CLOSED-FORM LIKELIHOOD EXPANSIONS FOR MULTIVARIATE DIFFUSIONS. BY YACINE AÏT-SAHALIA 1 Princeton University

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1 The Annals of Statstcs 2008, Vol. 36, No. 2, DOI: / Insttute of Mathematcal Statstcs, 2008 CLOSED-FORM LIKELIHOOD EPANSIONS FOR MULTIVARIATE DIFFUSIONS B ACINE AÏT-SAHALIA 1 Prnceton Unversty Ths paper provdes closed-form expansons for the log-lkelhood functon of multvarate dffusons sampled at dscrete tme ntervals. The coeffcents of the expanson are calculated explctly by explotng the specal structure afforded by the dffuson model. Examples of nterest n fnancal statstcs and Monte Carlo evdence are ncluded, along wth the convergence of the expanson to the true lkelhood functon. 1. Introducton. Dffusons and, more generally, contnuous-tme Markov processes are generally specfed n economcs and fnance by ther evoluton over nfntesmal nstants, that s, by wrtng down the stochastc dfferental equaton followed by the state vector. However, for most estmaton strateges relyng on dscretely sampled data, we need to be able to nfer the mplcatons of the nfntesmal tme evoluton of the process for the fnte tme ntervals at whch the process s actually sampled, say daly or weekly. The transton functon plays a key role n that context. Unfortunately, the transton functon s, n most cases, unknown. At the same tme, contnuous-tme models n fnance, whch untl recently have been largely unvarate, now predomnantly nclude multple state varables. Typcal examples nclude asset prcng models wth multple explanatory factors, term structure models wth multple yelds or factors and stochastc volatlty or stochastc mean reverson models (see Sundaresan [28] for a recent survey). Motvated by ths trend and the need for effectve representaton methods, I construct closedform expansons for the log-transton functon of a large class of multvarate dffusons. Because dffusons are Markov processes, the log-lkelhood functon of observatons from such a process sampled at fnte tme ntervals reduces to the sum of the log-transton functon of successve pars of observatons. A closed form expanson for the latter therefore makes quas-lkelhood nference feasble for these models. The paper s organzed as follows. Secton 2 sets out the model and assumptons. In Secton 3, I ntroduce the concept of reducblty of a dffuson and provde a necessary and suffcent condton for the reducblty of a multvarate dffuson. In an earler work (Aït-Sahala [2]), I constructed explct expansons for the tran- Receved May 2004; revsed January Supported n part by NSF Grants SES and DMS AMS 2000 subject classfcatons. Prmary 62F12, 62M05; secondary 60H10, 60J60. Key words and phrases. Dffusons, lkelhood, expansons, dscrete observatons. 906

2 LIKELIHOOD FOR DIFFUSIONS 907 ston functon of unvarate dffusons based on Hermte seres. The natural extenson of the Hermte method to the multvarate case s applcable only f the dffuson s reducble, whch all unvarate, but few multvarate, dffusons are. So, ths paper proposes an alternatve method, whch determnes the coeffcents n closed form by requrng that the expanson satsfes the Kolmogorov equatons descrbng the evoluton of the process up to the order of the expanson tself. When a dffuson s reducble, the coeffcents of the expanson are obtaned as a seres n the tme varable, whch I show n Secton 4. When the dffuson s not reducble, the expanson nvolves a double seres n the tme and state varables, descrbed n Secton 5. Secton 6 then studes the convergence of the lkelhood expanson and the resultng maxmzer to the theoretcal (but ncomputable) maxmum lkelhood estmator. Secton 7 contans examples of multvarate dffusons and Monte Carlo smulaton results. Proofs are n Secton 8 and Secton 9 concludes the paper. 2. Setup and assumptons. Consder the multvarate tme-homogenous dffuson (1) d t = μ( t )dt σ( t )dw t, where t and μ( t ) are m 1 vectors, σ( t ) s an m m matrx and W t s an m 1 vector of ndependent Brownan motons. Independence s wthout loss of generalty snce arbtrary correlaton structures between the shocks to the dfferent equatons can be modeled through the ncluson of off-dagonal terms n the σ matrx, whch, furthermore, need not be symmetrc. In tme-nhomogeneous dffusons, the coeffcents are allowed to depend on tme drectly, as n μ( t,t) and σ( t,t), beyond ther dependence on tme va the state vector. The tmenhomogeneous case can be reduced to the tme-homogenous case by treatng tme as an addtonal state varable and so t suffces to return to the model specfed n (1). The objectve of ths paper s to derve closed-form approxmatons to the log of the transton functon p (x x 0, ), that s, the condtonal densty of t = x gven t = x 0 nduced by the model (1). From an nference perspectve, the prmary use of ths constructon s to make feasble the computaton of the MLE. Assume that we parametrze (μ, σ ) as functons of a parameter vector θ and observe at dates {t = = 0,...,n}, where >0 s fxed. The Markovan nature of (1), whch the dscrete data nhert, mples that the log-lkelhood functon has the smple form n ( (2) l n (θ, ) l ( 1), ), where l ln p and where the asymptotcally rrelevant densty of the ntal observaton, 0, has been left out. In practce, the ssue s that for most models of nterest, the functon p, hence l, s not avalable n closed form. I wll use the followng notaton. Let S, a subset of R m, denote the doman of the dffuson, assumed, for smplcty, to be of the followng form.

3 908. AIT-SAHALIA ASSUMPTION 1. S s a product of m ntervals wth lmts x and x,where possbly x = and/or x =, n whch case, the ntervals are open at nfnte lmts. I wll use T to denote transposton and, for a functon η(x) = (η 1 (x),..., η d (x)) T, dfferentable n x, I wll wrte η(x) for the Jacoban matrx of η, that s, the matrx η(x) =[ η (x)/ x j ],...,d;j=1,...,m. For x R m, x denotes the usual Eucldean norm. If a =[a j ],j=1,...,m s an m m nvertble matrx, then I wrte a 1 for the matrx nverse, wth elements [a 1 ] j.det[a] and tr[a] denote the determnant of a and ts trace, respectvely. If a =[a ],...,m s a vector, tr[a] denotes the sum of the elements of a. a = dag[a ],...,m denotes the m m dagonal matrx wth dagonal elements a. When a functon η(x) s nvertble n x, I wrte η nv (y) for ts nverse. In some nstances, t may be more natural to drectly parametrze the nfntesmal varance covarance matrx of the process (3) v(x) σ(x)σ T (x) than σ(x) tself. Every characterzaton of the process, such as ts transton probablty, depends, n fact, on (μ, v). In partcular, t can be shown that, should there exst a contnuum of solutons n σ to equaton (3), then the transton probablty of the process s dentcal for each of these σ (see Remark and Secton 5.3 n Stroock and Varadhan [27]). Ths s also qute clear from nspecton of the nfntesmal generator A of the process, whch depends on v rather than σ.for functons f(,x)that are sutably dfferentable on ts doman, A has the acton f (x, ) m A f = f (x, ) μ (x) 1 v j (x) 2 f(x, ) (4). x 2 x x j,j=1 The doman of A ncludes at least those functons that, for each x 0 S, are once contnuously dfferentable n n R, twce contnuously dfferentable n x S and have compact support. As ths wll play a role n the lkelhood expansons, defne (5) D v (x) 1 2 ln(det[v(x)]). I wll assume that ths matrx v satsfes the followng regularty condton: ASSUMPTION 2. of S. The matrx v(x) s postve defnte for all x n the nteror Further assumptons are requred to ensure the exstence and unqueness of a soluton to (1) and to make the computaton of lkelhood expansons possble. I wll assume the followng. ASSUMPTION 3. μ(x) and σ(x) are nfntely dfferentable n x on S.

