On Delayed Logistic Equation Driven by Fractional Brownian Motion

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1 On Delayed Logiic Equaion Driven by Fracional Brownian Moion Nguyen Tien Dung Deparmen of Mahemaic, FPT Univeriy No 8 Ton Tha Thuye, Cau Giay, Hanoi, 84 Vienam dungn@fp.edu.vn ABSTRACT In hi paper we ue he fracional ochaic inegral given by Carmona e al. [] o udy a delayed logiic equaion driven by fracional Brownian moion which i a generalizaion of he claical delayed logiic equaion. We inroduce an approximae mehod o find he explici expreion for he oluion. Our propoed mehod can alo be applied o he oher model and o illurae hi, wo model in phyiology are dicued. Inroducion The claical logiic equaion wa propoed by Verhul [] o decribe populaion growh in a limied environmen and unil now i ha ill been very popular in populaion dynamic. However, in ome cae where here i a lag in ome of he procee involved Huchinon [3] poined ou ha he logiic equaion would be inappropriae for he decripion of populaion growh. A an example, he inroduced he delayed logiic equaion or Wrigh equaion dx = a bx τ X d, o model a ingle populaion whoe per capia rae of growh a ime. When modeling yem which do no noiceably affec heir environmen, ochaic procee are ofen ued o model he environmenal flucuaion, hu leading o ochaic differenial equaion wih or wihou delay. In he cae where noie i a Brownian moion, Alvarez and Shepp [4] and Guillouzic [5] udied he following ochaic logiic equaion, repecively dx = a bx X d + σx dw, X = x >, dx = a bx τ X d + σx dw, 3 where W i a andard Brownian moion. In he la decade here ha been an increaed inere in ochaic model baed on oher procee raher han he Brownian moion, much of he lieraure ha poined ou ha fracional Brownian moion or fracal noie provide a naural heoreical framework o model many phenomena ariing in biology for example, ee [6, 7]. Thi naurally lead u o inveigae logiic equaion driven by fracional Brownian moion, ha i o replace Brownian moion in equaion, 3 by a fracional Brownian moion. In [8] he auhor olved he ochaic differenial equaion wih polynomial drif and fracal noie which i a generalizaion of logiic equaion driven by fracional Brownian moion dx = ax bx n d + σx dw H 4

2 The udy of he ochaic differenial equaion depend on he definiion of he ochaic inegral involved. Among he many definiion of he fracional ochaic inegral we chooe, in hi paper, a definiion given by Carmona e al. [] o conider he delayed logiic equaion driven by fracional Brownian moion { dx = a bx τ X d + σx dw H, [,T ], X = φ, [ r,], where φ C[ r,], 5 where W H i a fracional Brownian moion fbm of he Liouville form wih Hur index H, ee he definiion in he following ex, a,b are poiive real number and σ i a real number. In he conex of populaion dynamic, τ characerize he reacion ime of he populaion o environmenal conrain, while b cale hee conrain and a i he Malhuian growh rae. Recenly, Ferrane el a. udied ome pecial form of he delayed ochaic differenial equaion driven by fracional Brownian moion in [9, ], however, all of hem do no cover 5. Moreover, heir mehod canno be ued in hi paper becaue our equaion 5 conain X in fracional ochaic inegral. Thi paper i organized a follow: In Secion, we recall a definiion of fracional ochaic inegral given in []. Secion 3 conain he main reul of hi paper ha we propoe an approximae mehod o find he explici oluion o he equaion 5. In Secion 4, we dicu ome oher model in phyiology o more clearly ee advanage of he approximae mehod ued in previou ecion. The concluion i given in Secion 5. Preliminarie In hi ecion we recall ome fundamenal reul abou he mehod of approximae fbm by emimaringale and a definiion of ochaic inegral wih repec o fracional Brownian moion. A fracional Brownian moion fbm of he Liouville form wih Hur index H, i a cenered Gauian proce defined by W H = K,dW 6 where W i a andard Brownian moion and he kernel K, = α, α = H. I i well known ha we canno ue claical Iô calculu o analyze model driven by fracional Brownian moion becaue fbm i neiher emimaringale nor Markov proce, excep for he cae where Hur index H =. For every ε > we define From [, ] we know ha W H,ε where ϕ ε = W H,ε = α + ε u α dw u. Moreover, W H,ε K + ε,dw = i a emimaringale wih he following decompoiion W H,ε = ε α W + + ε α dw. 7 ϕ ε d, 8 converge in L p Ω, p > uniformly in [,T ] o W H a ε : E W H,ε W H p c p,t ε ph. Le u recall ome elemen of ochaic calculu of variaion. For h L [,T ],R, we denoe by Wh he Wiener inegral T Wh = hdw.

