On the discretization schemes for the CIR (and Bessel squared) processes

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1 On the dscretzaton schemes for the CIR (and Bessel squared processes Aurélen Alfons CERMICS, projet MATHFI, Ecole Natonale des Ponts et Chaussées, 6-8 avenue Blase Pascal, Cté Descartes, Champs sur Marne, Marne-la-vallée, France. e-mal : alfons@cermcs.enpc.fr December 27, 25 Abstract In ths paper, we focus on the smulaton of the CIR processes and present several dscretzaton schemes of both the mplct and explct types. We study ther strong and weak convergence. We also examne numercally ther behavour and compare them to the schemes already proposed by Deelstra and Delbaen [5 and Dop [6. Fnally, we gather all the results obtaned and recommend, n the standard case, the use of one of our explct schemes. 1 Introducton The am of ths paper s to present an overvew on the dscretzaton schemes that can be used for the smulaton of the square-root dffusons of Cox-Ingersoll-Ross type. These processes, ntally ntroduced to model the short nterest rate (Cox, Ingersoll and Ross [4, are now wdely used n modellng because they present nterestng features lke the nonnegatvty and the mean reverson. Moreover, some standard expectatons can be analytcally calculated whch can be useful especally for calbratng the parameters. Thus, they have also been used n fnance to model the stochastc volatlty of the stock prce (Heston [9 or the credt spread (Brgo and Alfons [3. We wll use n ths paper the followng notaton for ths dffuson: (X t wll denote a Cox-Ingersoll-Ross (CIR for short process of parameter (k, a, σ, x f { Xt = x + t (a kx sds + σ t Xs dw s, t [, T x, σ, a, k R. (1 Under the above assumpton on the parameters that we wll suppose vald through all the paper, t s well known that ths SDE has a nonnegatve soluton, and ths soluton s

2 Dscretzaton schemes for the CIR processes 2 pathwse unque (see for example Rogers and Wllams [13. Let us recall here that under the assumpton (see for example Lamberton and Lapeyre [11 2a > σ 2 and x > (2 the process s always postve. When k >, t s common to defne θ = a/k and rewrte the SDE dx t = k(θ X t dt + σ X t dw t. Indeed, θ appears as the asymptotc mean of X t toward whch the process s attracted. In practce, ths more ntutve parametrzaton s preferred. In the sequel, (F t, t wll denote the natural fltraton of the Brownan moton W, and we wll consder the regular grd t n = T. Except n cases where t s mportant n to remnd the dependency n n, we wll wrte t rather than t n. It s well known that the ncrements of the CIR process are non-central ch-squared random varables that can be smulated exactly. Thus, we can nductvely smulate a random vector dstrbuted accordng to the law of (X t,..., X tn (see Glasserman [7, pp However, the exact smulaton n general requres more tme than a smulaton wth approxmaton schemes. It may also be restrctve f one wshes to correlate ths dffuson wth another dffuson va the Brownan motons as n Brgo and Alfons [3 where two correlated CIR processes are consdered. At least for both these reasons, studyng approxmaton schemes s relevant. It s mportant to remark frst that the natural way to smulate ths process, that s the explct Euler-Maruyama scheme ˆX t n +1 = ˆX t n + T n (a k ˆX t n + σ ˆX t n (W t+1 W t wth ˆX t n = x can lead to negatve values snce the Gaussan ncrement s not bounded from below. Thus, ths scheme s not well defned. To correct ths problem, Deelstra and Delbaen [5 have proposed to consder: ˆX n t +1 = ˆX n t + T n (a k ˆX n t + σ ˆXn t 1 ˆXn t > (W t +1 W t whle Dop proposes n [6: ˆX t n +1 = ˆX t n + T n (a k ˆX t n + σ ˆX t n (W t+1 W t. However, we can as proposed n Brgo and Alfons [3 obtan the postvty usng an mplct scheme. More precsely, f we rewrte the CIR process wth the postcpated stochastc

3 Dscretzaton schemes for the CIR processes 3 ntegral, we get, snce d X, W s = σ 2 ds: t t X t = x + (a kx s ds + σ Xs dw s { = x + lm (a kx t+1 T n n + σ (W Xt+1 t W +1 t ;t <t ;t <t σ ( X t+1 } X t (W t+1 W t ;t <t { = x + lm (a σ2 n 2 kx t +1 T n + σ } (W Xt+1 t W +1 t. ;t <t It s then natural to consder the followng mplct scheme that s well defned under the hypothess (2 at least when the tme step s small enough: ;t <t ˆX n t +1 = ˆX n t + (a σ2 2 k ˆX n t +1 T n + σ ˆXn t+1 (W t+1 W t. More precsely, when ˆX t n and T n 1/k (where y = max( y,, ˆXn t +1 can then be chosen as the unque postve root (snce 2a > σ 2, P ( < of the second-degree polynomal P (x = (1 + k T n x2 σ(w t+1 W t x ( ˆX t n + (a σ2 T, and we get 2 n ˆX t n +1 = σ(w t +1 W t + σ 2 (W t+1 W t 2 + 4( ˆX n t + (a σ2 2(1 + k T n T (1 + k T 2 n n 2. (3 Ths scheme s well defned and t s also easy to check that t preserves the monotoncty property satsfed by the CIR process: f x < x are two ntal condtons, the scheme satsfes ˆX t n < ˆX t n. Ths s an nterestng example of mplct scheme on the dffuson coeffcent whose general form s gven by Mlsten and al. (22 snce t leads to an analytcal formula. In the same sprt, we can look at the SDE that drves the square-root: d X t = a σ2 /4 2 dt k Xt dt + σ X t 2 2 dw t and consder the scheme obtaned by mplctng the drft. Ths gves also a second-degree equaton n ˆXn t +1 : ( 1 + kt 2n ˆX n t +1 [ σ 2 (W t +1 W t + ˆX t n ˆXn t+1 a σ2 /4 T 2 n =

4 Dscretzaton schemes for the CIR processes 4 that has also only one postve root when σ 2 < 4a and T n < 2/k, and t gves: σ ˆX t n 2 +1 = (W t +1 W t + ˆXn t + ( σ 2 (W t +1 W t + ˆXn t 2 + 4(1 + kt 2(1 + kt 2n a σ2 /4 2n 2 (4 In ths case ˆX t n +1 s stll an ncreasng functon of ˆXn t so that the monotoncty property s satsfed. One can wonder whether we can get other schemes lookng at the mplct scheme (mplct on the drft and the dffuson coeffcents wth the SDE satsfed by X α. It s not hard to see that the only two values of α that gve a second-degree equaton are 1 and 1/2. The other powers do not lead to analytcal formulas and requre a numercal resoluton. It s then nterestng to make a rough Taylor expanson of order 1 of these schemes,.e. we fx ˆX t n and only conserve the terms n T, (W n t +1 W t and (W t+1 W t 2. We get respectvely for the frst scheme (3 and the second (4: ( ˆX t n +1 ˆX t n 1 k T + σ ˆX t n n (W t+1 W t + σ 2 /2(W t+1 W t 2 + (a σ 2 /2 T n ˆX n t +1 ˆX n t ( 1 k T n + σ ˆX t n (W t+1 W t + σ 2 /4(W t+1 W t 2 + (a σ 2 /4 T n Ths ndcates us a famly of explct schemes E(λ for λ a σ 2 /4 that ensure nonnegatve values but not the property of monotoncty: ( ( ˆX t n +1 = 1 kt ˆX t n 2n + σ(w 2 t +1 W t 2(1 kt (5 2n +(a σ 2 /4T/n + λ[(w t+1 W t 2 T/n. It s well defned for kt/n 2. The expanson of the scheme (3 corresponds then to λ = σ 2 /4 whle the scheme (4 to λ =. It s nterestng here to notce that the mplct scheme on the square-root and the explct scheme E( have the same expanson (up to order 1 as the Mlsten scheme for (1 (whch can lead to negatve values lke the Euler scheme when k > and for k =, E( s exactly the Mlsten scheme. Let us menton also that we could have consdered as well the schemes obtaned by replacng the factor 1 kt by 1 kt/n n (5. 2n Ths paper ams to get results on the weak and strong convergence of these schemes. Let us menton here that Deelstra and Deelbaen have proven n [5 a strong convergence result for ther scheme. Dop also gets a strong convergence result n [6 but under some strong assumptons on the coeffcents. She also obtans a weak convergence rate that depends on parameters. We ntroduce a framework n Secton 2 that wll allow us to study smultaneously several schemes presented above. In Secton 3, we wll thus establsh a result of strong convergence for the schemes that satsfy an hypothess denoted by (H S. T n 2.