4 LIKELIHOOD FOR DIFFUSIONS 909 Assumpton 3 ensures the unqueness of solutons to (1). Indeed, Assumpton 3 mples n partcular, that the coeffcents of the stochastc dfferental equaton are locally Lpschtz under ther assumed (once) dfferentablty, whch can be seen by applyng the mean value theorem. Ths ensures that a soluton, f t exsts, wll be unque (see, e.g., Theorem n Karatzas and Shreve [21]). The nfnte dfferentablty assumpton n x s unnecessary for that purpose, but t allows the computaton of expansons of the transton densty, whch, as we wll see, nvolves repeated dfferentaton of the coeffcent functons μ and σ. There exst models of nterest n fnance, such as Feller s square root dffuson used n the Cox, Ingersoll and Ross model of the term structure, that fal to satsfy the Lpschtz condton snce they volate the dfferentablty requrement of Assumpton 3 at a boundary of S : for nstance, σ(x) = σ 0 x 1/2 s not dfferentable at the left boundary 0 of S. It s then possble to weaken Assumpton 3 to cover such cases (see Watanabe and amada [30] and amada and Watanabe [32]). The next assumpton restrcts the growth behavor of the coeffcents near the boundares of the doman. ASSUMPTION 4. The drft and dffuson functons satsfy lnear growth condtons, that s, there exsts a constant K such that for all x S and, j, (6) μ (x) K(1 x ) and σ j (x) K(1 x ). Ther dervatves exhbt at most polynomal growth. TheroleofAssumpton4 s to ensure the exstence of a soluton to the stochastc dfferental equaton (1) by preventng explosons of the process n fnte expected tme. Whle t can be relaxed n specfc examples, t s not possble to do so n full generalty. In dmenson one, however, fner results are avalable (see the Engelbert Schmdt crteron n Theorem of Karatzas and Shreve [21]), allowng lnear growth to be mposed only when the drft coeffcent pulls the process toward an nfnty boundary (see Proposton 1 of Aït-Sahala [2]). In all dmensons, the lnear growth condton n Assumpton 4 s only an ssue near the boundares of S. In the specal case where S s compact, the growth condton (boundedness, n fact) follows from the contnuty of the functons. The addtonal assumpton that the dervatves of the drft and dffuson functons grow at most polynomally smplfes matters n lght of the exponental tals of the transton densty p. Fnally, the dffuson process s fully defned by the specfcaton of the functons μ and σ and ts behavor at the boundares of S. In many examples, the specfcaton of μ and σ predetermnes the boundary behavor of the process, but ths wll not be the case for models that represent lmtng stuatons. For nstance, n Cox, Ingersoll and Ross processes wth affne μ and v, the behavor at the 0 boundary depends upon the values of the parameters. When ths stuaton occurs for a partcular model, the behavor of the lkelhood expanson near such a boundary wll be specfed exogenously to match that of the assumed model.

5 910. AIT-SAHALIA 3. Reducble dffusons. Whenever possble, I wll frst transform the dffuson nto one that s more amenable to the dervaton of an expanson for ts transton densty. For that purpose, I ntroduce the followng defnton. DEFINITION 1(Reducblty). The dffuson s sad to be reducble to unt dffuson (or reducble, n short) f and f only f there exsts a one-to-one transformaton of the dffuson nto a dffuson whose dffuson matrx σ s the dentty matrx. That s, there exsts an nvertble functon γ(x), nfntely dfferentable n on S, such that t γ( t ) satsfes the stochastc dfferental equaton (7) on the doman S. d t = μ ( t )dt dw t By Itô s lemma, when the dffuson s reducble, the change of varable γ satsfes γ(x)= σ 1 (x). Every scalar (.e., one-dmensonal) dffuson s reducble, by means of the smple transformaton t (8) du t γ( t ) = σ(u), where the lower bound of ntegraton s an arbtrary pont n the nteror of S. The dfferentablty of γ ensures that μ satsfes Assumpton 3. Ths change of varable, known as the Lampert transform, played a crtcal role n the dervaton of closed-form Hermte approxmatons to the transton densty of unvarate dffusons n Aït-Sahala [2]. How to deal wth the case where 1/σ (u) cannot be ntegrated n closed form s dscussed after Proposton 2 below. Whenever a dffuson s reducble, an expanson can be computed for the transton densty p of by frst computng t for the densty p of the reduced process and then transformng back nto, essentally proceedng by extendng the unvarate method. However, not every multvarate dffuson s reducble. Whether or not a gven multvarate dffuson s reducble depends on the specfcaton of ts σ matrx, n the followng way. PROPOSITION 1 (Necessary and suffcent condton for reducblty). The dffuson s reducble f and only f σ k (x) σ j (x) (9) σ lj (x) = σ lk (x) x l=1 l x l=1 l for each x n S and trplet (,j,k)= 1,...,msuch that k>j.if σ s nonsngular, then the condton can be expressed as (10) [σ 1 ] j (x) x k = [σ 1 ] k (x) x j.

6 LIKELIHOOD FOR DIFFUSIONS 911 Smlar restrctons on the σ matrx arse n dfferent contexts; see Doss [11] who studed the queston of when the soluton of the SDE can be expressed as a functon of the Brownan moton W and the soluton of an ODE and the concept of commutatve nose n Secton 10.6 of Cyganowsk, Kloeden and Ombach [8] where they show that restrctng the σ matrx leads to a smplfcaton of the Mlshten scheme for. In the bvarate case m = 2, condton (10) reduces to [σ 1 ] 11 (x) x 2 [σ 1 ] 12 (x) x 1 = [σ 1 ] 21 (x) x 2 [σ 1 ] 22 (x) x 1 = 0. For nstance, consder dagonal systems: f σ 12 = σ 21 = 0, then the reducblty condton becomes [σ 1 ] 11 / x 2 = [σ 1 ] 22 / x 1 = 0. Snce [σ 1 ] = 1/σ n the dagonal case, reducblty s equvalent to the fact that σ depends only on x for each = 1, 2. Ths s true more generally n dmenson m. Note that ths s not the case f off-dagonal elements are present. Another set of examples s provded by the class of stochastc volatlty models. Consder the two models where ether σ(x)= ( σ11 (x 2 ) 0 0 σ 22 (x 2 ) ) or σ(x)= ( a(x1 ) a(x 1 )b(x 2 ) 0 c(x 2 ) In the frst model, the process s not reducble n lght of the prevous dagonal example, as ths s a dagonal system where σ 11 depends on x 2.However,nthe second, the process s reducble. 4. Closed-form expanson for the lkelhood functon of a reducble dffuson. When the dffuson s reducble, I propose two approaches to construct a sequence of explct expansons for the log-lkelhood functon. The frst s based on computng the coeffcents of a Hermte expanson for the densty of the transformed process, p. The coeffcents are found n the form of a seres expanson n, the tme separatng successve observatons. The second approach takes the form of the Hermte seres and determnes ts coeffcents by solvng the Kolmogorov partal dfferental equatons whch characterze the transton functon p. In both cases, gven a seres for p, I obtan a seres for the orgnal object of nterest, p, by reversng the change of varable and the Jacoban formula. The two approaches yeld the same fnal seres Multvarate Hermte expansons. To motvate the form of the expanson that I wll propose n the multvarate case, n both the reducble and rreducble cases, consder the followng natural multvarate counterpart to the unvarate Hermte expanson of Aït-Sahala [2]. Let φ(x) denote the densty of the m-dmensonal multvarate Normal dstrbuton wth mean zero and dentty covarance matrx. For each vector h = (h 1,...,h m ) T N m, recall that tr[h] =h 1 h m and let H h (x) denote the assocated Hermte polynomals, whch are defned by H h (x) = (( 1) tr[h] /φ(x)) tr[h] φ(x)/dx h 1 1 dxh m m and ).

