THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, Hui-Hsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No. 04 August 2013
Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif 1, Hui-Hsiung Kuo 2, Hbib Ouerdine 3 nd Benedykt Szozd 4 1 University of Tunis El Mnr, nrjeskhlif@yhoo.fr 2 Louisin Stte University, kuo@mth.lsu.edu 3 University of Tunis El Mnr, hbib.ouerdine@fst.rnu.tn 4 Arhus University, szozd@imf.u.dk Abstrct In this pper we use the new stochstic integrl introduced by Ayed nd Kuo (2008) nd the results obtined by Kuo et l. (2012b) to find solution to drift-free liner stochstic differentil eqution with nticipting initil condition. Our solution is bsed on well-known results from clssicl Itô theory nd nticiptive Itô formul results from Kuo et l. (2012b). We lso show tht the solution obtined by our method is consistent with the solution obtined by the methods of Mllivin clculus, e.g. Buckdhn nd Nulrt (1994). Keywords: dpted stochstic processes, nticipting stochstic differentil equtions, Brownin motion, Itô integrl, instntly independent stochstic processes, liner stochstic differentil equtions, stochstic integrl AMS Subject Clssifiction: 60H05, 60H20 1 Introduction The im of the present pper is to estblish solution to liner stochstic differentil eqution with n nticipting initil condition of certin form, nmely dx t = α t X t db t + β t X t dt t [, b (1.1) X = p(b b B ). In the cse with X = x R, it is well-known fct tht the unique solution is given by ( ) X t = x exp α s db s + βs 1 2 α2 s ds. (1.2) For detils, see for exmple, (Kuo, 2006, Section 11.1). The significnce of our result lys in the fct tht the solution X t of Eqution (1.1) is n nticipting stochstic process nd it cnnot be obtined by the clssicl tools from the Itô theory of 1
stochstic integrtion. Insted, we use the integrl of dpted nd instntly independent processes introduced by Ayed nd Kuo (2008, 2010) nd further developed by Kuo et l. (2012,b, 2013). In contrst to results obtined by Buckdhn nd Nulrt (1994) nd Esunge (2009), our results do not rely on Mllivin clculus or white noise nlysis nd re nchored in bsic probbility theory. The reminder of this pper is orgnized s follows. In Section 2 we recll ll the necessry definitions nd previous results used in the rest of the pper. Section 3 contins simple exmple tht illustrtes our methods nd Section 4 presents our min result, Theorem 4.1. We conclude with severl exmples in Section 5. 2 Preliminry Definitions In this section we fix the nottion nd recll severl definitions used in the reminder of this work. We denote by C k (R) the spce of ll functions f : R R tht re k times continuously differentible, nd by C (R) the spce of functions whose derivtives of ll orders exist nd re continuous. The spce of ll smooth functions whose Mclurin series converges for ll x R is denoted by M, tht is M = f C (R) f(x) = k=0 f (k) (x) x k for ll x R, k! where f (k) (x) stnds for the k-th derivtive of f(x). We denote by S(R) the Schwrtz clss of rpidly decresing functions, tht is S(R) = f C (R) sup x n f (m) (x) <, for ll m, n N. (2.1) x R It is well known fct tht S(R) is closed under the Fourier trnsform, which we define s ˆf(ζ) = R f(x)e 2πixζ dx, with the inverse Fourier trnsform given by f(x) = ˆf(ζ)e 2πixζ dζ. In this setting, we hve the following property of the Fourier R trnsform ( d ) dx f(x) (ζ) = 2πiζ ˆf(x). (2.2) Let (Ω, F, P ) be complete probbility spce, B t be stndrd Brownin motion on (Ω, F, P ) nd (F t ) t [0, ) be right-continuous, complete filtrtion such tht: 1. for ech t [0, ), the rndom vrible B t is F t -mesurble; 2. for ll s nd t such tht 0 s < t, the rndom vrible B t B s is independent of F s. Following Ayed nd Kuo (2008), we sy tht stochstic process X t is instntly independent with respect to (F t ) t [0, ) if for ech t [0, ), the rndom vrible X t is independent of F t. Recll tht if f t is dpted nd ϕ t is instntly independent with respect to (F t ), the Itô integrl of the product of f nd ϕ is defined s the limit b f t ϕ t db t = lim n 0 n f ti 1 ϕ ti (B ti B ti 1 ), (2.3) i=0 2
