A BSDE approach to the Skorokhod embedding problem for the Brownian motion with drift

Transcription

1 A BSDE approach o he Skorokhod embedding problem for he Brownian moion wih drif Sefan Ankirchner and Gregor Heyne and Peer Imkeller Insiu für Mahemaik Humbold-Universiä zu Berlin Uner den Linden 6 99 Berlin Germany December 2, 27 Absrac We solve Skorokhod s embedding problem for Brownian moion wih linear drif (W + κ by means of echniques of sochasic conrol heory. The search for a sopping ime T such ha he law of W T + κt coincides wih a prescribed law µ possessing he firs momen is based on soluions of backward sochasic differenial equaions of quadraic ype. This new approach generalizes an approach by Bass [3] of he classical version of Skorokhod s embedding problem using maringale represenaion echniques. Key words and phrases: Skorokhod embedding; Brownian moion; diffusion; sopping ime; conrol heory; BSDE; quadraic growh; Malliavin calculus. 2 AMS subjec classificaions: Primary 6G4, 6G44; secondary 6J65, 6F5, 6F7. Inroducion Brownian moion owes is cenral imporance in he enire realm of sochasic dynamics o is appearance in he Donsker-Varadhan invariance principle. The laer underpins he fac ha he law of he dynamics of any linearly inerpolaed random walk S k = k i= X i wih i.i.d. cenered incremens X i wih variance σ 2, speeded up in ime wih a facor n and rescaled in space by he facor nσ converges o he law of a Brownian moion, irrespecive of wha he individual law of he X i may be. In his famous 96 embedding formulaion Skorokhod [2] approaches he invariance principle from a reverse angle. He embeds he random walk by law ino a given Brownian moion W in he following way. He firs chooses a sopping ime T of W such ha W T possesses he law µ of X, hen a sopping ime T 2 of he Brownian moion W T + W T such ha W T2 +T W T possesses again he law µ of X 2. Consequenly he law of W T2 +T coincides wih he one of S 2. This clearly provides a recipe o find a sequence of sopping imes (T n n N such ha he law of S n finds iself embedded ino he Brownian moion as he law of W a ime T + + T n for any n N. Is main ingredien, he consrucion of a sopping ime T for a given Brownian moion W such ha he law of W T realizes a given cenered probabiliy measure µ possessing a second momen is oday

2 2 generally known as Skorokhod s embedding problem. Since is original appearance i has been rephrased many imes, and a variey of approaches have been developed. For an overview see Oblój [9]. The embedding problem may obviously be generalized from Brownian moion o ransien diffusion processes. In his seing (see [9] Hall [7] developed a firs soluion in 969. The subjec was reaed again by Falkner and Grandis [6], and Peskir [] in 2. According o [9] he mehodology of he embeddings considered in hese approaches may be summarized as follows. Viewed in he opics of is scale funcion, he given diffusion urns ino a coninuous local maringale. Using he ime change echnique by Dambis, Dubins and Schwarz he laer may be ransformed ino a Brownian moion, for which he version of Skorokhod s embedding due o Azéma and Yor (see [2], or [9], Chaper 5 applies. Then he previous seps have o be aken in he reverse direcion. In he presen paper we sar wih he observaion ha he Skorokhod embedding problem may be viewed as he weak version of a sochasic conrol problem: a Brownian moion has o be seered in such a way ha i akes he prescribed law of some random variable. This observaion will be pu ino a concep of solving he Skorokhod embedding problem in he framework of a paricular ransien diffusion, he Brownian moion wih linear drif, which is based on he powerful ool of backward sochasic differenial equaions (BSDE. The precise formulaion of he problem we solve by means of BSDE echniques is his. Problem (*. Le µ be a probabiliy measure on R wih x dµ(x < and (W +κ+c a Brownian moion wih linear drif of slope κ (w.l.o.g κ > and saring poin c defined on a probabiliy space (Ω, F, P. If (F denoes he filraion generaed by W, compleed by he P -null ses of F, find an (F sopping ime T such ha W T +κt +c has law µ and ET <. Our mehod o solve (* for Brownian moion wih linear drif is in spiri much relaed o an approach of he original Skorokhod embedding problem for Brownian moion by Bass [3], where he soluion algorihm consiss in he following seps.. Find a σ(w -measurable random variable ξ wih law µ. 2. Use he maringale represenaion propery of Brownian moion o wrie ξ E(ξ as a sochasic inegral Z sdw s. 3. Use random ime change of he Dambis, Dubins and Schwarz ype wih he scale process Z2 s ds o ransform he maringale Z sdw s ino a Brownian moion B. Then here is a sopping ime S of B such ha B S = ξ, whence B S has law µ. 4. To finally ransform his embedding wih respec o B ino one wih respec o W, use explici descripions of soluions of an associaed parial differenial equaion which provides a represenaion of he sopping ime as a funcional of he corresponding Brownian moion. Working wih Brownian moion wih linear drif insead requires a modificaion of his algorihm where maringale represenaion comes ino play. We replace i - in he erms of Peng [] - wih nonlinear maringale represenaion appearing in he disguise of BSDE. To undersand his modificaion more precisely, noe firs ha o find an explici represenaion ξ E(ξ = Z s dw s