5 Dscretzaton schemes for the CIR processes 5 Then we analyze the weak error n Secton 4, establshng a convergence result wth a 1/n rate for schemes satsfyng an hypothess denoted by (H W. Moreover, an expanson of the weak error s gven for the schemes E(λ. Secton 5 presents numercal results. We study n partcular the strong convergence speed numercally and also calculate the computng tme requred by the several schemes. All the propertes put n evdence by our analyss are lsted n the concluson, and E( seems to be the scheme that gathers the most nterestng propertes. 2 Notatons and prelmnary lemmas 2.1 Some results on the CIR process Lemma 2.1. The moments of (X t t [,T are unformly bounded by a constant that depends only on the parameters (k, a, σ, x, T, and the order of the moment p N. More precsely, settng ũ p (t, x = E[X p t, there exsts smooth functons ũ j,p (t that depend on (k, a, σ such that: p ũ p (t, x = ũ j,p (tx j. j= Proof: We have ũ (t, x = 1 and n the case p = 1, ũ 1 (t, x = x + t (a kũ 1(s, x ds than can be solved: ũ 1 (t, x = x e kt + a 1 e kt k wth the conventon that 1 e kt = t for k =. Let us consder p 2 and assume the result k true for 1 j p 1. One has d(ũp(t,x = [ap + 1p(p dt 2 1σ2 ũ p 1 (t, x kpũ p (t, x. Hence, we have t ũ p (t, x = (e kt (x p p + [ap p(p 1σ2 (e ks p ũ p 1 (s, x ds and we get the nducton relatons t j p 1, ũ j,p (t = (e kt p [ap p(p 1σ2 (e ks p ũ j,p 1 (sds ũ p,p (t = (e kt p. Ths gves the desred result, and we remark ncdentally that ũ j,p (t can be wrtten as a polynomal of e kt or t dependng on whether we are n the case k or k =. 2.2 Introducton of the notatons O(1/n δ and O(1/n δ In ths secton, we ntroduce Landau type notatons for sequences of random varables that wll consderably smplfy formulas later. To allow the multplcaton of two O, we suppose

6 Dscretzaton schemes for the CIR processes 6 the exstence of moments of any order. The results presented here are elementary, and wll largely be used later. Defnton 2.2. Let us consder a doubly ndexed famly of random varables Z = (Z n γ n,γ wth n N and γ Γ n a nonempty set. We wll say that Z s of order δ R - and use the notaton Z n γ = O(1/nδ - f there exsts a famly of postve random varables (A n γ γ,n that have moments of any order unformly bounded (.e p N, κ(a, p >, n N, sup γ Γ n E[(A n γ p κ(a, p and such that: Z n γ A n γ/n δ Ths s clearly equvalent to the followng property: p N, κ(p >, n N, sup γ Γ n E[(n δ Z n γ p κ(p When n partcular the (Z n γ γ,n are determnstc, ths s equvalent to the boundedness of (n δ Z n γ γ,n and we use the standard notaton Z n γ = O(1/nδ. Remarks It s obvous but mportant to observe that Zγ n = O(1/nδ mples that E[Zγ n = O(1/nδ. 2. Typcally we wll use n the paper ths defnton for Γ n = {t, t 1,.., t n }. 3. A smple but fundamental example s W t n +1 W t n = O(1/ n whch s clear snce n Wt n W +1 t n law = N (, T has moments of any order. Proposton 2.4. If (Zγ n n N,γ Γ n and (Z γ n n N,γ Γ are two famles such that n Zn γ = O(1/n δ and Z γ n =, we have: O(1/nδ 1 c R, czγ n = O(1/n δ 2 d R, Zγ n /n d = O(1/n δ+d 3 Zγ n + Z γ n 4 d >, (Zγ n d = O(1/n dδ 5 Zγ nz n γ = O(1/nδ+δ where the famles n 3 and 5 are ndexed n Γ n Γ n. In partcular, f we have a famly of functons h n : Γ n Γ n, we have also: 3 Z n γ + Z n h n(γ = O(1/nnf(δ,δ 5 Z n γ Z n h n(γ = O(1/nδ+δ. Proof : 1 and 2 are obvous. To prove 3, let us assume for example that δ δ. Then, t s not hard to see that Z n = O(1/n δ. Snce a sum of L p random varables s L p, we conclude easly. 4 comes mmedately from the defnton whle 5 requres the use of Cauchy-Schwarz nequalty to get the boundedness of the moments. By Jensen s nequalty, we also easly check the followng result. Lemma 2.5. Let us consder a famly (G γ γ Γ n of σ-algebras and (Zn γ n N,γ Γ n a famly of random varables such that Z n γ = O(1/nδ, then E(Z n γ G γ = O(1/nδ.

7 Dscretzaton schemes for the CIR processes On the moments of the dscretzaton schemes Frst of all, we need the followng lemma to control the moments of the schemes presented here. Lemma 2.6. Let us suppose that ( ˆX t n s an nonnegatve adapted scheme (.e. ˆXn t s F t -measurable such that for all n N, ˆX n t = x n 1, ˆXn t+1 (1 + b/n ˆX n t + σ n t ˆX n t (W t+1 W t + O(1/n where (σt n s also supposed to be adapted wth σt n unformly bounded moments, that s ˆX t n = O(1. = O(1 and b >. Then, ( ˆX n t has Proof : Let us frst remark that t s suffcent to study the case b =. Indeed, (1 + b/n ˆXn t satsfes the condton above wth b = : we have for x [, n, 1 (1 + b/n x e b and thus on the one hand, (1 + b/n 1 /2 σt n s adapted and thanks to Proposton 2.4 s a O(1, and on the other hand (1 + b/n 1 O(1/n = O(1/n. We observe then that ˆX t n = O(1 (1 + b/n ˆXn t = O(1. By Defnton 2.2, there s A n = O(1 such that we can rewrte the nequalty (wth b = as follows : ˆX t n +1 ˆX t n + σt n ˆX t n (W t+1 W t + A n /n. We denote n ths proof κ(a, p = supe[ A n p. We are gong to check by on p that,n p N, supe [( ˆX nt p <. It s easy to check that E[ ˆX t n x + κ(a, 1 snce we have,n E[ ˆX t n +1 E[ ˆX t n + κ(a, 1/n. Let us assume for any q p 1, there s a postve constant κ(q such that E[( ˆX t n q κ(q. Snce ( ˆX t n +1 p p! ( ˆX l 1!l 2!l 3! t n ( l 1+l 2 /2 σt n (W t+1 W t l 2 (A n /n l 3, t s suffcent to l 1 +l 2 +l 3 =p [ control E(l 1, l 2, l 3 = E ( ˆX t n ( l 1+l 2 /2 σt n (W t+1 W t l 2 (A n /nl 3 for l 1 + l 2 + l 3 = p. If l 1 + l 2 /2 p 3/2, we have necessary l 3 + l 2 /2 3/2 and Hölder nequalty gves [ ((σ E(l 1, l 2, l 3 (κ(p 1 1/α n E t (W t+1 W t l β 1/β 2 (A n /n l 3 C(l 1, l 2, l 3 n 3/2 where α = p 1 l 1 +l 2 /2 and 1/α + 1/β = 1. Thus, there s a postve constant Cte such that : E[( ˆX n t +1 p E[( ˆX n t p + p n E[( ˆX n t p 1 A n + p n E[( ˆX n t p 1/2 σ n t E(W t+1 W t + p(p 1 E[( 2n ˆX t n p 1 (σt n 2 + Cte/n.