whenever the limit exists in probbility. Note tht if ϕ 1, then the integrl defined in Eqution (2.3) reduces to the ordinry Itô integrl for dpted processes. This kind of integrl ws introduced by Ayed nd Kuo (2008, 2010) nd studied further by Kuo et l. (2012,b, 2013). Following the nottion of Kuo (2006), we denote by L 2 d (Ω [, b) the spce of ll dpted stochstic processes X t such tht E[ b X2 t db t <. It is well-known fct tht the Itô integrl is well-defined for processes from L 2 d (Ω [, b). As in the Itô theory of stochstic integrtion, the key tool used in this work will be the Itô formul. We stte below one of the results of Kuo et l. (2012b) where the uthors provide severl formuls of this type. Multidimensionl version nd further generliztions of Itô formuls together with n nticiptive version of the Girsnov theorem cn be found in Kuo et l. (2013+). Theorem 2.1 (Kuo et l. (2012b, Corollry 6.2)). Suppose tht θ(t, x, y) = τ(t)f(x)ϕ(y), where τ C 1 (R), f C 2 (R), nd ϕ M. Let X t = where α, β L 2 d (Ω [, b). Then α s db s + β s ds, θ θ(t, X t, B b B ) = θ(, X, B b B ) + x (s, X s, B b B ) dx s + 1 2 θ 2 x (s, X s, B 2 b B ) (dx s ) 2 2 θ + x y (s, X s, B b B ) (dx s )(db s ) + θ t (s, X s, B b B ) ds. (2.4) Equivlently, we cn write the Eqution (2.4) in differentil form s dθ(t, X t, B b B ) = θ x (t, X t, B b B ) dx t + 1 2 θ 2 x (s, X t, B 2 b B ) (dx t ) 2 + 2 θ x y (t, X t, B b B ) (dx t )(db t ) + θ t (t, X t, B b B ) ds. (2.5) 3 A Motivtionl Exmple In this section, we present n exmple tht illustrtes the method for obtining solution of Eqution (1.1). We begin with the simplest possible cse of Eqution (1.1), 3
tht is we set α 1, β 0 nd p(x) = x, nd restrict our considertions to the intervl [0, 1. Thus we wish to find solution to dx t = X t db t, t [0, 1 (3.1) X 0 = B 1. The nturl guess for the solution of Eqution (3.1) is obtined by putting B 1 for x in Eqution (1.2) to obtin X t = B 1 exp B t 1 2 t. Using the Itô formul, it is esy to show tht the process X t is not solution of Eqution (3.1), but it is solution of dx t = X t db t + e Bt 1 2 t dt, (3.2) which is obviously different from Eqution (3.1). The filure of this pproch comes from the fct tht we do not ccount for the new fctor in the eqution, nmely B 1. To ccount for B 1 in Eqution (3.1), we cn introduce correction term to X t tht will counterct the dt term ppering in Eqution (3.2). Now, we will use the following s n nstz for the solution of Eqution (3.1) X t = (B 1 ξ(t)) exp B t 1 2 t, (3.3) where ξ(t) is deterministic function. The reson for this prticulr choice is simple. We see tht the difference between Equtions (3.2) nd (3.1) is the term expb t 1 t dt, nd to counterct this, we need to introduce nother dt-term with 2 the opposite sign. Looking t the Itô formul in Theorem 2.1, we see tht we hve to introduce correction fctor tht depends only on t. We use the Itô formul from Theorem 2.1 with θ(t, x, y) = (y ξ(t))e x 1 2 t, nd to obtin θ t = ξ (t)e x 1 2 t 1 2 (y ξ(t))ex 1 2 t, θ x = (y ξ(t))e x 1 2 t, θ xx = (y ξ(t))e x 1 2 t, θ xy = e x 1 2 t, dθ(t, B t, B 1 ) = (B 1 ξ(t)) e Bt 1 2 dbt + 1 (B 2 1 ξ(t)) e Bt 1 2 dt ) + e Bt 1 2 t dt (ξ (t)e Bt 1 2 t + 12 (B 1 ξ(t))e Bt 1 2 t dt ) = (B 1 ξ(t)) e Bt 1 2 dbt + (e Bt 1 2 t ξ (t)e Bt 1 2 t dt. So for X t = θ(t, B t, B 1 ) to be the solution of Eqution (3.1), function ξ(t) hs to stisfy the following ordinry differentil eqution ξ (t) = 1, t [0, 1 (3.4) ξ(0) = 0. 4