3 A WEAK SOLUTION OF THE SKOROKHOD EMBEDDING PROBLEM 3 amouns o finding he soluion (Y, Z of a BSDE wih generaor f =, i.e. Y = ξ Z s dw s. Now observe ha he random ime change in he quadraic variaion scale S = Z2 s ds which urns he maringale Z s dw s ino a Brownian moion will of course urn his scale process ino a linear drif. This is he reason why for generalizing he second sep in Bass [3] o Brownian moion wih linear drif of slope κ we have o solve he nonlinear maringale represenaion problem in erms of he BSDE Y = ξ Z s dw s κ Z 2 s ds ( wih he quadraic generaor f(s, y, z = κz 2. This paricular BSDE has been sudied in he lieraure already, for example in [5]. Alernaively, we migh hink of eliminaing firs he linear drif wih slope κ by means of Girsanov s heorem providing an equivalen measure Q, hen following he algorihm proposed by Bass [3], and finally reurning o he original measure P. Maringale represenaions wih respec o Q will hen correspond exacly o solving non-linear BSDEs ( wih respec o P. Here is an ouline of he following presenaion of he Skorokhod embedding problem for Brownian moion wih linear drif along hese lines of reasoning. In Secion we use BSDE relaed echniques o exend seps - 3 of he algorihm skeched above o our seing. This way, we consruc a Brownian moion B and a corresponding sopping ime S, wih B S + κs + c = ξ (Lemma.. An explici soluion of Equaion ( and herefore closed formulas for Y and Z in erms of an associaed parial differenial equaion will be given in Secion 2 (Lemma 2.2. An adapaion of argumens from [3] yields an ODE providing a represenaion of he sopping ime S in he filraion of B (Theorem 2.3. To finally obain he desired embedding ino W in Secion 3, B has o be replaced wih W in he ODE. We close he presenaion by describing he domain of possible saring poins and discussing some inegrabiliy properies of he sopping ime. A weak soluion of he Skorokhod embedding problem Le µ be a probabiliy measure on R which is no idenical o a Dirac measure. In his secion we will show ha saring from a soluion of a simple quadraic BSDE one can consruc a Brownian moion and a sopping ime such ha he disribuion of he Brownian moion sopped a his ime is equal o µ. Le us firs recall he definiion of a BSDE. Le ξ be an F -measurable random variable, and le f : Ω [, ] R R R be a measurable funcion such ha for all y, z R he mapping f(,, y, z is predicable. A soluion of he BSDE wih erminal condiion ξ and generaor f is defined o be a pair of predicable processes (Y, Z such ha almos surely we have Z2 s ds <, f(s, Y s, Z s ds <, and for all [, ] Y = ξ Z s dw s + f(s, Y s, Z s ds. The soluion processes (Y, Z are ofen shown o saisfy some inegrabiliy properies. To his end one usually verifies wheher hey belong o he following funcion spaces. Le p. We

