 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, 10, 17, 19, 22]. Here, the key dffculty s to obtan a contnuous dependence

3 ERROR BOUNDS 1863 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 types of results are not known for numercal schemes n general. We menton here the nce paper of Krylov [22] where such knds 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. The second approach conssts of consderng some approxmaton of the equaton or the assocated control problem and obtanng the other bound ether by probablstc arguments (as Krylov frst dd usng pecewse constant controls, [21, 20]) or by buldng a sequence of approprate smooth supersolutons 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 and mprovng the arguments of [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, as mentoned before, n certan stuatons the frst approach yelds better rates [22]. The results n [2] apply to statonary HJB equatons set n the entre 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 slghtly more dffcult from a mathematcal pont of vew. We ntroduce a new and very general monotoncty assumpton for the scheme, ts generalty allowng us to handle more general tme-dscretzatons of HJB equatons than have appeared n the lterature so far. We relax the assumptons on the controls compared wth [2], the new assumptons beng more natural, and we also present several techncal mprovements and smplfcatons n the proofs. However, the most mportant dfference between the two papers les n our opnon n the formulaton of the consstency requrements of the man (abstract) result and n the new applcatons we are able to handle. Here we ntroduce a new, more general formulaton of consstency that emphaszes more the nonsymmetrcal 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. Other mportant contrbutons of ths paper are the new applcatons we consder: () Fnte dfference methods (FDMs) usng the θ-method for tme dscretzaton (Crank Ncholson type schemes), () semdscrete splttng methods, and () approxmaton by pecewse constant controls. In the frst two cases error bounds have not appeared before, and n the last two cases the results are obtaned by a new method based on semgroup technques whch we develop here. The results for fnte dfference approxmatons can be compared wth the ones obtaned by Krylov n [20, 21]. As n [2], we get the rate 1/5 for monotone FDMs whle the correspondng result n [21] s 1/21. Of course, n specal stuatons the

4 1864 GUY BARLES AND ESPEN R. JAKOBSEN rate can be mproved to 1/2, whch s the maxmal rate under our assumptons. We refer to [22] for the most general results n that drecton, and to [11] for the optmalty of the rate 1/2. The results for semdscrete splttng methods are new, whle the result we get for the control approxmaton s 1/10, whch s worse than the 1/6 obtaned by Krylov n [21]. It would be nterestng to understand why Krylov s dong better than we are here but not n the other cases. We conclude ths ntroducton by explanng the notaton 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,orthe space of N P matrces, we set w 0 = Furthermore, for δ (0, 1], we set [w] δ = sup (t,x) (s,y) sup w(t, y). (t,y) Q w(t, x) w(s, y) ( x y + t s 1/2 ) δ and w δ = w 0 +[w] δ. Let C b (Q )andc 0,δ (Q ), δ (0, 1], denote respectvely the space of bounded contnuous functons on Q and the subset of C b (Q )nwhchthenorm 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 denote 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+2 ρ 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 obtanng the general results on the rate of convergence of approxmaton/numercal schemes and are of an ndependent nterest. Secton 3 s devoted to statng and provng the man result on the rate of convergence. Fnally we present applcatons to classcal fnte dfference schemes, the splttng method, and 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 been studed n [13, 7], and a vscosty solutons theory of swtchng systems can be found n [28, 16, 15]. 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,

5 ERROR BOUNDS 1865 where the soluton v =(v 1,...,v M )snr M,andfor I,(t, x) Q T, r = (r 1,...,r M ) R M, p t R, p x R N,andX S N, F s gven by { } F (t, x, r, p t,p x,x)=max p t +supl 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 {r j + k}. j 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 converges locally unformly to the soluton of the followng HJB equaton: (2.2) u t +supl (x, u, Du, D 2 u)=0 n Q T, α à u(0,x)=u 0 (x) n R N, 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, andhölder 1/2 n t. Ths 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, ), thenw 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.