7 912. AIT-SAHALIA can be computed explctly to an arbtrary order tr[h] (see, e.g., Chapter 5 of Mc- Cullagh [23] or Wthers [31]). The polynomals are orthonormal n the sense that R m H h(x)h k (x)φ(x) dx = h 1! h m! f h = k and 0 otherwse. The Hermte seres approxmaton of p s n the form (11) p (J ) (y y 0, )= m/2 φ ( ) y y0 1/2 h N m :tr[h] J η h (, y 0 )H h ( y y0 1/2 and the Hermte coeffcents η h (, y 0 ) can be computed as n the unvarate case: by orthonormalty of the Hermte polynomals, the coeffcents η h are gven by the condtonal expectaton 1 η h (, y 0 ) = h 1! h m! E[ ( H h 1/2 ( t y 0 ) ) ] (12) t = y 0. Ths expresson s then amenable to computng an expanson n usng the generator (4). To evaluate that condtonal expectaton, the determnstc Taylor expanson E 1 [f(, 0, ) 0 = y 0 ] (13) = K k=0 k k! Ak f(y,y 0,δ) y=y0,δ=0 O( K1 ) can be used, where A s the nfntesmal generator of the process, the functon f s suffcently dfferentable n (y, δ) and ts terates by applcaton of A up to K tmes reman n the doman of A, as n Aït-Sahala [2]. The result wll be a small-tme expanson, n the same sprt as n Azencott [4] and Dacunha-Castelle and Florens-Zmrou [9], except that the expansons gven here are n closed form nstead of relyng on moments of functonals of Brownan brdges (whch are to be computed by smulaton). Replacng the unknown η h n (11) by ther expansons n to order K gves rses to an expanson of p (J ) where the coeffcents are gathered n ncreasng powers of, whch I denote p (J,K). If we gather the terms n the rght-hand sde of (11) accordng to powers of, we can rewrte p (J,K) n the form of a truncated seres n, (14) p (J,K) (y y 0, )= m/2 φ ( 1/2 (y y 0 ) ) K k=0 c (J,k) (y y 0 ) k k!. For the log-transton densty and for any gven J, or n the unvarate case where the convergence of the Hermte seres s establshed as J, the resultng expanson has the form ) (15) l (K) (y y 0, )= m 2 ln(2π ) C( 1) (y y 0 ) K k=0 C (k) (y y 0) k k!,

8 LIKELIHOOD FOR DIFFUSIONS 913 whose coeffcents C (k),k= 1, 0, 1, 2,...,K, are combnatons of the coeffcents of (11) obtaned by dentfyng the terms n the expanson n of the log of (14). The method just descrbed s the natural extenson to the multvarate settng of the Hermte approach employed n the unvarate case n Aït-Sahala [2]. Extensons of the unvarate Hermte expanson results n two dfferent unvarate drectons have been recently developed for tme-nhomogeneous dffusons (Egorov, L and Hu [13]) and for models drven by Lévy processes other than Brownan moton (Schaumburg [25] andu[33]). DPetro [10] has extended the methodology to make t applcable n a Bayesan settng. The Hermte method requres, however, that the dffuson be reducble snce the straght Hermte expanson wll not n general converge f appled to p drectly nstead of p. And as dscussed above, whle all unvarate dffusons are reducble, so that such a exsts, not all multvarate dffusons are. Ths necesstates an alternatve method, whch I now develop Connecton to the Kolmogorov equatons. An alternatve method to obtan an explct expanson for l s to take nspraton from the form of the soluton gven by the expanson (15) and to use the Kolmogorov equatons to determne ts coeffcents, wthout any further reference to the Hermte expanson. As s often the case when a dfferental operator s nvolved, t s easer to verfy that a gven functonal form, n ths case the expanson n the form (15), s the rght soluton. Consder the forward and backward Kolmogorov equatons (see, e.g., Secton 5.1 of Karatzas and Shreve [21]) 2 p (y y 0, ) (16) (17) p (y y 0, ) p (y y 0, ) = = {μ (y)p (y y 0, )} y 1 2 μ (y 0 ) p (y y 0, ) y y 2 2 p (y y 0, ). The soluton p nherts the smoothness n (,y,y 0 ) of the coeffcents μ (see, e.g., Secton 9.6 n Fredman [14]), so we are enttled to look for an approxmate soluton n the form of a smooth expanson. The fact that the Hermte expanson turns out to have exactly the rght form for solvng the forward and backward equatons term by term s an nterestng feature of these expansons. Focusng for now on the forward equaton (16), the equvalent form for the loglkelhood l (whch s the object of nterest for MLE and whch wll turn out to lead to a smple lnear system) s (18) l (y y 0, ) = 1 2 μ (y) y 2 l (y y 0, ) y 2 y 2 0 μ (y) l (y y 0, ) y 1 ( ) l (y y 0, ) 2. 2 y,

9 914. AIT-SAHALIA Suppose that we substtute the postulated form of the soluton (15) nto(18). Snce l (K) (y y 0, ) = C( 1) (y y 0 ) 2 m l (K) (y y 0, ) = 1 C ( 1) (y y 0 ) y y 2 l (K) (y y 0, ) y 2 = 1 2 C ( 1) (y y 0 ) y 2 K 1 2 k=1 K C ( 1) k=0 K k=0 C (k) (y y 0) k 1 (k 1)! (y y 0 ) y 2 C ( 1) k k! (y y 0 ) y 2 k k!, equatng the coeffcents of 2 on both sdes of (18) mples that the leadng coeffcent n the expanson, C ( 1), must solve the nonlnear equaton (19) C ( 1) (y y 0 ) = 1 ( C ( 1) (y y 0 ) 2 y ) T ( C ( 1) (y y 0 ) y Because the densty must approxmate a Gaussan densty as 0, the approprate soluton s the one wth a strct maxmum at y = y 0, namely ). (20) C ( 1) (y y 0 ) = 1 2 y y 0 2 = 1 2 (y y 0 ) 2. Consderng now the coeffcents of 1 on both sdes of (18), we see that C (0) (y y 0) (y y 0 ) = μ (y)(y y 0 ). y Integratng along a lne segment between y 0 and y, we obtan 1 C (0) (y y ( (21) 0) = (y y 0 ) μ y0 u(y y 0 ) ) du, wth ntegraton constants determned n the proof of the theorem below usng boundary condtons and the backward equaton. The hgher-order coeffcents are obtaned usng the same prncple, and we have the followng result. THEOREM 1. The coeffcents of the log-densty expanson l (K) (y y 0, ) are gven explctly by (20), (21) and, for k 1, 0 (22) C (k) (y y 0) = k 1 0 G (k) ( ) y0 u(y y 0 ) y 0 u k 1 du.