4 A WEAK SOLUTION OF THE SKOROKHOD EMBEDDING PROBLEM 4 denoe by H p he se of all R-valued predicable processes ζ such ha E ζ p d <, and by S p he se of all R-valued predicable processes δ saisfying E (sup s [,] δ s p <. Le us now come back o problem of finding a Brownian moion B and a sopping ime T such ha B T + κt + c has he disribuion µ. Le ˆF be he disribuion funcion of µ and ˆF (y = inf{x : ˆF (x y}. Wih Φ denoing he disribuion funcion of he sandard normal disribuion we define he funcion g : R R by g(x = ˆF (Φ(x and se ξ as ξ = g(w = ˆF (Φ(W. Noe ha g is nondecreasing, lef-coninuous, measurable and no idenically consan. Hence ξ is a σ(w -measurable random variable wih law µ. As menioned in our ouline ξ will now serve as he erminal value in he following quadraic BSDE Y = ξ Z s dw s κ Z 2 s ds,, (2 wih κ >. As is shown in he nex lemma, Y can be inerpreed as a ime changed Brownian moion wih drif κ ha is sopped as i runs ino he random variable ξ. Lemma.. Suppose (Y, Z is a soluion of (2, E(ξ < and Z H 2. Then here exiss a Brownian moion B and a sopping ime S, wih E(S <, such ha B S + κs + Y = ξ, i.e. he process B s + κs + Y sopped a S has law µ. Proof. Since Z H 2, he process M = Z s dw s,, is a maringale. We exend our probabliliy space such ha i accommodaes anoher Brownian moion B, which is independen of M. Le S = M. We ime-change M using he sopping imes { inf{ : M > s}, s < S, τ s =, s S, and obain he Brownian moion B s = B s B s S + M τs, s <. B is adaped o he filraion (G s defined by (G s = (F τs. Since S = M is a (G s sopping ime we find B S + κs + Y = M + κ Z2 s ds + Y = ξ. Furhermore E(S = E( Z 2 s ds = E( κ Y κ Z s dw s κ Y = κ E(ξ κ Y (3 <, and hence he proof is complee. Remark.2. We remark ha i is a priori no clear ha he sopping ime S of Lemma. is also a sopping ime wih respec o (F B s, he righ coninuous compleion of he filraion (σ(b u : u s. However, i will follow from he funcional dependence of S on he pahs of he Brownian moion B shown in he nex secion. In Lemma. i is assumed ha here exiss a soluion (Y, Z of (2. The nex goal of his secion herefore is o show ha one can explicily consruc a soluion of (2, if ξ saisfies he inegrabiliy condiion E(e 2κξ <. We will also see ha he consruced soluion is such ha Z H 2, and hence i saisfies he assumpions of Lemma..

5 A WEAK SOLUTION OF THE SKOROKHOD EMBEDDING PROBLEM 5 If e 2κξ is inegrable, hen we can define he maringale N = E [exp( 2κ ξ F ],. The maringale represenaion propery implies ha here exiss a predicable and locally square inegrable process H such ha N = N + H sdw s. I is sraighforward o show ha he wo processes defined by Y = 2κ ln N, and Z = H,, (4 2κ N are a soluion of (2. Moreover, we have he following. Lemma.3. Le ξ and e 2κξ be inegrable. Then Y S, Z H 2 and (Y, Z solves (2. Proof. By Io s formula, applied o ln N, we have [ 2κ [ln N ln N ] = H s dw s 2κ N s 2 ( Hs N s 2 ds], (5 which means ha Y Y = Z sdw s + κ Z2 s ds. Since Y = ξ, his implies ha (Y, Z is a soluion of (2. In order o show ha Z belongs o H 2, observe ha κ Z 2 s ds = 2κ [ln N ln N ] and by Jensen s inequaliy Using his in (6, we obain κ Z s dw s, (6 ln N = ln E(exp( 2κξ F 2κE(ξ F. Z 2 s ds E(ξ F + 2κ ln N Z s dw s. (7 There exiss an increasing sequence of sopping imes τ n wih τ n, a.s. and such ha E τ n Zs 2 ds <. This implies E τ n Zs 2 ds κ E(ξ + ln N 2κ 2, and monoone convergence yields ha Z belongs o H 2. Now noice ha Equaion (6 implies sup Y Y + κ Zs 2 ds + sup Z s dw s, (8 and wih he Burkholder-Davis-Gundy Inequaliy we obain ha for a consan C R +, ( E sup Y C + E Zs 2 ds <, (9 which shows ha Y belongs o he space S. Remark.4. We wan o commen on he uniqueness of soluions of quadraic BSDE, such as (2. Kobylanski [8] proved exisence and uniqueness under he resricion ha he erminal variable ξ is bounded. Recenly Briand and Hu [4] gave a uniqueness resul under he assumpions ha he generaor f is convex in he variable z and ha exp(p ξ is inegrable for all p. The inegrabiliy condiion imposed on µ in Lemma 2.2 ranslaes o E(exp( 2κξ < for our choice of ξ, i.e. we canno draw on he uniqueness resuls jus menioned. In fac, refer o [] Secion.3. for an example of a BSDE wih generaor 2 z2, ξ fulfilling E(exp(γ ξ < for an arbirary γ > and wo differen soluions.