6 1866 GUY BARLES AND ESPEN R. JAKOBSEN In order to obtan the rate of convergence for the swtchng approxmaton, we use a regularzaton procedure ntroduced by Krylov [20, 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, v ε (0,x)=v 0 (x) n R N, where v ε =(v1,...,v ε M ε ), F ε (t, x, r, p t,p x,m) { =max p t + sup α A 0 s ε 2, e ε } L α (t + s, x + e, r,p x,x); r M r, and L and M are defned below (1.2) 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 exsts a unque soluton v ε : Q T +ε 2 R of (2.3) satsfyng v ε ε vε v 0 C, 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, ), thenw 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 [23] together wth the regularzaton procedure of Krylov [20]. Consder frst system (2.3). It follows that, for every 0 s ε 2, e ε, t v ε +supl α (t + s, x + e, v ε (t, x),dv ε,d 2 v ε ) 0 n Q T +ε 2, 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 +supl α (t, x, w,dw,d 2 w )=0 n Q ε T, I, α A 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.

7 ERROR BOUNDS 1867 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 vj ε 0 k,, j I. Then, by the defnton and propertes of v ε,wehave 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 +supl 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. Sncev ε s a subsoluton of (2.4), ths means that t v ε +supl α (x, v ε,dv ε,d 2 v ε ) C k α A ε 2 n Q ε T, I. 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). Ths result extends and mproves earler results by Krylov [19, 20], Barles and Jakobsen [1, 2, 17]. 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 ths assumpton and gve results for C 0,β solutons, β (0, 1). In addton, to get a lower bound on the error, we need the followng techncal assumpton that allows us to approxmate the HJB equaton (1.1) by another HJB equaton where the supremum (maxmum) s over a fnte set:

8 1868 GUY BARLES AND ESPEN R. JAKOBSEN (A2). The control set A s a separable metrc space, and the coeffcents σ α, b α, c α, f α are contnuous n α for all x, t. Ths assumpton s used by Krylov [20, 21], and t s more natural and general than the one used n [2] (Assumpton (A3) page 8). Next we ntroduce the assumptons for the scheme (1.3). (S1) (Monotoncty). There exst 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. Assumptons (S1) and (S2) mply a comparson result for the scheme (1.3); see Lemma 3.2 below. Assumpton (S1) s very general and allows one to handle more general tme dscretzatons than n prevous papers; see Remark 3.4 below. Remark 3.1. In (S1) and (S2) (and n Theorem 3.1 below) we may replace C b (G h )by any subset of ths space as long as (1.3) s well-defned for functons n ths set. E.g. n Secton 4 we replace t by C b (R N ) and n Secton 5 by C({0, 1,...,n T }; C 0,1 (R N )) (snce G h = {0, 1,...,n T } R N there). Let us now state the key consstency condtons. (S3)() (Subconsstency). There exsts a functon E 1 ( K,h,ε) such that for any sequence {φ ε } ε>0 of smooth functons satsfyng β 0 t D β φ ε (x, t) Kε 1 2β 0 β n Q T, for any β 0 N, β =(β ) N N, where β = N =1 β, the followng nequalty holds: 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)() (Superconsstency). There exsts a functon E 2 ( K,h,ε) such that for any sequence {φ ε } ε of smooth functons satsfyng β 0 t D β φ ε (x, t) Kε 1 2β 0 β n Q T, for any β 0 N, β N N, the followng nequalty holds: 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. The typcal φ ε we have n mnd n (S3) s 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. Ths functon satsfes the dervatve bounds of (S3). 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 ).Letudenote the soluton of (1.1) (1.2), andleth be suffcently small. (a) (Upper bound) If (S3)() holds, then there exsts a constant C dependng only on µ, K n (S1), (A1) such that u u h e µt (u 0 u 0,h ) C mn ε>0 for h small enough and K = u 1. ( ε + E 1 ( K,h,ε) ) n G h,