10 The functons G (k) (23) and, for k 2, (24) are gven by G (1) (y y 0) = G (k) (y y 0) = 1 2 LIKELIHOOD FOR DIFFUSIONS k 1 μ (y) y { 2 C (0) (y y 0) y 2 μ (y) C(k 1) (y y 0 ) y 1 2 h=0 ( ) k 1 C (h) h μ (y) C(0) (y y 0) y [ C (0) (y y 0) y (y y 0) y ] 2 } 2 C (k 1) (y y 0 ) y 2 C (k 1 h) (y y 0 ) y. Theorem 1 provdes the explct form of l (K) that solves the Kolmogorov equatons to the desred order K. Ths does not necessarly mply that l (K) s a proper Taylor expanson of l at the desred order K 1 ; ths wll be establshed as part of Theorem 3 below Change of varable. Gven l, the expresson for l s gven by the Jacoban formula l (x x 0, )= 1 2 ln(det[v(x)]) l (, γ (x) γ(x 0 )) (25) = D v (x) l (, γ (x) γ(x 0 )), whch I mmc at the level of the approxmatons of order K n, thereby defnng l (K) as l (K) (x x 0, )= D v (x) l (K) (, γ (x) γ(x 0 )) (26) = m 2 ln(2π ) D v(x) C( 1) (γ (x) γ(x 0 )) K k=0 C (k) (γ (x) γ(x 0)) k k! from l (K) gven n (15), usng the coeffcents C (k),k = 1, 0,...,K,gvenn Theorem 1. By constructon, l (K) solves the Kolmogorov equatons for at the same order. 5. Closed-form expanson for the log-lkelhood functon of an rreducble dffuson. In the reducble case, the two approaches (Hermte and soluton of the Kolmogorov equatons) concde. When the dffuson s rreducble, however, one

11 916. AIT-SAHALIA no longer has the opton of frst transformng to, computng the Hermte expanson for and then, va the Jacoban formula, transformng t nto an expanson for. But, t remans possble to postulate an approprate form of an expanson for l and then to determne that ts coeffcents satsfy the Kolmogorov equatons to the relevant order, as follows. Mmckng the form of the expanson n obtaned n the reducble case, namely (26), leads to the postulaton of the followng form for an expanson of the log-lkelhood (27) l (K) (x x 0, )= m 2 ln(2π ) D v(x) C( 1) (x x 0) K k=0 C (k) (x x 0) k k! and solvng for the coeffcents usng the Kolmogorov equatons. The expanson exsts because the log-transton functon nherts the smoothness of the coeffcents (μ, v) (see, e.g., Secton 9.6 of Fredman [14]). When wrtten drectly for the process, as requred n the rreducble case, the equatons take the form (28) (29) l (x x 0, ) l (x x 0, ) = =,j= μ (x) x 1 2,j=1 μ (x) l (x x 0, ) x,j=1,j=1 2 v j (x) x x j v j (x) l (x x 0, ) x x j v j (x) 2 l (x x 0, ) x x j l (x x 0, ) v j (x) l (x x 0, ), x x j μ (x 0 ) l (x x 0, ) x 0,j=1,j=1 v j (x 0 ) 2 l (x x 0, ) x 0 x 0j l (x x 0, ) v j (x 0 ) l (x x 0, ). x 0 x 0j

12 LIKELIHOOD FOR DIFFUSIONS 917 The soluton method s as follows: as n the reducble case, substtutng the postulated soluton (27) nto(28) provdes an equaton at each order n whch s solved for the correspondng coeffcent of the expanson. Whle the dfferental equaton for l s nonlnear, t can be transformed nto a lnear one by exponentaton and so the expanson l (K) constructed n ths way wll approxmate l. Start wth the equaton of order 2 whch determnes the leadng order coeffcent C ( 1). Whle the leadng coeffcent C( 1) n the case of a reducble dffuson s smply C ( 1) (x x 0) = γ(x) γ(x 0 ) 2 /2, the stuaton s more nvolved when the dffuson s not reducble. The equaton that determnes the coeffcent C ( 1) s obtaned by equatng the terms of order 2 n (28), yeldng C ( 1) (x x 0) = 1 ( ( 1) C (x x ) T ( 1) 0) C v(x)( (x x ) 0) (30). 2 x x The soluton of ths equaton s not explct n general, although t has a nce geometrc nterpretaton as mnus one half the square of the shortest dstance from x to x 0 n the metrc nduced n R m by the matrx v(x) 1 (see [29]) Tme and state expanson. The analyss of the coeffcent C ( 1) suggests that t wll generally be mpossble to explctly characterze the coeffcents of the expanson (27)snce(30) wll not n general admt an explct soluton. Ths s where the next step n the analyss comes nto play. The dea now s to derve an explct approxmaton n (x x 0 ) of the coeffcents C (k) (x x 0), k = 1, 0,...,K. In other words, I localze the log-lkelhood functon n both and x x 0. The key dfference between what can be done n the reducble specal case of Theorem 1 and n the general case of Theorem 2 s that the coeffcents of the expanson n can be obtaned drectly by (20) (22) wth no need for an expanson n the state varable. How ths works can be seen by once agan consderng the coeffcent C ( 1). Consder a quadratc [n (x x 0 )] approxmaton of the soluton to the equaton (30) determnng C ( 1). The constant and lnear terms are necessarly zero snce the matrx v(x) s nonsngular. Wrte the second-order expanson as C ( 1) (x x 0) = (1/2)(x x 0 ) T V(x x 0 ) o( x x 0 2 ). Equaton (30) mples the equaton V = Vv(x 0 )V, whose soluton s V = v 1 (x 0 ). As a consequence, the leadng term comng from the expanson of C ( 1) (x x 0) n x x 0 s (1/2 )(x x 0 ) T v(x 0 ) 1 (x x 0 ) so that the leadng term n the expanson for the log-densty wll correspond to that of a Normal wth mean x 0 and covarance matrx v(x 0 ). More generally, I wll derve a seres n (x x 0 ) for each coeffcent C (k), at some order j k n (x x 0 ). That expanson s to be denoted by C (j k,k).oneremanng queston s the choce of the order j k [n (x x 0 )] correspondng to a gven