8 Dscretzaton schemes for the CIR processes 8 Usng once agan the Hölder nequalty to bound E[( ˆX n t p 1 A n from above, have for a constant C > E[( ˆX n t +1 p E[( ˆX n t p + C n (E[( ˆX n t p + 1 = E[( ˆX n t p (1 + C n + C n and then we easly conclude that E[( ˆX t n p + 1 (x p + 1e C. Now, we present a qute general framework that ncludes, as we wll see, the mplct scheme (3 and the explct schemes E(λ. The hypotheses that are stated below wll be useful later to get results of strong and weak convergence. Hypothess (H S We wll say that ( ˆX t n satsfes (H S f t s a nonnegatve adapted scheme such that: ˆX t n +1 = ˆX t n + T n (a k ˆX t n + σ ˆX t n (W t+1 W t + m n t +1 m n t + O(1/n 3/2 (6 where m n t +1 m n t s a martngale ncrement (.e. E[m n t +1 m n t F t = of order 1: m n t +1 m n t = O(1/n. (7 ˆX n t + k T n ˆX t n + σ ˆXn t (W t+1 W t + If t s satsfed, we get mmedately that ˆX t n +1 O(1/n usng that a k ˆX t n a + k ˆX t n. Therefore, we can apply the Lemma 2.6 and deduce that ˆX t n = O(1. We defne n that case the dscrete martngale (Mt n by { M n t = (8 M n t +1 M n t = σ ˆXn t (W t+1 W t + m n t +1 m n t. Thanks to Proposton 2.4 and Remark 2.3, we get Corollary 2.7. Under hypothess (H S, ˆX n t has unformly bounded moments, and we have: (M n t +1 M n t 2 = σ 2 ˆXn t (W t+1 W t 2 + O(1/n 3/2 ˆX n t +1 ˆX n t = O(1/ n. However, as we wll see when studyng the weak error, t can be useful to make a stronger assumpton to get a faster convergence. Hypothess (H W We say that a scheme ( ˆX t n satsfes (H W f t already satsfes (H S and moreover ˆX t n +1 = ˆX t n + T n (a k ˆX t n + σ ˆX t n (W t+1 W t + m n t +1 m n t + O(1/n 2 (9 E [( ˆX nt+1 ˆX nt 2 F t = σ 2 ˆXn t T/n + O(1/n 2. (1 The absence of term of order 3/2 n (1 and the knowledge of the expanson of the scheme (9 up to order 2 play a key role to get a weak error at most proportonal to the tme step.

9 Dscretzaton schemes for the CIR processes 9 Remark 2.8. Let us suppose that there s a functon ψ n (x, w whch s even wth respect to ts second argument w such that: m n t +1 m n t = ψ n ( ˆX n t, (W t+1 W t + O(1/n 3/2. Then, ( ˆX n t +1 ˆX n t 2 = σ 2 ˆXn t (W t+1 W t 2 + 2σ ˆXn t ψ n ( ˆX n t, (W t+1 W t (W t+1 W t + O(1/n 2, and therefore condton (1 s automatcally satfed thanks to Lemma Study of the expanson of the dfferent schemes In ths secton we examne each scheme presented n the ntroducton and our am s to dscuss whether t satsfes or not Hypotheses (H S and (H W defned before Expanson of the mplct scheme (3 We assume here that 2a > σ 2, and expand the relaton that defnes the mplct scheme (3: ˆX n t +1 = 1 (2σ 2 (W 4(1 + kt/n 2 t+1 W t 2 + 4( ˆX nt + (a σ2 T/n(1 + kt/n 2 +2σ(W t+1 W t σ 2 (W t+1 W t 2 + 4( ˆX t n + (a σ2 T/n(1 + kt/n (11 2 Let us now observe that σ 2 (W t+1 W t 2 + 4( ˆX t n + (a σ2 2 T/n(1 + kt/n 2 ˆX t n (1 + kt/n σ 2 (W t+1 W t 2 + 4(a σ2 2 (1 + kt/nt/n = O(1/ n, (12 usng Proposton 2.4. Thus, we have ˆX n t +1 = 1 1+kT/n ˆX n t + 1 (1+kT/n 3/2 σ ˆX t n (W t+1 W t + O(1/n whch gves that ˆX t n = O(1 usng Lemma 2.6. Once we know ths, we can contnue the expanson thanks to Proposton 2.4 and t s not hard to get: where ˆX n t +1 ˆX n t = T n (a k ˆX n t + σ2 2 [(W t +1 W t 2 T/n + M n t +1 M n t + O(1/n 2 (13 M n t s a dscrete F t -martngale defned by M n t = and M t n +1 = M t n + σ(w ( t +1 W t σ 2(1 + kt/n 2 (W 2 t+1 W t ˆX t n + (a σ2 2 T (1 + kt/n. n

10 Dscretzaton schemes for the CIR processes 1 Indeed, we have E( M n t +1 F t = M t n σ T + 2 2πn(1 + kt/n 2 xe x2 σ2 T 2 n x2 + 4( ˆX t n + (a σ2 T/n(1 + kt/ndx 2 }{{} = M n t. Moreover, we have ( M t n +1 M t n 2 = σ 2 (W t+1 W t 2 ˆXn t + O(1/n 2 and n partcular M t n +1 M t n = O(1/ n. Now, we can defne the martngale (m n t by m n t +1 m n t = σ2 2 [(W t +1 W t 2 T/n + M t n +1 M t n σ ˆX t n (W t+1 W t and t s easy from (13 to see that the propertes (6 and (9 are satsfed. Inequalty (12 gves us that M t n +1 M t n σ ˆXn t (W t+1 W t = O(1/n and therefore property (7 s satsfed by m n snce σ2 [(W 2 t +1 W t 2 T/n = O(1/n. We have frst shown thus that (H S s satsfed. Now, usng the Proposton 2.4, we get that: ( ˆX n t +1 ˆX n t 2 = σ 2 (W t+1 W t 2 ˆXn t +[σ 2 ((W t+1 W t 2 T/n + 2(a k ˆX n t T/n( M n t +1 M n t + O(1/n 2 and that the term of order 3/2, [σ 2 ((W t+1 W t 2 T/n + 2(a k ˆX t n T/n( M t n +1 M t n, has a null condtonal expectaton respect to F t snce t can be wrtten as an odd functon respect to the Brownan ncrement. Ths shows that we have (1 and (H W s also satsfed by ths mplct scheme Expanson of the mplct scheme (4 Let us assume here that 4a > σ 2. Expandng (4, we get: [ ˆX t n 1 +1 = 2 4(1 + kt 2n 2 ( σ (W t +1 W t + ( 2 ( σ 2 (W t +1 W t + ˆX t n kt a σ 2 /4 T/n 2n 2 ( σ ˆX t n 2 (W t +1 W t + 2 ( ˆX t n kt 2n a σ2 /4 T. 2 n Thus, usng the nequalty x 2 + x { 2x x 2 + y 2 + y/2 f x > for y, we get that f x [ (σ 2 ( ˆX t n 1 +1 (1 + kt 2n 2 2 (W t +1 W t + ˆX t n kt a σ 2 /4 T/n 2n 2 and we can therefore apply Proposton 2.6 to deduce that ˆX t n has bounded moments. Unfortunately, f we try now to get an expanson of ˆXn t up to order 3/2 by expandng the