9 ERROR BOUNDS 1869 (b) (Lower bound) If (S3)() and (A3) hold, then there exsts a constant C dependng only on µ, K n (S1), (A1) such that ( ) u u h e µt (u 0 u 0,h ) 0 C mn ε>0 ε 1/3 + E 2 ( K,h,ε) n G h, for h small enough and K = u 1. 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 asymmetry 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. 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 (some approprate subset of) C b (G h ) must be proved for each partcular scheme S. We refer to [19, 20, 1, 17] 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 β 0 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 2β 0 β t k β + KK c ε 1 2 β 0 β β x k. β I 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,.e., 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: β Ī

10 1870 GUY BARLES AND ESPEN R. JAKOBSEN (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 )whereh represents 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 ). Remark 3.4. Assumptons (S1 ) and (S1) are more general than the correspondng assumptons used n earler papers by Krylov and ourselves [20, 17]. They are satsfed for all monotone fnte dfferences n tme approxmatons of (1.1), e.g. monotone Runge-Kutta methods and monotone mult-step methods, both explct and mplct methods. Note that whereas many Runge-Kutta methods lead to monotone schemes for (1.1) (possbly under a CFL condton), t seems that the most commonly used mult-step methods (Adams-Bashforth, BDS) do not. We refer to [25] for a mult-step method that yelds a monotone approxmaton of (1.1). Usng the generalty of (S1), we provde the frst error bounds for Crank Ncholson type schemes for (1.1) n Secton 4. In the proof of Theorem 3.1 we need the followng 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 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, where g 1,g 2 C b (G h ).Then 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 note that t suffces to prove the lemma n the case (3.1) u(0,x) v(0,x) 0 n Gh, 0 (3.2) 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) 0nG0 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{u v ψ b }. G h We have to prove that M(0) 0 and we argue by contradcton assumng M(0) > Consder b 0forwhchM(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.

11 ERROR BOUNDS 1871 Snce M(b) > 0 and (3.1) holds, t n > 0 for all suffcently large n. Forsuchn, 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) wherewehavedroppedthedependencent n,x n of u, v and ψ b to smplfy the 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 >0ofM( b) =0. Forδ>0 satsfyng b δ>0, we have M( b δ) > 0andM( b δ) 0asδ 0. But, by 3 we have b δ λm( b δ), whch s a contradcton for δ small enough snce b >0. Proof of Theorem 3.1 (a). We only sketch t snce t reles on the regularzaton procedure of Krylov used n Secton 2. More detals can be found n [19, 20, 1, 17]. The man steps are: 1. Introduce the soluton u ε of u ε t + sup F (t + s, x + e, u ε (t, x),du ε,d 2 u ε )=0 nq T +ε 2, 0 s ε 2, e ε 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 on 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 unform C 0,1 estmate on u ε ; see step 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. ProofofTheorem3.1(b). Here, unfortunately, we cannot follow the scheme of the proof of (a) snce we do not know how to obtan a sequence of approxmate, global, smooth supersolutons of (1.1). Instead we buld approxmate supersolutons whch are smooth at the rght ponts. Frst we consder the case when A s fnte usng a swtchng system approach smlar to the approach n [2]. Then we do the general case usng our new Assumpton (A2), and fnally we prove a key lemma used n the frst part of the proof.