6 2 PATH DEPENDENCE OF THE SKOROKHOD STOPPING TIMES 6 2 Pah dependence of he Skorokhod sopping imes The aim of his secion is o find he analyic relaion of he pahs of he Brownian moion B and he sopping ime S consruced in Lemma. of he previous secion. Describing S as a funcional of he Brownian pahs, will allow us, in he nex secion, o solve he Skorokhod embedding problem no only for B, bu for any arbirary Brownian moion. We firs show ha he BSDE soluion processes (Y, Z, defined in (4, are deerminisic funcions of W. To his end we inroduce he auxiliary funcion F (, x = E[exp( 2κ g(w W + x], [, and x R. ( Noe ha F is finie for all < < and x R, and F (, = E[exp( 2κg(W ] <. This can be deduced from he following fac: If I is a bounded inerval in R, J a compac subse of (,, hen here exiss a consan < K < such ha exp( (z x 2 /2 exp( z 2 /2, ( for all x I, J, z > K. The nex lemma collecs some furher properies of he funcion F we will need in order o wrie he processes Y and Z as funcions of W. Some of he properies are only needed laer and will enable us o deduce an ODE ha esablishes he link beween he Brownian pahs and he Skorokhod sopping ime. The lemma follows from sraighforward adapaions of Lemma and Lemma 2 in [3], and herefore he proof is omied. Lemma 2.. (Properies of F The funcion F defined in ( has he following properies.. F (, x is sricly decreasing in x, for <. 2. F C,2 ((, R. 3. For (, he funcion F x can be expressed as he Lebesgue-Sieljes inegral F x (, x = ϕ (z x dḡ(z, where ϕ s is he densiy of he N (, s disribuion and ḡ he nondecreasing funcion defined by ḡ(z = exp( 2κg(z. 4. On compac subses of (, R, F (resp. F x is bounded above (resp. bounded below, bounded below away from (resp. bounded above away from, and uniformly Lipschiz in and x. 5. For each (,, le F (, be he inverse of F (,. Then on compac subses of is domain, F is uniformly Lipschiz in and joinly coninuous in and y. 6. The funcion h(, x = F x (, x 2κ F (, x, (2 is well defined for all < < and x R. On compac subses of (, R, h is bounded above, bounded below away from, and uniformly Lipschiz in and x.