13 918. AIT-SAHALIA order k (n ). For that purpose, note that 0 = O p ( 1/2 ),so (31) C (k) ( 0 ) k C (j k,k) ( 0 ) k = O p ( 0 j k k ) and therefore settng j k /2 k = K 1, that s, = O p ( j k/2k ) (32) j k = 2(K 1 k), for k = 1, 0,...,K,wll provde an approxmaton error due to the expanson n (x x 0 ) ofthesameorder K1 for each of the terms n the seres (27). The resultng expanson wll then be of the form (33) l (K) (x x 0, )= m 2 ln(2π ) D v(x) C(j 1, 1) (x x 0 ) K k=0 C (j k,k) (x x 0 ) k k!. Ths double expanson [n and n (x x 0 )] can be vewed, n probablty, as an expanson n only once the process s nserted n the lkelhood, n lght of (31). In general, the functon need not be analytc at = 0, hence ths should be nterpreted strctly as a seres expanson. Fnally, note that the term D v (x) whch arses n the reducble case from the Jacoban transformaton s ndependent of and so could be bult nto the C (0) coeffcent. Dong so, however, would subject t to beng expanded n x x 0, whch s unnecessary snce D v (x) s known. If D v (x) were beng expanded along wth C (j 0,0), we would lose the property that l (K) also solves the backward equaton (29) to the correspondng order n Determnaton of the coeffcents n the rreducble case. What remans to be done s to explctly compute the expanson C (j k,k) n x x 0 of each coeffcent C (k).let ( 1, 2,..., m ) denote a vector of ntegers and defne I k ={ ( 1, 2,..., m ) N m : 0 tr[] j k } so that the form of C (j k,k) s C (j k,k) (34) (x x 0 ) = β (k) I k (x 0 )(x 1 x 01 ) 1 (x 2 x 02 ) 2 (x m x 0m ) m. The coeffcents are determned one by one, startng wth the leadng term C (j 1, 1). Gven C (j 1, 1), the next term C (j 0,0) s calculated explctly, and so on. Based on (32), the hghest-order term (k = 1) s expanded to a hgher degree of precson j 1 than the successve terms. Ths s qute natural, gven that C (j 1, 1) s an nput to the dfferental equaton determnng C (j 0,0), andsoon.

14 LIKELIHOOD FOR DIFFUSIONS 919 In order to state the man result pertanng to the closed-form solutons C (j k,k), I defne the followng functons of the coeffcents and ther dervatves: (35) (36) G (0) (x x 0) = m 2 m G (1) 1 2 (x x 0) =,j=1,j=1,j=1 1 2,j=1 μ (x) C( 1) (x x 0) x v j (x) C ( 1) (x x 0) x x j v j (x) 2 C ( 1) (x x 0) x x j v j (x) C( 1) (x x 0) x μ (x) x 1 2,j=1 ( (0) C μ (x) (x x 0) x,j=1 2 v j (x) x x j D v (x) x j, D ) v(x) x ( (0) v j (x) C (x x 0) x D ) v(x) xj x j { 2 C (0) v j (x) (x x 0) 2 D v (x) x x j x x j ( C (0) (x x 0) D ) v(x) x x ( (0) C (x x 0) x j D v(x) x j )} and, for k 2, (37) G (k) (x x 0) = 1 2 μ (x) C(k 1) (x x 0 ) x,j=1,j=1 v j (x) 2 C (k 1) (x x 0 ) x x j v j (x) C (k 1) (x x 0 ) x x j

15 920. AIT-SAHALIA 1 2,j=1 {( (0) C v j (x) (x x 0) x k 2 h=0 ( ) k 1 h C(h) 2 D ) (k 1) v(x) C (x x 0 ) x x j (x x 0) x C (k 1 h) (x x 0 ) x j Note that the computaton of each functon G (k) requres only the ablty to dfferentate the prevously determned coeffcents C ( 1),..., C(k 1). The same apples to ther expansons. The followng theorem can now descrbe how the coeffcents C (j k,k), that s, the coeffcents β (k), I k, are determned. THEOREM 2. For each k = 1, 0,...,K, the coeffcent C (k) (x x 0) n (27) solves the equaton (38) where (39) (40) f ( 2) f ( 1) (x x 0 ) = 2C ( 1) (x x 0 ) =,j=1 f (k 1) (x x 0 ) = 0, (x x 0),j=1 v j (x) C( 1) (x x 0) x }. C ( 1) (x x 0) x j, v j (x) C( 1) (x x 0) C (0) (x x 0) G (0) x x (x x 0) j and, for k 1, f (k 1) (x x 0 ) = C (k) (x x 0) (41) 1 v j (x) C( 1) (x x 0) k x,j=1 C (k) (x x 0) G (k) x (x x 0), j where the functons G (k),k= 0, 1,...,K, are gven above. The coeffcents β(k) solve a system of lnear equatons, whose soluton s explct. G (k) nvolves only the coeffcents C(h) for h = 1,...,k 1, so ths system of equatons can be utlzed to solve recursvely for each coeffcent, meanng that the equaton f ( 2) = 0 determnes C ( 1) ; gven C( 1),G(0) becomes known and the equaton f ( 1) = 0 determnes C (0) ; gven C( 1) and C (0),G(1) becomes known and the equaton f (0) = 0 then determnes C(1), and so on.

16 LIKELIHOOD FOR DIFFUSIONS 921 Each of these equatons can be solved explctly n the form of the expanson C (j k,k) of the coeffcent C (k), at order j k n (x x 0 ). The coeffcents β (k) (x 0 ), I k,ofc (j k,k) are determned by settng the expanson f (j k,k 1) of f (k 1) to zero. The key feature that makes ths problem solvable n closed form s that the coeffcents solve a successon of systems of lnear equatons: frst determne β (k) for tr[] =0, then β (k) for tr[] =1 and so on, all the way to tr[] =j k.note, n partcular, for k = 1, β ( 1) = 0fortr[] =0, 1 (.e., the polynomal has no constant or lnear terms) and the terms correspondng to tr[]=2 (wth, of course, j 1 2) are β ( 1) (x 0 )(x 1 x 01 ) 1 (x m x 0m ) m I 1 :tr[]=2 = 1 2 (x x 0) T v 1 (x 0 )(x x 0 ), whch are the antcpated terms, gven the Gaussan lmtng behavor of the transton densty when s small. Thus, wth j 1 3, we only need to determne the terms β ( 1) correspondng to tr[]=3,...,j 1. Note that the soluton β ( 1) depends only on the specfcaton of the v matrx (the drft functons are rrelevant). For k = 0, β (0) = 0fortr[]=0, so the polynomal has no constant term. For k 1, the polynomals have a constant term (for k 1, β (k) 0fortr[]=0 n general). To obtan an expanson for the densty p nstead of for the log-densty l, one can ether take the exponental of l (K) or, alternatvely, expand the exponental n to obtan the coeffcents c for the expanson of p from the coeffcents C for the expanson of l. In general, lke a Hermte expanson, nether wll ntegrate to one wthout dvson by the ntegral over S of the densty expanson. Postvty s guaranteed, however, f one smply exponentates the log-transton functon Applyng the rreducble method to a reducble dffuson. Theorem 2 s more general than Theorem 1, n that t does not requre that the dffuson be reducble. As dscussed above, n exchange for that generalty, the coeffcents are avalable n closed form only n the form of a seres expanson n x about x 0. The followng proposton descrbes the relatonshp between these two methods when Theorem 2 s appled to a dffuson that s, n fact, reducble. PROPOSITION 2. Suppose that the dffuson s reducble and let l (K) denote ts log-lkelhood expanson calculated by applyng Theorem 1. Suppose, now, that we also calculate ts log-lkelhood expanson, l (K), wthout frst transformng nto the unt dffuson, that s, by applyng Theorem 2 to drectly. Then, each coeffcent C (j k,k) (x x 0 ) from l (K) s an expanson n (x x 0) at order j k of the coeffcent C (k) (x x 0) = C (k) (γ (x) γ(x 0)) from l (K).