12 1872 GUY BARLES AND ESPEN R. JAKOBSEN 1. The case A = {α 1,...,α M }. We buld the almost smooth supersolutons out of the solutons of the followng swtchng system approxmaton of (1.1): F ε (t, x, v ε, t v ε,dv ε,d 2 v ε )=0 n Q T +2ε 2, I:= {1,...,M}, (3.3) v ε (0,x)=v 0 (x) n R N, where v ε =(v1, ε,vm ε ), v 0 =(u 0,...,u 0 ), F ε (t, x, r, p t,p x,x) (3.4) { } =max p t + mn 0 s ε 2, e ε Lα (t + s ε 2,x+ e, r,p x,x); r M r, and L and M are defned below (1.1) and (2.1) respectvely. When k and ε are small, the soluton of ths system s expected to be close to the soluton of (1.1) (remember that A = {α 1,...,α M }!). In fact we have the followng result. Lemma 3.3. Assume (A1). There exsts a unque soluton v ε of (3.3) satsfyng v ε 1 K, v ε vj ε 0 k, and, for k small, max u I vε 0 C(ε + k 1/3 ), where u solves (1.1) and (1.2),, j I,and K, C only depend on T and K from (A1). Proof. 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 0 C. Herev 0 s the soluton of (3.3) wth ε = 0,.e. of (2.3) n Secton 2. Theorem 2.3 then mples that u v 0 0 Ck 1/3. Fnally, an argument gven n the proof of Theorem 2.3 n Secton 2 shows that 0 max v ε mn v ε k n Q T +ε2. The constants C depend only on T and K from (A1). In order to smplfy the arguments n ths proof (to have the smplest possble formulaton of Lemma 3.5 below), we have defned the solutons of the equaton wth shaken coeffcents (3.3) n a slghtly larger doman than Q T. The (almost) smooth supersolutons of (1.1) we are lookng for are bult out of v ε by mollfcaton. But to have ntal data at t = 0 after mollfcaton of v ε,we have to shft v ε n tme and consder v ε (t, x) :=v ε (t ε 2,x). The functon v ε s defned on Q ε T := ( ε2,t + ε 2 ] R N, and by Lemma 3.3 t satsfes v ε 1 K and hence (3.5) v ε v ε 0 Kε for I. We mollfy v ε and defne v ε := ρ ε v ε for I, where ρ ε s the mollfer defned at the end of the ntroducton. Note that v ε s defned on Q T +ε 2, and n vew of Lemma 3.3, nequalty (3.5), and propertes of mollfers t satsfes v ε 1 K, andfork small, (3.6) v ε, v ε,j C(k + ε) nq T +ε 2, max u v ε, C(ε + k 1/3 )nq T, I where, j I and the constants C only depend on T and K from (A1). Now we are n a poston to defne our almost smooth supersolutons w, w := mn v ε for I. I

13 ERROR BOUNDS 1873 The way we have bult v ε, t turns out that w wll be a supersoluton of (1.1) n Q T when ε s small compared wth k. Ths s a consequence of the followng lemma. Lemma 3.4. Assume (A1) and 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 α j (t, x, v εj (t, x),dv εj (t, x),d 2 v εj (t, x)) 0. The proof of ths lemma wll be gven at the end of ths secton. The key consequence s the next lemma, whch shows that w s an approxmate supersoluton of the scheme (1.3). Ths result s the cornerstone of the proof of the lower bound, and ts proof should explan the name almost smooth supersoluton for w. Lemma 3.5. Assume (A1) and 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. 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 α j (t, x, v εj (t, x),dv εj (t, x),d 2 v εj (t, x)) E 2 ( K,h,ε), and the proof s complete after an applcaton of Lemma 3.4. It s now straghtforward to derve the lower bound; we smply choose k = 8sup [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 nequaltes (3.6), for small enough ε (and hence small enough k) wehave w u 0 C(ε + k + k 1/3 ), and therefore u h u e µt (u h,0 u 0 ) te µt E 2 ( K,h,ε)+C(ε + k + k 1/3 ) n G h, for some constant C. In vew of our choce of k, we get the lower bound by mnmzng w.r.t ε. Note that here ε wll be an ncreasng functon of h and therefore the requrement that ε s small mples that h must be small as well. Ths concludes theproofnthecasewhena s fnte. 2. The case of general A. By (A2) A s a separable metrc space and hence has a countable dense subset A. Furthermore, by (A2) the coeffcents are contnuous n α and therefore sup L α (t, x, r, p, X) =supl α (t, x, r, p, X). A A In other words, under assumpton (A2) we may replace A by A n equaton (1.1).