11 Dscretzaton schemes for the CIR processes 11 square-root, we get a term n 1 q ˆXn t O(1/n 3/2 whch s hard to manage. Despte the good numercal convergence of ths scheme, our approach n ths paper dd not enable us to obtan theoretcal results for t Expanson of the explct scheme E(λ Let us assume here that 4a σ 2 and consder λ [, a σ 2 /4. Expandng (5, we get ˆX t n +1 ˆX t n = (a k ˆX t n T ( 2 ( 2 n + k2 4 ˆX T t n + kσ2 2 kt/(2n T n 8 (1 kt/(2n 2 n ( +σ ˆX t n σ 2 (W t+1 W t + 4(1 kt/(2n + λ ((W 2 t+1 W t 2 T/n σ ˆX t n (W t+1 W t + (k + k2 T ˆX n T t 4 n n + O(1/n. We can then apply Lemma 2.6 to deduce that ˆX t n = O(1. We have then an expanson analogous to that obtaned for the mplct scheme, that s ˆX t n +1 ˆX t n = T n (a k ˆX t n + σ ˆX t n (W t+1 W t + m n t +1 m n t + O(1/n 2 (14 where m n t s a F t -martngale defned by m n t = and m n t +1 m n t = ( σ2 4 + λ[(w t +1 W t 2 T/n. It s n ths case straghtforward to see that we have the propertes (6 and (9 and that the martngale ncrements satsfy (7 and (1 thanks to Remark 2.8. Hence, explct scheme E(λ fulflls the condtons of (H S and (H W. 3 Strong convergence In all ths secton, we consder a scheme ( ˆX t n that satsfes the hypothess (H S. We wll prove the strong convergence for t, followng the method proposed by Deelstra and Delbaen [5 that reles on Yamada s functons. Thus, we frst need to buld a contnuous adapted extenson of our scheme n order to use then Itô s formula. For that purpose, we need to explct the O terms and frst defne Zt n = O(1/n 3/2 as: ˆX n t +1 = ˆX n t + T n (a k ˆX n t + M n t +1 M n t + Z n t We can suppose that Zt n s F t -measurable. Indeed, f t were not the case, t would be suffcent then to consder the martngale ncrement M n t +1 M n t = M n t +1 M n t + Z n t E[Z n t F t and Z t n = E[Zt n F t nstead of respectvely Mt n +1 Mt n and Zt n. Thus, we have M t n +1 M t n + Z t n = Mt n +1 Mt n + Zt n and, thanks to Lemma 2.5, we get that Z t n = O(1/n 3/2, and also M n t +1 M n t = σ ˆXn t (W t+1 W t + O(1/n.

12 Dscretzaton schemes for the CIR processes 12 Now, we apply the martngale representaton theorem to the martngales {E(m n t +1 F t m n t, t [t, t +1 } to get the exstence of an F t -adapted process (Rt n, t T such that E(m n t +1 F t m n t = t t R n s dw s. In partcular, we know that t t Rs ndw s = O(1/n and so ( t t Rs ndw s 2 = O(1/n 2 whch gves us that, for t [t, t +1 : t t E[(R n s 2 ds = O(1/n 2. (15 Now, we are able to buld a contnuous extenson ( ˆX t n, t T F t-adapted of our dscretzaton scheme. Indeed, we defne for t [t, t +1 : ˆX t n = ˆX t n + (t t (a k ˆX t n + n t T Zn t + (σ ˆX t n + Rs n dw s. t Thus, namng η(t the functon defned on [, T by η(t = t for t [t, t +1, we can rewrte our scheme as follows: t ˆX t n = x + (a k ˆX η(s n + n t T Zn η(s (σ ds + ˆXn η(s + Rn s dw s. (16 Let us now ntroduce a famly of Yamada s functons (see Karatzas and Shreve [1 ψ ɛ,m parametrzed by two postve numbers ɛ and m. Snce we have ɛ 1 ɛe σ2 m du = m, there σ 2 u exsts a contnuous functon ρ ɛ,m wth a compact support n ɛe σ2m, ɛ[ such that ρ ɛ,m (x 2 for x > and ɛ σ 2 xm ɛe σ2 m ρ ɛ,m (udu = 1. We then consder ψ ɛ,m (x = x y ρ ɛ,m (ududy that can be vewed as a sequence of smooth approxmaton of x x when m s large and ɛ tends to. Indeed functons ψ ɛ,m thus satsfes: x ɛ ψ ɛ,m (x x, ψ ɛ,m (x 1, ψ ɛ,m (x = ρ ɛ,m( x Followng the method used by Deelstra and Delbaen [5, we frst wrte and then apply Itô s formula : ψ ɛ,m ( ˆX n t X t = 2 σ 2 x m. ˆX n t X t ɛ + ψ ɛ,m ( ˆX n t X t (17 t + (kx s k ˆX n η(s + n T Zn η(s ψ ɛ,m ( ˆX n s X sds t t (σ ˆXn η(s + Rn s σ X s ψ ɛ,m( ˆX n s X s dw s (σ ˆXn η(s + Rn s σ X s 2 ψ ɛ,m ( ˆX n s X sds =: I 1 (t, n + I 2 (t, n + I 3 (t, n.

13 Dscretzaton schemes for the CIR processes 13 The absolute value of the frst ntegral can be bounded usng that ψ ɛ,m 1 : I 1 (t, n k t For the thrd ntegral, we have that ( X s ˆX n s + ˆX n s ˆX n η(s ds + t (σ ˆXn η(s + Rn s σ X s 2 2(σ 2 X s ˆX n η(s + (Rn s 2 Therefore, usng that ψ ɛ,m (x x 2 σ 2 m I 3 (t, n 2t m + 2eσ2m ɛm n T Zn η(s ds. 2(σ 2 X s ˆX n s + σ2 ˆX n s ˆX n η(s + (Rn s 2. (18 and ψ t ɛ,m 2eσ 2 m σ 2 ɛm we get: ( ˆX n s ˆX n η(s + 1 σ 2 (Rn s 2 ds. Usng Lemma 2.1, Corollary 2.7 and (15, we check that E[I 2 (t, n =. Now, takng the expectaton n (17, we get ( t t [, T, E( ˆX t n X t ɛ + k E( ˆX s n X s ds + 2T m + 2e σ2 m σ 2 ɛm + k Cte n for some Cte >, usng that ˆX s n ˆX η(s n = O(1/ n and n T Zn η(s = O(1/ n. Gronwall s lemma leads then to [ ( t [, T, E( ˆX t n X t e k T ɛ + 2T m + 2e σ2 m σ 2 ɛm + k Cte. (19 n Now, takng m = 1 4σ 2 ln(n and ɛ = 1/ ln(n, we get that sup E( ˆX t n X t = O(1/ ln(n. (2 t T Now, we would lke to exchange the supremum and the expectaton. Doob s nequalty [ t gves E[ sup I 2 (s, n C E (σ ˆXn η(s + Rs n σ X s 2 (ψ ɛ,m( ˆX s n X s 2 ds. We use s t that ψ ɛ,m 1 and the nequalty (18, and then control each terms thanks to relatons (2 and (15, and observng that ˆX s n ˆX η(s n = O(1/ n: E[ sup I 2 (s, n = O(1/ ln(n. s t We can then use the same controls as before for I 1 and I 3 to conclude that ( E sup ˆX t n X t = O(1/ ln(n. (21 t T We sum up our results n the proposton that follows.

14 Dscretzaton schemes for the CIR processes 14 Proposton 3.1. Let us consder a dscretzaton scheme ( ˆX n that satsfes the hypothess (H S. Then, there exsts a postve constant C dependng on T and on the parameters (k, a, σ, x but not on n such that: 4 Weak convergence sup E( ˆX t n X t C/ ln(n n ( E sup ˆX t n X t = C/ ln(n. n In ths secton, we wll establsh a result that gves the convergence rate of E[f( ˆX T n to E[f(X T. We wll use the method ntroduced by Talay and Tubaro (199 to study that weak error and get also a convergence rate n 1/n provded that f s regular enough. We thus ntroduce the notaton Xt x to denote the CIR process wth ntal value x, and we frst need to establsh the followng techncal result. Proposton 4.1. Let us consder f : R + R a C q functon wth q 2, such that there s A > and m q, m N such that x, f (q (x A(1 + x m. Then u : [, T R + R defned by u(t, x = E[f(XT x t has successve dervatves x l l t u(t, x for l, l N and l + 2l q, that satsfy the followng property: C >, (t, x [, T R +, max l+2l q l x l t u(t, x C(1 + x m+q+l (22 and s a classcal soluton of the PDE: { t u(t, x + (a kx x u(t, x + σ2 2 x 2 xu(t, x = u(t, x = f(x. (23 More generally, let us assume that (f θ, θ Θ s a famly of C q functons wth q 2, such that there s A > and m q, m N such that θ Θ, x, f (q θ (x A(1 + x m and l < q, f (l ( A. (24 For τ T, we consder u θ,τ (t, x = E[f θ (Xτ t x for t τ and x. Then there s a constant C > that does not depend on τ and θ such that θ θ Θ, τ [, T, (t, x [, τ R +, max l+2l q l x l t u θ,τ(t, x C(1 + x m+q+l (25 The proof of ths proposton, manly based on the analytcal formula avalable for the transton densty of the CIR process s made n the Appendx A. We are now able to prove the man results of ths secton:

15 Dscretzaton schemes for the CIR processes 15 Proposton 4.2. Let f : R + R be a C 4 functon such that A, m >, x, f (4 (x A(1 + x m. Let us suppose moreover that the scheme ( ˆX n satsfes the hypothess (H W. Then, the weak error s n 1/n: E[f( ˆX n T = E[f(X T + O(1/n. More generally, f (f θ, θ Θ s a famly of C 4 functons satsfyng condton (24 for q = 4, E[f θ ( ˆX n t n j = E[f θ(x t n j + O(1/n where O(1/n has to be understood n the sense of Defnton 2.2 wth (θ, t n j Γ n = Θ {t n,..., t n n}. Proof : We have E[f( ˆX n T = E[u(T, ˆX n T and E[f(X T = u(, x so that: E[f( ˆX T n E[f(X T = E[u(T, ˆX n 1 T n u(, x = E[u(t +1, ˆX t n +1 u(t, ˆX t n. Let us consder (t, x and (s, y n [, T R +. We can apply the Taylor formula to t u(t, y up to order 2 and get: 1 u(s, y = u(t, y + (s t t u(t, y + (s t 2 (1 τ t 2 u(t + τ(s t, ydτ. Now, we apply Taylor formula to y u(t, y and y t u(t, y and we fnally get u(s, y = l+2l <4 x l (s t l tl (y x l u(t, x l!l! = 1 +(s t(y x 2 (1 ξ x 2 tu(t, x + ξ(y xdξ (y x4 + 3! 1 (1 ξ 3 x 4 u(t, x + ξ(y xdξ. Proposton 4.1 allows us then to get: u(s, y l (s x t l tl (y x l u(t, x l!l! l+2l <4 1 + (s t 2 (1 τ t 2 u(t + τ(s t, ydτ C(1 + max(x, y 6+m [ (s t 2 + s t (y x 2 + (y x 4 and we apply ths bound to (t, ˆX n t and (t +1, ˆX n t +1. Proposton 2.4 and Corollary 2.7 gve mmedately that C(1+max( ˆX n t, ˆX n t +1 6+m [(T/n 2 +( ˆX n t +1 ˆX n t 2 T/n+( ˆX n t +1 ˆX n t 4 = O(1/n 2 and therefore: u(t +1, ˆX n t +1 u(t, ˆX n t = <l+2l <4 x l t l u(t, ˆX ( ˆX n t n (T/nl t +1 ˆX t n l + O(1/n 2. (26 l!l!

16 Dscretzaton schemes for the CIR processes 16 Now we expand the powers of ( ˆX n t +1 ˆX n t up to order 2 usng the Hypothess (H W : ˆX t n +1 ˆX t n = T n (a k ˆX t n + σ ˆX t n (W t+1 W t + m n t +1 m n t + O(1/n 2 ( ˆX n t +1 ˆX n t 3 = σ 3 ( ˆX n t 3/2 (W t+1 W t 3 + O(1/n 2 Therefore, we get that [ E ˆXn t +1 ˆX t n F t E [( ˆX nt+1 ˆX nt 3 F t and accordng to (1, E [( ˆX nt+1 ˆX nt 2 F t = T n (a k ˆX n t + O(1/n 2 = O(1/n 2 = σ 2 ˆXn t T/n + O(1/n 2. The bound (22 and Lemma 2.6 ensure that x l t l u(t, ˆX t n = O(1 for l + 2l < 4. Thus, usng Lemma 2.5, we can deduce from (26 : [u(t +1, ˆX nt+1 u(t, ˆX nt F t E = <l+2l <4 x l l t u(t, ˆX (T/n l E [( ˆX nt+1 ˆX nt l F t n t + O(1/n 2 l!l! = t u(t, ˆX n t T/n + x u(t, ˆX n t T n (a k ˆX n t + 2 xu(t, ˆX n t σ2 2 ˆX n t T/n + O(1/n 2 = O(1/n 2 snce u solves the [ PDE (23. Therefore, there s a constant C > that does not depend on such that E u(t +1, ˆX t n +1 u(t, ˆX t n C/n 2 and so, we fnally get that E[f( ˆX T n E[f(X T C/n whch s the desred result. Now let us explan why ths proof can be generalzed easly to the case of the famly of functons f θ that satsfy (24 and all tmes t n j. We apply as before Taylor formula to functons u θ,t n j and thanks to Proposton 4.1, the bounds we have on ts dervatves do not depend on (θ, t n j and we get for < j n as n (26 u θ,t n j (t +1, ˆX n t +1 u θ,t n j (t, ˆX n t = <l+2l <4 x l l t u θ,t n(t j, ˆX ( ˆX n t n (T/nl t +1 ˆX t n l + O(1/n 2 l!l! wth the dfference that the O symbol s now meant wth Γ n = {(t n, tn j, < j n} Θ nstead of {t n, n} before. Then, the proof s the same, notcng that we stll have x l t l u θ,t n j (t, ˆX t n = O(1 for l + 2l q.

17 Dscretzaton schemes for the CIR processes 17 Remark 4.3. We desred to get a weak error n 1/n as n the case of the Euler scheme for stochastc dfferental equatons wth coeffcents regular enough (C 4 and bounded dervatves. Usng the argument of Talay and Tubaro, we need then a control on u(t +1, ˆX n t +1 u(t, ˆX t n up to order 2. Ths s why we assume to know the relaton (9 between ˆX t n +1 and ˆX t n up to order 2. Expandng u(t +1, ˆX t n +1 u(t, ˆX t n, we see that the term of order 1/2 has a null expectaton, the term of order 1/n s null snce u solves the PDE (23, but we need to requre condton (1 so that the term of order 3/2 has a null expectaton. If we had only assumed that the scheme satsfes (H S, we would have obtaned a weak error n 1/ n. Now, we would lke to expand further the weak error, n partcular to justfy the use the Romberg method that manly reles on the followng remark: f we know that there s c 1 R such that E[f( ˆX T n = E[f(X T + c 1 /n + O(1/n 2 2n, then 2E[f( ˆX T E[f( ˆX T n = E[f(X T + O(1/n 2 converges thus faster toward the desred expectaton. If we want to adapt the prevous proof, we see that we need to add the followng assumptons to get a weak error up to order ν N : f s regular enough (C 4ν and ts dervatves have a polynomal growth. We know the relaton between ˆX n t +1 and ˆX n t up to order ν + 1. Moreover, f we wsh to have as for the Euler scheme an error that expands only on the nteger orders: E[f( ˆX n T = E[f(X T + c 1 /n + c 2 /n c ν 1 /n ν 1 + O(1/n ν, we need to make assumptons of the same knd as (1 for any power of ( ˆX n t +1 ˆX n t to get terms of order nteger + one half wth null expectaton. However these assumptons would be hardly readable, and practcally, they would be clearly satsfed only by the explct schemes E(λ. That s why we prefer to state here drectly the result for the explct schemes E(λ. Proposton 4.4. Let ν N and f : R + R that we suppose C and such that q, A q >, m q N, f (q (x A q (1 + x mq. Let ( ˆX n be the explct scheme E(λ wth λ a σ 2 /4. Then, the weak error has an expanson up to order ν: E[f( ˆX n T = E[f(X T + c 1 /n + c 2 /n c ν 1 /n ν 1 + O(1/n ν where c 1 = T T E[ψ E(λ(t, X t dt wth ψ E(λ defned below n (28. Proof : Wth the same argument as n Proposton 4.2, frst usng the Taylor expanson respect to t and then to x, we get that there s C(ν > and M(ν N: u(s, y x l tl (y x l l t u(t, x(s l!l! l+2l <2ν+2 ν+1 C(ν(1 + max(x, y M(ν s t ν+1 j (y x 2j. j=