7 2 PATH DEPENDENCE OF THE SKOROKHOD STOPPING TIMES 7 Noe ha our funcion F (resp. F x is he analogue of he funcion b (resp. a in [3]. In he nex lemma we give a represenaion of he soluion (Y, Z of (2, as funcions of he Brownian moion W. Recall ha ξ = g(w and according o our assumpion κ >. Lemma 2.2. Le µ be such ha x dµ(x < and exp( 2κ x µ(dx <. Le F be defined as in (. Then he processes Y and Z, defined in (4, saisfy for almos all ω, and Y = 2κ ln F (, W, for all [, ], (3 Z = 2κ F x (, W, for all (,. (4 F (, W Proof. Noe ha he maringale N = E[e 2κξ F ] can be wrien as N = F (, W for all [, ]. Thus, he very definiion of Y implies (3. By Lemma 2., we have F C,2 ((, R, and hence wih Io s formula we find for ε > and < N N ε ε F x (u, W u dw u = ε F u (u, W u du + 2 ε F xx (u, W u du. On he lef hand side we have a local maringale, whereas on he righ hand side we have a process of bounded variaion. Hence we know ha ε F u(u, W u du + 2 ε F xx(u, W u du =, P-a.s. and herefore N = N ε + ha almos surely we have Z = 2κ ε F x(u, W u dw u. Again, he very definiion of Z, implies on (,. F x(,w F (,W The nex heorem will use he informaion gahered so far o show ha he sopping ime S, defined in Lemma., is acually an (F B s sopping ime. We remark ha he proof is a generalisaion of he proof of Proposiion 3 in [3]. Theorem 2.3. Le (Y, Z be defined as in (4. Then he associaed sopping ime S defined in Lemma. is an (F B s sopping ime. Proof. Le M and τ s be defined as in he proof of Lemma., and se S = M. By he properies of our explici soluion we find ha he sopping imes τ s reduce o τ s = S (s, for all s < S, and saisfy he equaion s τ s = S (S (s = h 2 (τ s, W τs, (5 for all < s < S. From he forward version of (2 we ge W τs = F (τ s, exp( 2κ(M τs + κ τ s Z2 u du + Y = F (τ s, exp( 2κ(B s + κs + Y and hus, for all < s < S, we can wrie (5 as s τ s = h 2 (τ s, F (τ s, exp( 2κ(B s + κs + Y. (6 Therefore, for each ω, (6 is a differenial equaion, which is solved a leas by τ s, wih < s < S. Sandard resuls of he heory of ordinary differenial equaions show ha our soluion is P-almos surely unique. Moreover i can be consruced via Picard ieraion and herefore τ s is measurable wih respec o Fs B. Because by monoone convergence we have lim S = S i is sufficien o show {S s} Fs B, for (, in order o see ha S = S is even an (Fs B sopping ime. Bu by measurabiliy of τ s we have {S s} = {τ s < } c Fs B.

8 3 THE SOLUTION OF THE SKOROKHOD EMBEDDING PROBLEM 8 3 The soluion of he Skorokhod embedding problem Now we are finally se o presen a soluion o Problem (*. Theorem 3.. Le κ > and µ a probabiliy measure on R such ha x dµ(x < and exp( 2κ x µ(dx <, and define c = 2κ ln exp( 2κ x µ(dx. Le W be a Brownian moion on a probabiliy space (Ω, P, F and (Fs W is righ coninuous and compleed filraion. Then here exiss an (Fs W sopping ime T, wih finie expecaion, such ha he process W s + κs + c sopped a T has law µ. Proof. Le Y and Z be defined as in (4, and le B and S be he associaed Brownian moion and sopping ime consruced in Lemma.. We insanly deduce c = Y = 2κ ln E(exp( 2κξ, and B S + κs + Y = ξ. In order o obain a sopping ime for W we use Equaion (6. By replacing B wih W we readjus he Brownian pahs and sill have he P-almos sure unique and global soluion, which we now call σ s, s σ s = h 2 (σ s, F (σ s, exp( 2κ(W s + κs + Y. Again wih Lemma 2. we know ha s σ s >, herefore σ s is sricly increasing and we can define σ for <. Se T = lim σ, allowing for T =. The same argumens as above yield {σ s} Fs W, which leads o T being a (Fs W sopping ime. The law of (B, S is he same as he law of (W, T, hence W T + κt + Y has disribuion µ. Since paricularly S d = T, we know from Equaion (3 E(T = κ E(ξ + ln E(exp( 2κ ξ. 2κ2 The domain of possible saring poins The following Corollary exends Theorem 3.. Corollary 3.2. Le µ saisfy x dµ(x < and exp( 2κ x dµ(x exp( 2κc, for κ > and for some c R. Le W be a Brownian moion on a probabiliy space (Ω, P, F and (F W s is righ coninuous and compleed filraion. Then here exiss an (F W s sopping ime T, wih finie expecaion such ha he process W s + κs + c sopped a T has law µ. Proof. We can choose c > c fulfilling exp( 2κ x µ(dx = exp( 2κ c. Then by Theorem 3. we obain a sopping ime T such ha W T + κt + c has law µ. Se a = c c > and b = κ. By sandard resuls abou firs hiing imes of Brownian moion we know ha he sopping ime T = inf{ : W +T W T = a + b}, a > and b <, has finie expecaion. Thus we define T = T + T and use he srong Markov propery of Brownian moion o show ha W T + κt + c has he same disribuion as W T + κt + c, i.e. has law µ. Obviously ET <.