Thus, with ξ(t) = t, process X t given in Eqution (3.3) is solution to stochstic differentil eqution (3.1), tht is X t = (B 1 t) exp B t 1 2 t (3.5) solves Eqution (3.1). We point out tht the solution in Eqution (3.5) coincides with the one tht cn be obtined by methods of Buckdhn nd Nulrt (1994), where in Proposition 3.2 uthors stte tht the unique solution of Eqution (3.1) hs the form X t = g(t, x) exp B t 1t, 2 x=b1 where g solves the following prtil differentil eqution g t (t, x) = g x (t, x), t (0, 1 In our cse, g(t, x) = x t. 4 Generl Cse g(0, x) = x. Theorem 4.1 gives the solution to Eqution (1.1) for certin clss of coefficients α t nd β t, nd initil conditions p(x) with x = B b B. The proof of this theorem uses the ide of correction term introduced in the previous section. Theorem 4.1. Suppose tht α L 2 ([, b) nd β L 2 d (Ω [, b). Suppose lso tht p M S(R). Then the stochstic differentil eqution dx t = α t X t db t + β t X t dt, t [, b (4.1) X = p(b b B ), hs unique solution given by where nd X t = [ p(b b B ) ξ(t, B b B ) Z t, (4.2) ξ(t, y) = α s p (y Z t = exp α s db s + s ) α u du ds, (4.3) ( ) βs 1 2 α2 s ds. Remrk 4.2. Before we proceed with proof of Theorem 4.1, let us remrk tht if = 0, α t α nd β t β, tht is the coefficients re constnt nd evolution strts t 0, we cn gin pply the results of Proposition 3.2 of Buckdhn nd Nulrt (1994). In our nottion, the bove mentioned proposition sttes tht the solution to Eqution (4.1) hs the form X t = g(t, B 1 ) exp αb t + ( β 1 2 α2) t, (4.4) 5
where g(t, x) is the solution of the following prtil differentil eqution g t (t, x) = αg x (t, x) t (0, b) g(0, x) = p(x). (4.5) Hence in order to show tht our solution nd the one given by Eqution (4.4) coincide, it is enough to show tht g(t, x) = p(x) ξ(t, x) solves Eqution (4.5). Note tht in the cse of constnt coefficients, g(t, x) = p(x αt). Now it is mtter of simple computtion to check tht g solves Eqution (4.5). Proof. The uniqueness of solution follows from linerity of Eqution (4.1) nd stndrd rguments. To prove the existence of solution, first observe tht Z t is solution of stochstic differentil eqution given by dz t = α t Z t db t + β t Z t dt, t [, b Z = 1. Consider dx t = d [( p(b b B ) ξ(t, B b B ) ) Z t = d [ p(b b B )Z t d [ ξ(t, Bb B )Z t, where ξ(t, y) = ξ n (t)y n, for ll t 0, y R. (4.6) n=0 Note tht since the function zξ n (t)y n stisfies the ssumptions of the Theorem 2.1, we cn write ) d(z t ξ(t, B b B )) = d (Z t ξ n (t)(b b B ) n = = n=0 d (Z t ξ n (t)(b b B ) n ) n=0 n=0 [ ξ n (t)(b b B ) n dz t + Z t ξ (t)(b b B ) n dt + Z t ξ n (t)n(b b B ) n 1 (dz t )(db t ) = ξ(t, B b B ) dz t + Z t ξ t (t, B b B ) dt + ξ y (t, B b B )(dz t )(db t ). (4.7) 6
Using Theorem 2.1 nd Eqution (4.7) we obtin dx t = p(b b B )dz t + p (B b B )(dz t )(db t ) [ ξ t (t, B b B )Z t dt + ξ(t, B b B ) dz t + ξ y (t, B b B )(dz t )(db t ) = [ p(b b B ) ξ(t, B b B ) dz t [ + p (B b B )(dz t )(db t ) ξ t (t, B b B )Z t dt ξ y (t, B b B )(dz t )(db t ). So for X t to be solution of Eqution (4.1), we need p (B b B )(dz t )(db t ) ξ t (t, B b B )Z t dt ξ y (t, B b B )(dz t )(db t ) = 0 (4.8) for ll t [, b. Note tht (dz t )(db t ) = ( α t Z t db t + β t Z t dt ) (db t ) Putting together Equtions (4.8) nd (4.9) yields = α t Z t dt. (4.9) p (B b B )α t Z t dt ξ t (t, B b B )Z t dt ξ y (t, B b B )α t Z t dt = 0, or equivlently, [ p (B b B )α t ξ t (t, B b B ) ξ y (t, B b B )α t Xt dt = 0. Hence it is enough to find ξ(t, y) such tht p (y)α t ξ ξ (t, y) (t, y)α t y t = 0, t [, b ξ(0, y) = 0. (4.10) Thus the problem of finding solution to the stochstic differentil eqution (4.1) hs been reduced to tht of finding solution to the deterministic prtil differentil eqution (4.10). In order to solve Eqution (4.10), we pply the Fourier trnsform to both sides of Eqution (4.10), to obtin p (ζ)α t t ξ(t, ζ) 2πiζ ξ(t, ζ)α t = 0. (4.11) Note tht Eqution (4.11) is n ordinry differentil eqution in t, with n integrting fctor exp 2πiζ α s ds. 7