17 922. AIT-SAHALIA In other words, applyng the rreducble method to a dffuson that s, n fact, reducble nvolves replacng the exact expresson for C (k) (x x 0) by ts seres n (x x 0 ). Of course, there s no reason to do so when the dffuson s reducble and the transformaton γ from to, defned n Defnton 1, s explct. However, Proposton 2 s relevant n the case where the dffuson s reducble, but the transformaton γ s not avalable n closed form. Ths can occur even n dmenson m = 1, where every dffuson s theoretcally reducble. For nstance, consder the specfcaton of the dffuson functon n the general nterest rate model proposed n [1], namely σ 2 (x) = θ 1 x 1 θ 0 θ 1 x θ 2 x θ 3,wheretheθ s denote parameters. In that case, γ(x),gvenn(8), nvolves ntegratng 1/σ and the result s not explct. Fortunately, one can use the rreducble method n that case and the result of applyng that method s gven by Proposton 2. Analternatvestouse the method that has been proposed by [5]. Fnally, the double seres n and (x x 0 ) produced by the rreducble method matches, when appled to a reducble dffuson, the expanson produced by the Hermte seres snce the coeffcents of the latter [a polynomal n (x x 0 ), by constructon] are obtaned as a seres n by computng ther condtonal expectatons, as descrbed n (13). But the nfntesmal generator of the process n (13) s by defnton, such that the resultng coeffcents solve, at each order n, the Kolmogorov equatons. Hence, the two seres match. 6. Convergence to the true log-lkelhood functon and the resultng approxmate MLE. Theorems 1 and 2 gve the expressons of the coeffcents of the expanson n the reducble and rreducble cases, respectvely. I now turn to the convergence of the resultng expanson to the object of nterest, showng that the seres constructed above s a Taylor expanson of the true, but unknown, log-lkelhood functon, and consderng ts applcaton to lkelhood nference. Suppose that (μ, σ ) are parametrzed usng a parameter vector θ and that (μ, σ ) and ther dervatves at all orders are three tmes contnuously dfferentable n θ. The dfferentablty of the coeffcents extends to l, n lght of the prevously cted results on the solutons of second-order parabolc partal dfferental equatons (Secton 9.6 of Fredman [14]), and to the expanson by constructon, gven that t conssts of sums and products of (μ, σ ) and ther dervatves. Let the parameter space be a compact subset of R r. Let θ 0 denote the true value of the parameter. Assume, for smplcty, that for fxed n and, θ l n (θ, ) defned n (2) has a unque maxmzer ˆθ n,. ˆθ n, s the exact (but ncomputable) MLE for θ. Consder, now, the approxmate MLE ˆθ (K) n, obtaned by maxmzng l (K) n (θ, ) (resp. l (K) n ), tself defned analogously to (2), but wth the expanson l (K) (resp. l (K) ) n the reducble (resp. rreducble) case nstead of the true logtranston functon l. We have the followng result.

18 LIKELIHOOD FOR DIFFUSIONS 923 THEOREM 3. For any n, (42) l (K) n (θ, ) l n (θ, ) 0 sup θ n P θ0 -probablty as 0. In the reducble case, the same holds for l (K) n. The approxmate MLE sequence ˆθ (K) n, exsts almost surely and satsfes ˆθ (K) n, ˆθ n, 0 n P θ0 -probablty as 0. Furthermore, suppose that as, we have ˆθ n, θ 0 n P θ0 -probablty and that there exsts a sequence of nonsngular r r matrces S n, such that (43) Sn, 1 ( ˆθ n, θ 0 ) = O p (1). There then exsts a sequence n 0 such that (44) S 1 n, n ( ˆθ (K) n, n ˆθ n, n ) = op (1). Intutvely, the reason that the log-approxmaton error (42) s small n probablty s as follows. For small, n a small neghborhood about x 0, the approxmaton error s small by constructon because l (K) (resp. l (K) ) s a Taylor expanson of l about = 0 (and about x = x 0, resp.). Away from x 0, the approxmaton error may not be small, unless l s analytc, but ths does not matter much because such an error s at most polynomal, whle the probablty of reachng ths regon n tme s exponentally small. Note, also, that t follows from (43) (44) that ˆθ (K) n, and ˆθ n, share the same asymptotc dstrbuton as. For nstance, (43) s verfed, n partcular, f the process s statonary wth postve defnte Fsher nformaton matrx F for a par of successve observatons, n whch case S n, canbetakentoben 1/2 F 1/2 (see Bllngsley [6] for the requred regularty condtons). 7. Examples. In ths secton, I apply the above results to a leadng multvarate dffuson example. The last example shows that the method of ths paper also apples to tme-nhomogeneous models. (45) 7.1. The Bvarate Ornsten Uhlenbeck model. Consder the model d t = β(α t )dt σdw t, where α =[α ],2, β =[β ],2 and σ =[σ,j ],j=1,2 and assume that β and σ have full rank. Ths s the most basc model capturng mean reverson n the state varables. Consder, now, the matrx equaton βλ λβ T = σσ T, whose soluton n the bvarate case s the 2 2 symmetrc matrx λ gven by (46) λ = 1 ( Det[β]σσ T (β tr[β])σ σ T (β tr[β]) T ). 2tr[β] Det[β]

19 924. AIT-SAHALIA When the process s statonary, that s, when the egenvalues of the matrx β have postve real parts, λ s the statonary varance covarance matrx of the process. That s, the statonary densty of s the bvarate Normal densty wth mean α and varance covarance λ. The transton densty of s the bvarate Normal densty p (x x 0, )= (2π) 1 Det[ ( )] 1/2 (47) exp ( ( x m(, x 0 ) ) T 1 ( ) ( x m(, x 0 ) )), where m(, x 0 ) = α exp( β )(x 0 α) and ( ) = λ exp( β )λexp( β T ), wth exp denotng the matrx exponental. Identfcaton of the contnuous-tme parameters from the dscrete data for ths partcular model s dscussed n Phlps [24], Hansen and Sargent [15] and Kessler and Rahbek [22]. If we wsh to dentfy the parameters n θ from dscrete data sampled at the gven tme nterval, then we must restrct the set of admssble parameter values. For nstance, we may restrct n such a way that the mappng β exp( β ) s nvertble, for nstance, by restrctng the admssble parameter matrces β to have real egenvalues. Ths wll be the case, for example, f we restrct attenton to matrces β whch are trangular (and, of course, have real elements). For the rest of ths dscusson, I wll assume that has been restrcted n such a way. By applyng Proposton 1, we see that the process s reducble and that γ(x)= σ 1 x,so (48) d t = (σ 1 βα σ 1 βσ t )dt dw t κ(η t )dt dw t, where η = σ 1 α =[η ],2 and κ = σ 1 βσ =[κ,j ],j=1,2. One can therefore apply Theorem 1 to obtan the coeffcents of the expanson: C ( 1) (y y 0 ) = 1 2 (y 1 y 01 ) (y 2 y 02 ) 2, C (0) (y y 0) = 1 2 (y 1 y 01 ) ( ) (y 1 y 01 2γ 1 )κ 11 (y 2 y 02 2γ 2 )κ (y 2 y 02 ) ( (y 1 y 01 2γ 1 )κ 21 ( ) ) y 2 y 02 2γ 2 κ22, C (1) (y y 0) = 1 ( 2 κ11 ( ) 2 ) (y 01 η 1 )κ 11 (y 02 η 2 )κ 12 ( κ22 ( ) 2 ) (y 01 η 1 )κ 21 (y 02 η 2 )κ (y 1 y 01 ) ( (y 01 η 1 )(κ 2 11 κ2 21 ) (y 02 η 2 )(κ 11 κ 12 κ 21 κ 22 ) ) 24 1 (y 1 y 01 ) 2 ( 4κ 2 11 κ κ 12 κ 21 3κ21 2 ) 1 2 (y 2 y 02 ) ( (y 01 η 1 )(κ 11 κ 12 κ 21 κ 22 ) (y 02 η 2 )(κ 2 12 κ2 22 ))