18 Dscretzaton schemes for the CIR processes 18 Smlarly, we get that u(t +1, ˆX n t +1 u(t, ˆX n t = <l+2l <2ν+2 x l l t u(t, ˆX ( ˆX n t n (T/nl t +1 ˆX t n l + O(1/n ν+1. l!l! and then E [u(t +1, ˆX nt+1 u(t, ˆX nt F t = <l+2l <2ν+2 +O(1/n ν+1. x l l t u(t, ˆX (T/n l E [( ˆX nt+1 ˆX nt l F t n t l!l! Let us frst expand (5 to get: ˆX t n +1 = ˆX t n + σ ˆX t n (W t+1 W t + (a k ˆX t n T ( n + ( + k2 4 ˆX t n (T/n 2 + σ2 4 (W t +1 W t 2 1 ( 1 kt 2n λ + σ ((W t+1 W t 2 T/n Snce 1 1 = (1 kt 2n 2 j 1 (j + 1(k/2j (T/n j, we get that ˆX t n +1 ˆX t n = σ ˆX t n (W t+1 W t + (a k ˆX t n T ( n + λ + σ2 ((W t+1 W t 2 T/n 4 + k2 4 ˆX t n (T/n 2 + σ2 4 (W ν 1 t +1 W t 2 (j + 1(k/2 j (T/n j + O(1/n ν+1. All the terms here are of nteger order but σ ˆXn t (W t+1 W t that s of order 1/2. Now, takng the power l of these expanson, we get usng Proposton 2.4 an expanson of ( ˆX n t +1 ˆX n t l up to order ν +1 (even ν +1+(l 1/2. What s mportant to remark s that the term of order nteger + one half comes from an odd power of σ ˆXn t (W t+1 W t and a product of the other terms. Snce all these other terms are even respect to (W t+1 W t, we fnally get that all the terms of order nteger + one half have a null condtonal expectaton. Thus, we see that we can wrte for l N E [( ˆX nt+1 ˆX nt l F t = j=1 ν φ l,j ( ˆX t n (T/n j + O(1/n ν+1 j l/2 where φ l,j are polynomal functons that we do not explct and satsfy φ l,j ( ˆX t n = O(1. Thus, (T/n l E [( ˆX nt+1 ˆX nt l F t = φ l,j ( ˆX t n (T/n j+l + O(1/n ν+1 l/2 j ν.

19 Dscretzaton schemes for the CIR processes 19 = l+2l 2j<2ν+2 φ l,j l ( ˆX t n (T/n j + O(1/n ν+1 and so, E [u(t +1, ˆX nt+1 u(t, ˆX nt F t ( ν (T/n j j=1 For ν = 2, one obtans: <l+2l 2j l x l t u(t, ˆX t n ( ˆX n φl,j l t + O(1/n ν+1. (27 l!l! = [ E ˆXn t +1 ˆX t n F t E [( ˆX nt+1 ˆX nt 2 F t E [( ˆX nt+1 ˆX nt 3 F t E [( ˆX nt+1 ˆX nt 4 F t E [( ˆX nt+1 ˆX nt 5 F t = (a k ˆX t n T/n + k2 ˆXn t + kσ 2 (T/n 2 + O(1/n 3 4 = σ 2 ˆXn t T/n + [(a k ˆX nt 2 + 2(λ + σ 2 /4 2 (T/n 2 + O(1/n 3 = 3σ 2 ˆXn t [a k ˆX t n + 2(λ + σ 2 /4 (T/n 2 + O(1/n 3 = 3σ 4 ( ˆX t n 2 (T/n 2 + O(1/n 3 = O(1/n 3 and so: E [u(t +1, ˆX nt+1 u(t, ˆX nt F t = (T/n 2 ψ E(λ (t, ˆX t n + O(1/n 3 where ψ E(λ (t, x = t u(t, x + k2 x + kσ 2 x u(t, x + (a kx x t u(t, x ( [ (a kx 2 + 2(λ + σ 2 /4 2 2 σ2 xu(t, x x 2 x tu(t, x + σ2 2 x [ a kx + 2(λ + σ 2 /4 3 xu(t, x + σ4 8 x2 4 xu(t, x. Therefore, summng and takng the expectaton, we get that E[f( ˆX T n = E[f(X T + (T/n 2 n 1 = E[ψ E(λ(t, ˆX t n + O(1/n 2. We then apply Proposton 4.2 to the famly of functons x ψ E(λ (t, x whch satsfes condton (24 thanks to (22 (we ncdentally remark that t s suffcent to have f C 8 to get the expanson wth ν = 2. It gves that E[ψ E(λ (t, ˆX t n = E[ψ E(λ (t, X t + O(1/n. Snce t E[ψ E(λ (t, X t s bounded on [, T, we have that (T/n n 1 = E[ψ E(λ(t, X t = T E[ψ E(λ(t, X t dt + O(1/n and then: T E[f( ˆX T n = E[f(X T + (T/n E[ψ E(λ (t, X t dt + O(1/n 2. (29 To get the expanson for ν = 3 and further, one has to check by nducton the desred result for any ν usng the same methodology.

20 Dscretzaton schemes for the CIR processes 2 5 Numercal results In ths secton, we wll analyze numercally the convergence of the dscretzaton schemes. For the theortcal study, an nterestng feature of the mplct schemes (3 and (4 and of the explct schemes E(λ, s ther automatc nonnegatvty for the followng parameters: Scheme Condton on (a, σ Implct (3 σ 2 2a Implct (4 σ 2 4a E(λ λ a σ 2 /4 (3 Indeed, contrary to the schemes usng a reflecton technque as those proposed by Deelstra- Delbaen or Dop, there s no need to control the reflecton. However, we can use the followng trck to extend schemes (3, (4 and E(λ to all the values of the parameters (k, a, σ: For the mplct schemes whch are defned wth second-degree polynomals, we wll set ˆX n t +1 = when the dscrmnant s negatve and else use formulas (3 and (4. For the explct schemes E(λ, we smply defne ˆX n t +1 as the postve part of the left-hand sde of (5 We wll use these extensons when needed for the smulatons presented n ths secton. 5.1 Numercal study of the strong convergence In ths paragraph we present a numercal analyss of the strong convergence of varous schemes. It does not seem possble to compute the lmt process on the same probablty space, and we overcome ths dffculty usng the followng lemma that says that t s suffcent to study the dfference between the values obtaned wth a scheme for a gven tme step and the ones obtaned wth the same scheme and a tme step twce smaller. Let us recall here that t n = T/n = t2 2n. Lemma 5.1. Let us consder a scheme ( ˆX t n that converges toward a contnuous process X t n the followng sense: [ E sup n ˆX t n X n t n. (31 n Then, for any α > and β, [ [ E sup n ˆX t n (ln nβ X n t n = O( E sup n n ˆX n α t n 2n ˆX t 2n 2 (ln nβ = O(. n α The condton (31 has been establshed n ths paper for the explct schemes E(λ and for the mplct scheme (3 and t has also been proved for the scheme of Deelstra-Delbaen [5. Under some restrctve condtons of the parameters, the scheme proposed by Dop converges wth a 1/ n rate [6. For the other parameters and for the Implct scheme