Hence Eqution (4.11) is equivlent to ( ξ(t, ζ) exp 2πiζ α s ds ) = p t (ζ)α t exp 2πiζ α s ds. (4.12) Integrtion with respect to t of both sides of Eqution (4.12) yields s ξ(t, ζ) exp 2πiζ α s ds = p (ζ) α s exp 2πiζ α u du ds + Ĉ(ζ), (4.13) for some function Ĉ(ζ) S(R). Thus, the Fourier trnsform of function ξ(t, y), tht is solution of Eqution (4.10), is given by ξ(t, ζ) = p (ζ) α s exp 2πiζ α u du ds + Ĉ(ζ) exp 2πiζ Now, we pply the inverse Fourier trnsform to get ξ(t, y) = p (ζ) α s exp 2πiζ α u du R s + Ĉ(ζ) exp 2πiζ α s ds R ( = α s p (ζ) exp πiζ y R s ( ) + Ĉ(ζ) exp πiζ y α u du dζ = R α s p (y s s ) ( α u du ds + C y s α s ds. (4.14) ds exp 2πiyζ dζ exp 2πiyζ dζ ) α u du dζ ds ) α s ds. Using the initil condition from Eqution (4.10), we see tht C(y) 0. Hence X t s in Eqution (4.2) is solution of Eqution (4.1). Remrk 4.3. Although very tedious, it is strightforwrd to check tht function ξ(t, y) in Eqution (4.3) cn be expressed in the form of Eqution (4.6). 5 Exmples Below we give severl exmples of stochstic differentil equtions with either deterministic or nticipting initil conditions. It is interesting to compre the solutions to see how nticipting initil conditions ffect the solutions. Exmple 5.1 (Adpted). Eqution dx t = X t db t + X t dt hs solution given by X 0 = x X t = x exp B t + 1 2 t. 8
Exmple 5.2 (Anticipting, compre with Exmple 5.1). Eqution dx t = X t db t + X t dt X 0 = B 1 hs solution given by X t = (B 1 t) exp B t + 1 2 t. Exmple 5.3 (Anticipting, compre with Exmple 5.1). Eqution dx t = X t db t + X t dt X 0 = e B 1 hs solution given by X t = e B 1 t exp B t 1 2 t Exmple 5.4 (Adpted). Eqution dx t = α t X t db t + β t X t dt X 0 = x hs solution given by X t = x exp α s db s + 0 0 ( ) βs 1 2 α2 s ds. Exmple 5.5 (Anticipting, compre with Exmple 5.4). Eqution dx t = α t X t db t + β t X t dt hs solution given by X t = ( B 1 Acknowledgments 0 X 0 = B 1 ) α s ds exp α s db s + 0 0 ( ) βs 1 2 α2 s ds. Hui-Hsiung Kuo is grteful for the support of Fulbright Lecturing/Reserch Grnt 11-06681, April 1 to June 30, 2012 t the University of Tunis El Mnr, Tunisi. Benedykt Szozd cknowledges the support from The T.N. Thiele Centre For Applied Mthemtics In Nturl Science nd from CREATES (DNRF78), funded by the Dnish Ntionl Reserch Foundtion. 9
References Wided Ayed nd Hui-Hsiung Kuo. An extension of the Itô integrl. Commun. Stoch. Anl., 2(3):323 333, 2008. Wided Ayed nd Hui-Hsiung Kuo. An extension of the Itô integrl: towrd generl theory of stochstic integrtion. Theory Stoch. Process., 16(1):17 28, 2010. R. Buckdhn nd D. Nulrt. Liner stochstic differentil equtions nd wick products. Probbility Theory nd Relted Fields, 99(4):501 526, 1994. Julius Esunge. A clss of nticipting liner stochstic differentil equtions. Commun. Stoch. Anl., 3(1):155 164, 2009. Hui-Hsiung Kuo. Introduction to stochstic integrtion. Universitext. Springer, New York, 2006. Hui-Hsiung Kuo, Anuwt Se-Tng, nd Benedykt Szozd. A stochstic integrl for dpted nd instntly independent stochstic processes. In Stochstic Processes, Finnce nd Control, Advnces in Sttistics, Probbility nd Acturil Science, chpter 3, pges 53 71. World Scientific, 2012. Hui-Hsiung Kuo, Anuwt Se-Tng, nd Benedykt Szozd. The Itô formul for new stochstic integrl. Communictions on Stochstic Anlysis, 6(4):604 614, 2012b. Hui-Hsiung Kuo, Anuwt Se-Tng, nd Benedykt Szozd. An isometry formul for new stochstic integrl. In Proceedings of Interntionl Conference on Quntum Probbility nd Relted Topics, My 29 June 4 2011, Levico, Itly, volume 29 of QP-PQ: Quntum Probbility nd White Noise Anlysis, pges 222 232, 2013. Hui-Hsiung Kuo, Yun Peng, nd Benedykt Szozd. Itô formul nd Girsnov theorem for the new stochstic integrl, 2013+. in preprtion. 10