20 LIKELIHOOD FOR DIFFUSIONS (y 2 y 02 ) 2 ( 4κ 2 22 κ2 21 2κ 12κ 21 3κ 2 12 ) 1 3 (y 1 y 01 )(y 2 y 02 )(κ 11 κ 12 κ 21 κ 22 ), C (2) (y y 0) = 12 1 ( 2κ κ22 2 (κ 12 κ 21 ) 2) 1 6 (y 1 y 01 )(κ 12 κ 21 ) ( (y 01 η 1 )(κ 11 κ 12 κ 21 κ 22 ) (y 02 η 2 )(κ 2 12 κ2 22 )) 1 12 (y 1 y 01 ) 2 (κ 12 κ 21 )(κ 11 κ 12 κ 21 κ 22 ) 1 12 (y 2 y 02 ) 2 (κ 21 κ 12 )(κ 11 κ 12 κ 21 κ 22 ) 1 6 (y 2 y 02 )(κ 21 κ 12 ) ( (y 01 η 1 )(κ 2 11 κ2 21 ) (y 02 η 2 )(κ 11 κ 12 κ 21 κ 22 ) ) 1 12 (y 1 y 01 )(y 2 y 02 )(κ 12 κ 21 )(κ 2 22 κ2 12 κ2 11 κ2 21 ). Because ths s essentally the only multvarate model wth a known closedform densty (other than multvarate models whch reduce to the superposton of unvarate processes), the Ornsten Uhlenbeck process can serve as a useful benchmark for examnng the accuracy of the expansons and the resultng MLE. Table 1 reports the results of 1,000 Monte Carlo smulatons comparng the dstrbuton of the maxmum-lkelhood estmator ˆθ (MLE) based on the exact transton densty for ths model, around the true value of the parameters θ (TRUE), to the dstrbuton of the dfference between the MLE ˆθ (MLE) and the approxmate MLE ˆθ (2) based on the expanson wth K = 2 terms shown above. To ensure full dentfcaton, the off-dagonal term κ 21 s constraned to be zero. As dscussed above, ths guarantees that the egenvalues of the mean reverson matrx are both real and avods the alasng problem altogether. The constrants κ 11 > 0andκ 22 > 0 make the process statonary, so standard asymptotcs gve the asymptotc dstrbuton of ˆθ (MLE) (the nverse of Fsher s nformaton s computed as E[ 2 l / θ θ T ] 1 ). TABLE 1 Monte Carlo smulatons for the bvarate Ornsten Uhlenbeck model Asymptotc Small sample Small sample ˆθ (MLE) θ (TRUE) ˆθ (MLE) θ (TRUE) ˆθ (MLE) ˆθ (2) Parameter θ (TRUE) Mean Stdev Mean Stdev Mean Stdev η η κ κ κ

21 926. AIT-SAHALIA Each of the 1,000 samples s a seres of n = 500 weekly observatons ( = 1/52), generated usng the exact dscretzaton of the process. The results n the table show that the dfference ˆθ (MLE) ˆθ (2) s an order of magntude smaller than the (nescapable) samplng error ˆθ (MLE) θ (TRUE). Hence, for the purpose of estmatng θ (TRUE), ˆθ (2) can be taken as a useful substtute for the (generally ncomputable) ˆθ (MLE). In other words, at least for ths model, K = 2 provdes suffcent accuracy for the types of stuatons and values of the samplng nterval one typcally encounters n fnance Comparng the accuracy of the reducble and rreducble methods. Usng nonlnear transformatons of the Ornsten Uhlenbeck process, we can assess the emprcal performance of the general method for rreducble dffusons. Let denote the process gven n (48) and defne t exp( t ) = γ nv ( t ). From Itô s lemma, the dynamcs of t are gven by ( ( 12 ( 1t κ 11 (η 1 ln( 1t )) κ 12 η2 ln( 2t ) )) ) d t = ( 12 ( 2t κ 21 (η 1 ln( 1t )) κ 22 η2 ln( 2t ) )) dt (49) ( ) 1t 0 dw 0 t. 2t By constructon, ths process has a known log-transton functon gven by l (x x 0, )= ln(xx 0 ) l (, ln(x) ln(x 0 )) and t s reducble by transformng t back to t = ln( t ) = γ( t ), wth D v (x) = ln( 1t 2t ) for that transformaton. But, n order to assess the accuracy of the rreducble method, we can drectly calculate the rreducble expanson (based on Theorem 2) for the model (49). We can then compare t to the closed-form soluton, but also to the reducble expanson obtaned usng the order 2 expanson gven n the prevous secton for l and then the Jacoban formula, l (x x 0, ) = ln( 1t 2t ) l (, ln(x) ln(x 0 )). Based on Proposton 2, we know that n ths stuaton, the rreducble expanson nvolves Taylor expandng the coeffcents of the reducble expanson n x about x 0. Monte Carlo smulatons wth the same desgn as n the prevous secton help document the effect of that further Taylor expanson on the accuracy of the resultng MLE. The results are gven n Table 2 and they show that the dfference ˆθ (MLE) ˆθ (2,rreducble), although larger than ˆθ (MLE) ˆθ (2,reducble), remans smaller than the dfference ˆθ (MLE) θ (TRUE) due to the samplng nose. In other words, replacng ˆθ (MLE) by ˆθ (2,rreducble) has an effect whch s not statstcally dscernble from the samplng varaton of ˆθ (MLE) around θ (TRUE). And, of course, ˆθ (MLE) s generally ncomputable, whereas ˆθ (2,rreducble) s computable Tme-nhomogeneous models. Tme-nhomogeneous models are of partcular nterest for the term structure of nterest rates. A large swathe of the term structure lterature has proposed models desgned to ft exactly the current bond