21 Dscretzaton schemes for the CIR processes 21 on the square-root (4, we can check numercally the condton (31 dong the comparson wth a scheme on whch ths comparson has been proved. [ Proof of the Lemma. If there s K > such that n N, E sup n ˆX t n (ln X n nβ t n K, n α then [ ( E sup n ˆX t n 2n (ln n β (ln n + ln 2β (ln nβ ˆX n t K + K. 2n 2 n α (2n α n α Recprocally, snce sup n ˆX t n X n t n [ get E sup n ˆX t n X n t n K l k= [ E sup n ˆX t n X n t n K l k= sup n (ln n+k ln 2 β (2 k n α + E k= ˆX n t 2k n 2 k [ sup n 2n ˆX t 2k+1 n 2 k+1 2n ˆX X t 2l+1 n t n 2 l+1 (ln n β + (k ln 2 β (ln nβ C β K (2 k n α n α 2n + sup ˆX X n t 2l+1 n t n, we 2 l+1 and wth l, for some constant K >, usng that k= kβ /2 k <. Now for[ the numercal study, we consder the standard tme nterval [, 1 (T = 1 and set S n = E sup n ˆX t n 2n ˆX. The fgures below show the convergence of S n t 2n n n functon 2 of the tme-step 1/n for dfferent parameters. Let us frst observe that the mplct scheme (4 and the explct scheme E( gve errors smaller than the others, for all the values of the parameters tested. Whch s also nterestng and nontrval s that the behavour of the convergence depends on the parameters. We notce that for the case 2a > σ 2 the schemes (4 and E( present an error whch looks lnear respect to the tme-step whle the others gve a square-root shape (see Fg. 1. Ths s not totally surprsng because we have seen that these schemes correspond to the Mlsten expanson, and we also know that under ths hypothess, X t never reaches so that the non-lpschtzan behavour of the square-root s less mportant. When 2a < σ 2 < 4a, the schemes (3, E(σ 2 /8, E(σ 2 /4, Deelstra-Delbaen and Dop, S n stll has a square-root behavour (see Fg. 2. Fnally, let us menton that for the last case σ 2 > 4a, the schemes (4 and E( stll gve the smaller value of S n. However, we have to say that when σ 2 >> 4a, the convergence s really slow. Lastly, concernng the mpact of λ for the explct schemes E(λ, we see (Fg. 1 and 2 that λ = σ 2 /4 s the parameter that gves a strong convergence analogous to the schemes of Dop, Deelstra-Delbaen and mplct (3; and the value of S n for E(σ 2 /8 s as one can expect between those of E( and E(σ 2 /4. To get an dea of the speed of convergence n functon of the parameters, we postulate that S n C/n α wth α > Thanks to the lemma, ths s equvalent to a strong convergence speed n 1/n α. To estmate α, we remark that log 1 (S n log 1 (S 1n n + α, and we have reported log 1 (S n log 1 (S 1n for n = 2 n Fgure 3. We have plotted the result n functon of the parameter σ 2 /(2a snce t s the one that plays a key role. Ths

22 Dscretzaton schemes for the CIR processes Imp (3, D-D, Dop and E(σ 2 / E(σ 2 /8.8 rag replacements.4 E( Imp ( Fgure 1: S n n functon of the tme-step 1/n for x = 1, k = 1, a = 1 and σ = Imp (3, D-D, Dop and E(σ 2 /4.8.6 E(σ 2 /8.4 rag replacements.2 E( Imp ( Fgure 2: S n n functon of the tme-step 1/n for x = 1, k = 1, a = 1 and σ = 3.

23 Dscretzaton schemes for the CIR processes Imp (4 E( Imp (3, D-D, Dop, E(σ 2 /8 and E(σ 2 /4.4.3 rag replacements Fgure 3: Speed convergence of S n : estmaton of the α parameter n functon of σ 2 /(2a for x = 1, k = 1 and a = 1. can be understood easly wth a tme-scalng. For the schemes (4 and E(, the estmated α s close to 1 for σ 2 < 2a and decreases from 1 to 1/2 for 2a < σ 2 < 4a whle for the other schemes, the estmated value of α s close to 1/2 for σ 2 < 4a. Intutvely, we can understand ths decrease because for σ 2 > 2a, X t can reach the orgn, and a non neglgble tme s spent n the neghbourhood of where the square root s non Lpschtz. Obvously, the speed of convergence may have a more complcated form than the one postulated, but our method gves nonetheless a good dea of ts behavour. 5.2 Numercal study of the weak convergence We have plotted n fgures 4,5,6 and 7, for fxed parameters of the CIR process, the approxmaton gven by the scheme or a Romberg extrapolaton of the expected value E[f(X 1 for the functon f(x = 5+3x4. Ths functon has been chosen to be senstve to varaton 2+5x for large and small values so that t catches the defaults of the schemes near and. We have taken two sets of parameters that llustrate the cases σ 2 2a and 2a σ 2 4a.

24 Dscretzaton schemes for the CIR processes 24 Implct (3 Implct (4 Dop Deelstra-Delbaen E( E(σ 2 /4 Exact σ = σ = Table 1: Smulaton tme (n s for 1 6 paths wth a tme step equal to 1 3 and parameters k = 1, a = 1 and x = 1. Let us recall here that we have proved here the O(1/n convergence only for regular functons and for the schemes satsfyng (H W, that s (3 wth σ 2 < 2a and E(λ wth λ a σ 2 /4. What comes out from the computatons (see Fgures 4 and 5 s that for the small values of σ (σ 2 2a, Fg. 4 all the schemes seem to have a behavour n O(1/n whle for the large values (σ 2 > 2a, Fg. 5, only the Explct schemes and the Deelstra-Delbaen scheme gve shapes compatble wth a behavour n n O(1/n. On the contrary, the scheme of Dop shows clearly a root shape whle the mplct schemes (3 and (4 seem to converge a lttle bt slower than K/n. Concernng the Romberg method to calculate E(f( ˆX 1, the fgure 6 show that n the both cases σ 2 2a and σ 2 > 2a, Dop s and mplct schemes (3 and (4 do not show a quadratc convergence. As expected, Explct schemes have a quadratc shape n all the cases even f, strctly speakng, we have not proved the speed convergence observed for E(σ 2 /8, k = 1, a = 1 and σ = 3 snce λ = σ 2 /8 > a σ 2 /4. Concernng the Deelstra- Delbaen scheme, let us frst say that for large tme-steps, negatve values may be frequent whch explans the strange behavour observed. However, for tme-steps small enough, the convergence seems compatble wth a quadratc convergence. 5.3 Computaton tme requred by the schemes In ths paragraph we compare the tme requred by the schemes and the exact method to smulate 1 6 paths wth a tme step equal to 1 3 on the tme nterval [, 1 (see Table 1. Concernng the exact smulaton of the ncrement of the CIR process, we have used the method proposed by Glasserman n [7 (see p As we could expect, ths method s more tme-consumng (up to a factor 1. Thus, t should be used to compute expectatons that depend on the values of the process (X t at a few fxed tmes. On the contrary, for expectatons that depends on all the path (such as ntegrals, dscretzaton schemes should be preferred. As we see n Table 1, the tme requred by the schemes presented are of the same order. Let us menton here that for the mplct scheme (4, one has to be careful and store at each step the value of ˆXn t so that only one square-root has to be computed at each tme step.

25 Dscretzaton schemes for the CIR processes E(σ 2 /8 E( Imp (3 D-D Dop rag replacements Imp ( Fgure 4: E(f( ˆX 2n 5+3x4 1 n functon of 1/n wth f(x = 2+5x for x =, k = 1, a = 1 and σ = Imp (3 E(σ 2 /8 E( 2.82 rag replacements D-D Dop Imp (3 Imp (4 E( E(σ 2 / Dop D-D 2.54 Imp ( Fgure 5: E(f( ˆX 2n 5+3x4 1 n functon of 1/n wth f(x = 2+5x for x =, k = 1, a = 1 and σ = 3.

26 Dscretzaton schemes for the CIR processes Imp (3 Dop D-D rag replacements E( Imp (4 E(σ 2 / Fgure 6: 2E(f( ˆX 2n 1 E(f( ˆX 1 n n functon of 1/n wth f(x = (5 + 3x4 /(2 + 5x for x =, k = 1, a = 1 and σ = Dop Imp ( rag replacements D-D Dop Imp (3 Imp (4 E( D-D E(σ 2 / Imp (4 E(σ 2 /8 E( Fgure 7: 2E(f( ˆX 2n 1 E(f( ˆX 1 n n functon of 1/n wth f(x = (5 + 3x4 /(2 + 5x for x =, k = 1, a = 1 and σ = 3.