14 1874 GUY BARLES AND ESPEN R. JAKOBSEN Now take a sequence of sets {A M } M=1 A such that A M A M+1 for M N and A M = A. M=1 Let u denote the soluton of (1.1) and (1.2) and let u M be the soluton of (1.1) and (1.2) when A s replaced by A M. By Proposton 2.1 we have the followng C 0,1 bounds: u 1 + u M 1 K, where K only depends on T and K from (A1) and not on M. Arzela Ascol s theorem then mples that a subsequence of {u M } M, also denoted {u M } M,converges locally unformly. It s an easy exercse to prove that the lmt functon solves (1.1) and (1.2), and hence by unqueness s equal to u. By part 1 of ths proof we have that u M h u M lower bound, where u M h solves a scheme (1.3) related to equaton (1.1) when A s replaced by A M. In fact we do not need (or want!) to ntroduce u M h at all. To see ths note that the almost smooth supersoluton w of part 1 s a supersoluton of (1.1) also for general A. Now by re-examnng the proof of Lemma 3.5 we see that we can replace u M h by u h n that lemma, and repeatng the rest of part 1 then leads to u h u M lower bound. The lower bound on u u h can then be obtaned by sendng M after notng that the lower bound mentoned above (see part 1) does not depend on M. 3. The proof of Lemma 3.4. Fx an arbtrary pont (t, x) Q T and set j := argmn I v ε (t, x). By defnton of M and j we have v εj (t, x) M j v ε (t, x) =max {v εj(t, x) v ε (t, x) k} k, j and by Hölder contnuty of v ε (Lemma 3.3) and propertes of mollfers we see that v j ε (t, x) M j v ε (t, x) k +2max[v ε ] 1 2ε. Usng agan Hölder contnuty of v ε, for any ( t, x) Q ε T,weget v j ε ( t, x) M j v ε ( t, x) k +2max[v ε ] 1 (2ε + x x + t t 1/2 ). Now we conclude that f x x <ε, t t <ε 2,andε (8 max [v ε] 1) 1 k,then v j ε ( 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 0 s ε 2, e ε Lα j ( t + s, x + e, v j ε ( 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, t v j ε (t s, x e)(t, x) (3.7) + L α j (t, x, v j ε (t s, x e),d v j ε (t s, x e),d 2 v j ε (t s, x e)) 0. In other words, for every 0 s<ε 2 and e <ε, v j ε (t s, x e) sa(vscosty) supersoluton at (t, x) of (3.8) χ t + L α j (t, x, χ, Dχ, D 2 χ)=0.

15 ERROR BOUNDS 1875 By mollfyng (3.7) w.r.t. (s, e), we see that v εj s also a (vscosty) supersoluton of(3.8)at(t, x) and hence a 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 (3.8); we refer to the proof of Theorem 2.3 and to the Appendx n [1] for the detals. We conclude the proof of Lemma 3.4 by notng that snce v εj ssmooth,tsnfacta classcal supersoluton of (1.1) at (t, x). Ths completes the proof of Theorem 3.1 (b). 4. Monotone fnte dfference methods In ths secton, we apply our man result to fnte dfference approxmatons of (1.1) based on the ϑ-method of approxmaton n tme and two dfferent approxmatons n space: one proposed by Kushner [12] whch s monotone when a s dagonally domnant and a (more) general approach based on drectonal second dervatves proposed by Bonnans and Zdan [5], but see also Dong and Krylov [10]. 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) as { } 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,wherethestencl S s a fnte subset of Z N \{0}, andwhere (4.2) C α h (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 R N and defne (4.3) L α hφ = N =1 [ a α 2 + ( a α+ j 2 + j aα j j 2 j ) + b α+ δ + b α δ ] φ,

16 1876 GUY BARLES AND ESPEN R. JAKOBSEN where b + =max{b, 0}, b =( b) + (b = b + b ), and δ ± 1 w(x) =± 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)}, 1 jw(x) = 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)}. The stencl s S = {±e, ±(e ± e j ):, j =1,...,N}, and t s easy to see that the coeffcents n (4.1) are Ch α (t, x, ±e )= aα (x) 2 x 2 a α j (x) 4 x 2 j + bα± (x) x, Ch α (t, x, e h ± e j h)= 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 dagonally 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)β T 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 (4.6) L α hφ = β S ā α β β φ + N =1 [ b α+ δ + b α δ ] φ, where β s an approxmaton of Dβ 2 gven by β w(x) = 1 β 2 {w(x + β x) 2w(x)+w(x β x)}. x2