22 LIKELIHOOD FOR DIFFUSIONS 927 TABLE 2 Monte Carlo smulatons for the exponental transformaton of the Ornsten Uhlenbeck model: comparng the reducble and rreducble methods Small sample Small sample Small sample ˆθ (MLE) θ (TRUE) ˆθ (MLE) ˆθ (2,reducble) ˆθ (MLE) ˆθ (2,rreducble) Parameter θ (TRUE) Mean Stdev Mean Stdev Mean Stdev η η κ κ κ prces, as well as other market data, such as bond volatltes or the mpled volatltes of nterest rate caps, for nstance. Calbratng such a model to tme-varyng market data gves rse to tme-varyng drft and dffuson coeffcents. Typcal examples of ths approach nclude the models of Ho and Lee [17], Black, Derman and Toy [7] and Hull and Whte [18], where the short-term nterest rate (or ts log) follows the dynamcs d 1t = (α(t) β(t) 1t )dt κ(t)dw 1t. Markovan specalzatons of the Heath, Jarrow and Morton [16] model wll also be, n general, tme-nhomogeneous. The unvarate results of Aït-Sahala [2] have been extended to cover such models by Egorov, L and u [13]. Wth expansons now avalable for tmehomogenous dffusons of arbtrary specfcatons and dmensons, a tme-nhomogeneous dffuson of dmenson m can be smply reduced to a tme-homogenous dffuson of dmenson m 1. Indeed, consder the state vector t = ( 1t,..., mt ). Now, defne tme as the addtonal state varable m1,t = t, whose dynamcs are d m1,t = dt, and consder the extended state vector as t = ( 1t,..., mt, m1,t ).Thssan(m 1)-dmensonal, tme-homogenous, dffuson. 8. Proofs Proof of Proposton 1. Suppose that a transformaton exsts and defne t γ( t ), where γ(x)= (γ 1 (x),..., γ m (x)) T. By Itô s lemma, the dffuson matrx of s σ ( t ) = γ( t )σ( t ). For σ to be Id, we must therefore have that γ( t ) = σ 1 (x) (recall that σ s assumed to be nonsngular). Thus, [σ 1 ] j (x) = γ (x) (50) x j and hence [σ 1 ] j (x) x k = x k ( γ (x) x j ) = x j ( γ (x) x k ) = [σ 1 ] k (x) x j,

23 928. AIT-SAHALIA for all (,j,k)= 1,...,m. Contnuty of the second-order partal dervatves s requred for the order of dfferentaton to be nterchangeable. Here, we have nfnte dfferentablty. Conversely, suppose that σ 1 satsfes (10). Then, for each = 1,...,m,use row of the matrx σ 1,σ 1 =[[σ 1 ] j ] j=1,...,m, to defne the dfferental 1-form ω = m j=1 [σ 1 ] j dx j and calculate ts dfferental, the dfferental 2-form dω. Condton (10) mples that dω = 0, that s the dfferental 1-form ω s closed on S. The doman S s sngly connected (or wthout holes). Therefore, by Poncaré s lemma (see, e.g., Theorem V.8.1 of Edwards [12]), the form ω s exact, that s there exsts a dfferental 0-form γ such that dγ = ω. In other words, for each row of the matrx σ 1, there exsts a functon γ defned by γ (x) = x j [σ 1 ] j (x) dx j (the choce of the ndex j s rrelevant) whch satsfes (50), has the requred dfferentablty propertes and s nvertble. The functon γ s then defned by each of ts d components γ,= 1,...,m, and because of Assumptons 2 and 3, t s nvertble and nfntely often dfferentable. By constructon, t γ( t ) has unt dffuson and therefore s reducble. To prove the equvalent characterzaton (9), apply Itô s lemma from to (nstead of from to ) and proceed as above Proof of Theorem 1. To show that (15) wth the coeffcents gven n the statement of Theorem 1 ndeed represent the Taylor expanson n of the logdensty functon l, at order K 1, t suffces to verfy that the dfference between the left- and rght-hand sdes n the Kolmogorov forward and backward partal dfferental equatons s of order K. Defne F (K) (y y 0, ) [resp. B (K) (y y 0, )] as the dfference between the leftand rght-hand sdes of the forward (resp. backward) equatons when l s replaced by l (K). The backward equaton for l s l (y y 0, ) = μ (y 0 ) l (y y 0, ) y 0 (51) 1 2 l (y y 0, ) 1 ( ) l (y y 0, ) y 2 0 Substtutng n the expanson (15) and equatng the coeffcents of 2 on both sdes of (51) yelds C ( 1) (y y 0 ) = 1 ( ( 1) C (y y 0 ) 2 y 0 ) T ( C ( 1) y 0 (y y 0 ) y 0 whch s satsfed by the (already determned) soluton (20), whch s therefore the desred soluton. ),

24 LIKELIHOOD FOR DIFFUSIONS 929 Startng wth the Gaussan leadng term (20), we have F (K) B (K) (y y 0, )= (y y 0, )= K 1 k= 1 K 1 k= 1 f (k) (y y 0) k k! O( K ) b (k) (y y 0) k k! O( K ) [wth the conventon that ( 1)!=0!=1]. The frst term n F (K) (52) f ( 1) (y y 0 ) = m 2 m 1 2 μ (y) C( 1) (y y 0 ) y C( 1) (y y 0 ) C (0) (y y 0) y y s 2 C ( 1) (y y 0 ) y 2 = (y y 0 )μ (y) (y y 0 ) C(0) (y y 0). y Solvng the equaton f ( 1) (y y 0 ) = 0forC (0) (y y 0) yelds the full soluton 1 C (0) (y y ( 0) = (y y 0 ) μ y0 u(y y 0 ) ) du,j=1, j 0 α (0) j y y 0 M (0) y j y 0j where the α (0) j and M (0) are ntegraton constants n the dfferental equaton f ( 1) = 0, hence arbtrary functons of y 0. The boundary condton that C (0) be fnte when passng through the axes y j = y j0 for all j = 1,...,m mposes the condton α (0) j = 0. To determne M (K), (51) gves, b ( 1) (y y 0 ) = (y y 0 )μ (y) (y y 0 ) C(0) (y y 0) y 0 and the canddate soluton (52) must satsfy b ( 1) = 0. Thus, we must have M (0) (y 0) (y y 0 ) = 0 y 0 for all y and y 0 and so M (0) (y 0) s constant. Snce the lmtng behavor of p s N(0,I)as 0and ( lm l (K) 0 (y y 0, ) m 2 ln(2π ) 1 ) 2 y y 0 2 = C (0) (y y 0),

25 930. AIT-SAHALIA we must have M (0) = 0 to ensure that as 0, the lmtng densty ntegrates to one [otherwse, t ntegrates to exp(m (0) )]. The next term s where G (1) C (0). f (0) (y y 0) = C (1) (y y 0) 1 { 2 C (0) 2 (y y 0 ) C(1) (y y 0) y μ (y) C(0) (y y 0) y (y y 0) y 2 [ C (0) (y y 0) y ] 2 } μ (y) y = C (1) (y y 0) (y y 0 ) C(1) (y y 0) G (1) y (y y 0), s gven n (23) and depends on the prevously determned C ( 1) Solvng the equaton f (0) (y y 0) = 0, whch s lnear n C (1), smlarly yelds the explct soluton C (1) (y y 0) = 1 0 G (1) ( ) y0 u(y y 0 ) y 0 du,j=1,j α (1) j y y 0 (y j y 0j ) 2 M(1) and whch ncludes generc ntegraton constants α (1) j and M (1). The soluton has α (1) j = 0 when mposng fnteness of l (K) when passng through the axes y j = y j0 for all j = 1,...,m.As for M (1), nvokng the backward equaton (51) yelds M (1) (y 0), M (1) (y 0) y 0 (y y 0 ) = 0, whose only soluton vald for all (y, y 0 ) s M (1) (y 0) = 0. Ths yelds the coeffcent C (1). More generally, the term f (k 1),k 1, s gven by f (k 1) where G (k)...,c (k 1) (y y 0 ) C(k) (y y 0) G (k) y (y y 0), sgvenn(24) and depends on the prevously determned C( 1),C (0),. Solvng the equaton f (k) (y y 0) = 0forC (k) (wth the same boundary (y y 0 ) = C (k) (y y 0) 1 k