27 Dscretzaton schemes for the CIR processes 27 Implct (3, σ 2 2a Implct (4, σ 2 4a Dop Deelstra Delbaen E(, σ 2 4a Nonnegatvty Y Y Y N Y Y Monotoncty Y Y N N N N Strong CV Y? Y Y Y Y Weak CV rate n 1/n Y? Y? Y Y Weak error expanson???? Y Y E(λ, < λ, λ a σ 2 /4 6 Concluson Table 2: Theoretcal results We have sum up n Table 2 the theoretcal results obtaned n ths paper and those of Dop, Deelstra and Delbaen [6, 5. We frst pont out whch scheme satsfy the algebrac propertes of postvty and monotoncty. Then, we examne among the several schemes whether t has been proved a result of strong convergence, a weak convergence rate n 1/n, an expanson of the weak error along the powers of 1/n. The star (Y means that the result has been establshed under some assumpton on the parameters whle the queston mark ndcates that no result has been shown yet. Let us menton here that Dop n [6 has also obtaned a strong convergence speed n 1/ n under some restrctve condtons on parameters. Table 3 presents the results of the numercal tests of Secton 5. All these results tend to show that the explct scheme E( s the one that gathers the most nterestng propertes. Moreover, t s really easy to mplement and s not more tme consumng than the other schemes. That s why n the general case, t s recommended to use ths scheme, at least for σ 2 4a. As a further work, t would be nterestng to get an accurate mathematcal study on the dependence of the strong convergence of E( on σ2 (see Fg. 3. It would be also 2a nterestng to study the behavour of the convergence of the varous schemes for large values of σ, (σ 2 4a. Snce none of the scheme studed n ths paper seems to be effcent for these large values of σ, desgnng a relevant scheme appears to be an nterestng challenge. Lastly, n a dfferent drecton, t would be nce to relax the condton of regularty on f for the weak error and prove estmates on the cumulated dstrbuton functon and the densty of X T, as n Bally and Talay [1, 2 or more recently Guyon [8. Acknowledgement. I am grateful to Benjamn Jourdan (ENPC-CERMICS for hs numerous and helpful comments. I also thank Chalnène Bassnah (Pars 13-Insttut Gallée for havng double checked some numercal results.

28 Dscretzaton schemes for the CIR processes 28 σ 2 [, 2a Implct (3 Implct (4 Dop Deelstra Delbaen Strong CV order 1/2 1 1/2 1/2 1 1/2 Weak CV rate n 1/n Y Y Y Y Y Y Romberg n 1/n 2 N N N Y Y Y E( E(λ, < λ, λ a σ 2 /4 σ 2 [2a, 4a Strong CV order 1/2 1/2 1/2 1/2 1/2 1/2 Weak CV rate n 1/n?? N Y Y Y Romberg n 1/n 2 N N N Y? Y Y Table 3: Numercal results A Proof of the Proposton 4.1 We wll focus for sake of smplcty on the case of one functon and one tme T before explanng how to extend the results to the case of a famly of functons that satsfy (24. We wll frst prove max l q l xu(t, x C(1 + x q+m (32 for some constant C >, and then (23, so that (22 wll outcome automatcally by an nducton on l, usng that for l 1 such that l + 2l q, ( x l l t u(t, x = l x (a kx x l 1 t u(t, x + σ2 2 x 2 1 x l t u(t, x = σ2 2 x l+2 x l 1 t u(t, x (l σ2 2 + a kx l+1 x l 1 t u(t, x + lk x l l 1 u(t, x. Let us set ũ(t, x = u(t t, x = E(f(Xt x. By Lemma 2.1, (32 holds for f(x = x p (p N and therefore for any polynomal. Now, usng the decomposton f(x = f(x P (x + P (x wth P (x = q l= f (l (x l /l!, we deduce that t s enough to prove (32 for f C q such that f(x A(1 + x m and f (l ( = for l q. Integratng successvely, we get easly that f (l (x A(1 + x m+q l and so, l q, f (l (x A(1 + x m+q. The densty of Xt x s known and s gven by: t p(t, x, z = e λtx/2 (λ t x/2 =! c t /2 Γ( + v/2 ( ct z 1+v/2 e c tz/2 2

29 Dscretzaton schemes for the CIR processes 29 where c t = We have for t > : where 4k, v = σ 2 (1 e kt 4a/σ2 and λ t = c t e kt. Let us remark here that I (f, c t = 4k, k > σ 2 4 c t c mn := σ, k = 2 T 4 k, k <. σ 2 (e k T 1 ũ(t, x = e λtx/2 (λ t x/2 I (f, c t! = c t /2 f(z Γ( + v/2 ( ct z 1+v/2 e c tz/2 dz. 2 Snce for l q, f (l (z A(1 + z m+q, we have ( ( m+q 2 N, I (f (l Γ( + m + q + v/2, c t A 1 +. (33 Γ( + v/2 Takng l =, the convergence of the above seres s ensured. Dervatng successvely n x, we get that for l q, c t t (, T, x R +, l xũ(t, x = e λtx/2 (λ t x/2 l! t(i (f, c t (34 = where t : R N R N s an the operator defned on sequences (I R N by t (I = λ t (I 2 +1 I = e kt c 2 t(i +1 I. Let us remark now that, snce f (l 1 ( =, an ntegraton by part gves for < l q and 1 I (f (l, c t = f (l 1 (c t /2 2 (z Γ( + v/2 f (l 1 (z (c t/2 2 ( 1 + v/2 Γ( + v/2 = c t 2 (I (f (l 1, c t I 1 (f (l 1, c t. ( ct z 1+v/2 e c tz/2 dz 2 Therefore, we get that t (I (f, c t = e kt I +1 (f (1, c t and fnally: ( ct z 2+v/2 e c tz/2 dz 2 t (, T, x R +, xũ(t, l x = e λtx/2 (λ t x/2 I +l (f (l, c t e klt.! = Usng (33, t gves mmedately that l xũ(t, x A The quotent Γ(+l+m+q+v/2 Γ(+l+p+v/2 ( 1 + 2m+q c m+q t = e λ t x/2 (λ tx/2! Γ(+l+m+q+v/2. Γ(+l+v/2 s a polynomal of degree m + q n, and we note β,..., β m+q

30 Dscretzaton schemes for the CIR processes 3 ts coeffcents n the bass {1,, ( 1,..., ( 1 ( (m + q + 1}. Thus, we get l A2m+q that xũ(t, x A + (β c m+q + β 1 λ t x + + β m+q (λ t x m+q and snce λ t c t e k T, t l xũ(t, x A + Ae(m+q k T ( β /c m+q mn + β 1 /c m+q 1 mn x + + β m+q x m+q. Ths allows us to conclude that there s a constant C > (that depends only on A, T and the parameters (x, k, a, σ such that l q, t (, T, x >, l xũ(t, x C(1 + xm+q. Proof of (23. We deduce from Lemma 2.1 that ũ (T t, x and ũ 1 (T t, x solve the PDE (23 and t s therefore suffcent to prove the result for functons f C 2 that satsfy f(x A(x 2 + x m. Let us now observe that dct = σ 2 c dt t λ t /4 and dλt = (σ 2 λ dt t /4 + kλ t. Then, t s no hard to get di (f,c t = (σ 2 /2 + a dt t (I (f, c t and that for any bounded d e sequence I, λ t x/2 (λ tx/2 I dt! = ( σ2 λ t + kx e λ t x/2 (λ tx/2 4! t (I. Combnng these = results, we get usng relaton (34: t ũ(t, x = ( σ2 λ t e λtx/2 (λ t x/2 + kx t (I (f, c t 4! = e λtx/2 (λ t x/2 + ( σ2! 2 + a t(i (f, c t = ( = (a kx x ũ(t, x + σ2 e λtx/2 (λ t x/2 λ tx 2! 2 + t (I (f, c t = = = (a kx x ũ(t, x + σ2 2 x e λtx/2 (λ t x/2! = = (a kx x ũ(t, x + σ2 2 x 2 xũ(t, x. λ t 2 ( t(i +1 (f, c t t (I (f, c t Fnally, the contnuty of f ensures that ũ(t, x = E(f(Xt x f(x when t thanks to Lebesgue s theorem. Let us explan now how to extend the result to a famly of functons f θ and get (25. Let us denote P θ (x = q 1 l= f (l l! θ (xq. Condton (24 ensures that the coeffcents of P θ are unformly bounded n θ. Wrtng f θ (x = P θ (x + (f θ (x P θ (x, one obtans (25 n the same way as (22. References [1 Bally, V. and Talay, D. (1996. The law of the Euler scheme for stochastc dfferental equatons I: convergence rate of the dstrbuton functon, Probab. Theory Related Felds, Vol. 14, pp