17 ERROR BOUNDS 1877 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, =1,...,N, Ch α (t, x, ±β) = āα β (t, x) β 2 x 2, 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 ) 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: u(t, x) =u(t t, x) (1 ϑ) t sup{ L α (4.8) hu c α u f α }(t t, x) α A ϑ t sup{ L α hu c α u f α }(t, x) n G + h. α A The cases ϑ =0andϑ = 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 {[ 1 S(h, t, x, r, [u] t,x )=sup α A t ϑcα + ϑ ] Ch α (t, x, β) r β S [ 1 t +(1 ϑ)cα (1 ϑ) ] Ch α (t, x, β) [u] t,x ( t, 0) β S [ ]} 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 ( tϑ c α (t, x) ) (4.10) 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 )).

18 1878 GUY BARLES AND ESPEN R. JAKOBSEN The (1 ϑ) t-term s a nonstandard 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. 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 Crank 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.

19 ERROR BOUNDS 1879 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[10]twasprovedthatfanoperatorL 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 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 (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, =1 j where the coeffcents are nonnegatve f and only f a s dagonally 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, 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 mples 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 semdscretzaton 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, (5.2) 1 [ ] 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 β 0 t D β φ γ β 0 tδ β, β I

20 1880 GUY BARLES AND ESPEN R. JAKOBSEN where β =(β 0,β ) I for β 0 N and β N N, and n a smlar way, 1 [ ] S h (t n,t n 1 ) 1 φ(t n 1,x) F (t, x, φ, Dφ, D 2 φ) t=tn t (5.3) K c β 0 t D β φ γ β t δ β 0, 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 C( u 0 u h,0 0 + t 1 10 r 1 ) u u h C( u 0 u h,0 0 + t 1 4 r 2 ) n R N,where { } δ β r 1 := mn β I 3(2β 0 + β, 1)γ β +1 { r 2 := mn β Ī δ β (2β 0 + β 1) γ β +1 where β denotes the sum of the components of β. Proof. We defne S(h, t n,x,u h, [u h ] tn,x) = 1 ( ) u h (t n,x) [u h ] tn,x, t where [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 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 β I }, β 0 t D β φ γ β 0 tδ β, whch leads to E 1 ( K,h,ε)= 1 2 Kε 1 4 t + K c ( Kε 1 2β 0 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 prove the lower bound. Remark 5.1. In vew of the consstency requrements (5.2) and (5.3), for schemes such as (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 β I

21 ERROR BOUNDS 1881 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 led 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 semgroups are obtaned and used to study the rate of convergence for splttng problems Semdscrete splttng. We consder an equaton of the form (5.4) u t + F 1 (D 2 u)+f 2 (D 2 u)=0 n Q T, where F j (X) =sup{ tr[a α j X] fj α }, 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 = tof (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)=0andu t + F 1 (D 2 u)=0. 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 )ntotself, they are monotone, and they satsfy the followng comparson prncple: S(t)φ 1 S(t)φ 2 (φ 1 φ 2 ) + 0, for φ 1,φ 2 W 1, (R N )andwhere 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, contrary to the case of fnte dfference 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. The upper and lower bounds are dfferent due to convexty of the equaton.

22 1882 GUY BARLES AND ESPEN R. JAKOBSEN Remark 5.3. To smplfy the presentaton we have only stated the prncpal error terms n Lemma 5.2,.e., the terms determnng the rate n Corollary 5.3 below. To see whch terms are prncpal, one must look also at the proof of Corollary 5.3. The h.o.t. category contans the terms that are not prncpal, both hgh and low order terms, and maybe a better name 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,andletu 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: 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 Corollares 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 nonncreasng. Defne F δ by F δ = F ρ δ,