9 3 THE SOLUTION OF THE SKOROKHOD EMBEDDING PROBLEM 9 I can be shown ha c = 2κ ln exp( 2κ x dµ(x is he larges saring poin ha allows o sop he Brownian moion wih drif of slope κ so ha i has disribuion µ. As saed in [6] for c =, an exponenial change of he space variable shows ha he exisence of a sopping ime T, such ha W T + κt + c has law µ already implies exp( 2κx dµ(x exp( 2κc. Inegrabiliy properies of he sopping ime The sopping ime T of Theorem 3., ha solves he embedding problem, has been shown o be inegrable. In he remainder we will show ha if µ has a finie momen of order p, hen T is also L p -inegrable. Theorem 3.3. Le p. If x p dµ(x < and exp( 2κ x µ(dx <, hen he sopping ime T of Theorem 3. solving he embedding problem, saisfies E(T p <. Proof. Le (Y, Z be defined as in (4, S = Z2 s ds, and B he Brownian moion consruced in Secion so ha B S +κs +Y ( has disribuion µ. Since (B, S has he same law as (W, T, p all we need o show is ha E Z2 s ds is finie. To his end noice ha Inequaliy (7 and he inequaliy a+b+c p 3 p ( a p + b p + c p, imply ( p Zs 2 ds ( κ p 3p ξ p + 2κ ln N p p + Z s dw s. The Burkholder-Davis-Gundy Inequaliy furher yields ha for a consan C R +, ( p ( E Zs 2 ds C E ξ p + ( p/2 2κ ln N p + E Zs 2 ds. (7 ( Now le C 2 R + be such ha a a2 p/2 2C + C 2 for all a. Then E Z2 s ds ( p 2C E Z2 s ds + C2, and combining his wih (7 we obain for some consan C 3 ( p ( C 3 2 E Z 2 s ds and hence E(T p <. + E ξ p + 2κ ln N p = C 3 ( + x p dµ(x + 2κ ln <, exp( 2κ x µ(dx Remark 3.4. If x p dµ(x < and exp( 2κ x µ(dx <, hen he process Y, defined in (4, belongs o S p. Proof. Observe ha from Equaion (6 we may deduce ( ( p sup Y p 3 p Y + κ p Zs ds 2 + sup Z s dw s p,

10 REFERENCES and hence he Burkholder-Davis-Gundy Inequaliy implies ha here exis consans C 4 and C 5 such ha ( p ( p/2 E sup Y p C 4 ( Y p + E Zs ds 2 + E Zs 2 ds ( p C 5 ( + E Zs 2 ds <. and hence he resul. References [] S. Ankirchner, P. Imkeller, and A. Popier. On measure soluions of backward sochasic differenial equaions. Preprin, arxiv:mah/ v, 27. [2] J. Azéma and M. Yor. Une soluion simple au probléme de skorokhod. Séminaire de Probabiliés, 3 (979:9 5. [3] R.F. Bass. Skorokhod imbedding via sochasic inegrals. Séminaire de probabiliés (Srasbourg, 7 (983: [4] P. Briand and Y. Hu. Quadraic BSDEs wih convex generaors and unbounded erminal condiions. Preprin, arxiv:mah/73423v, 27. [5] P. Briand, J. Lepelier, and J. San Marín. One-dimensional backward sochasic differenial equaions whose coefficien is monoonic in y and non-lipschiz in z. Bernoulli, 3(:8 9, 27. [6] P. Grandis and N. Falkner. Embedding in Brownian moion wih drif and he Azéma-Yor consrucion. Sochasic Process. Appl., 85 (2: [7] W.J. Hall. Embedding submaringales in Wiener processes wih drif, wih applicaions o sequenial analysis. J. Appl. Probabiliy, 6 (969: [8] M. Kobylanski. BSDE and PDE wih quadraic growh. The Annals of Probabiliy, 28(2 (2: [9] J. Oblój. The Skorokhod embedding problem and is offspring. Probabiliy Surveys, (24: [] S. Peng. Backward SDE and relaed g-expecaion. In Backward sochasic differenial equaions (Paris, , volume 364 of Piman Res. Noes Mah. Ser., pages Longman, Harlow, 997. [] G. Peskir. The Azéma-Yor embedding in Brownian moion wih drif. High dimensional probabiliy, II (Seale, WA, 999, 47 (2: [2] A. V. Skorokhod. Sudies in he heory of random processes. Translaed from he Russian by Scripa Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass., 965.