3 Le S denoe he dene ube of L Ω,F,P coniing of hoe clae of random variable of he form F = f Wh,...,Wh n, 9 where n N, f C b Rn,L [,T ],R,h,...,h n L [,T ],R. If F ha he form 9, we define i derivaive a he proce D W F := {D W F, [,T ]} given by D W F = n k= f Wh,...,Wh n h k. x k For any p we hall denoe by D,p W he cloure of S wih repec o he norm F,p := [ E F p] [ T p + E D W u F p du] p. I i well known from [, 3] ha for an adaped proce f belonging o he pace D, W we have f dw H,ε = = f K + ε,dw + f K + ε,dw + f ϕ ε d f u f Ku + ε,duδw + u D W f u Ku + ε,ddu, where he econd inegral in he righ-hand ide i a Skorokhod inegral we refer o [4] for more deail abou he Skorohod inegral. Hypohei H: Aume ha f i an adaped proce belonging o he pace D, W and ha here exi β fulfilling β+h > / and p > /H uch ha [ f u f + T i f E L, := up β <<u<t ii up f belong o L p Ω. <<T ] D W r f u D W r f dr u β In [, 3] he auhor proved ha for f H, i finie, f dw H,ε of converge o he ame erm where ε =. Then, i i naural o define converge in L Ω a ε. Each erm in he righ-hand ide Definiion.. Le f H. The fracional ochaic inegral of f wih repec o W H i defined by f dw H = f K,dW + f u f Ku,duδW + u du D W f u Ku,d, where K, = α, K, = α α. 3 The main reul In hi ecion we conider fracional Brownian moion of he Liouville form wih Hur index H > and he following noaion i ued: C and for a finie conan no depending on ε and whoe value may vary from one occurrence o anoher. Theorem 3.. Suppoe ha µ i an adaped ochaic proce and fulfil he following condiion up e <<T µd L p Ω for ome p > H,

4 µd e L q Ω for ome q > 4, 3 T µ,4 d <. 4 Then he equaion ha a unique oluion in H, which i given by dx = µx d + σx dw H, X = x > 5 X = X e µd+σw H. 6 Proof. Normally, he way o olve he ochaic differenial equaion i baed on he Iô differenial formula. However, in our conex hi eem impoible ince, from Definiion., he oluion of 5 i a ochaic proce belonging o he pace H and ha ha a form X = X + µx d + σx K,dW + σx u X Ku,duδW + In order o find he oluion of 5 we conider approximaion equaion driven by emimaringale where W H,ε u σd W X u Ku,ddu. dx ε = µx ε d + σx ε dw H,ε,X ε = X = x 7 i defined by 7 wih ε,. And hen we prove ha he limi of X ε of 5. From decompoiion 8 above equaion can be rewroe ino in L Ω a ε will be he oluion dx ε = µ + σϕ ε X ε d + σε α X ε dw. 8 We can ee ha 8 i a claical Black-Schole ype equaion driven by Brownian moion, o i i eay o find i oluion X ε = X e µ σ ε α d+σw H,ε. 9 Uing chain rule of Malliavin derivaive we have for all u D W u X ε = Thu X ε D, W and hen by he equaion 7 i equivalen o X ε = X + µx ε d + σx ε K + ε,dw + u D W u µd + σ u + ε α X ε σxu ε X ε Ku + ε,duδw + σd W u X ε K + ε,udud.

5 Hence, in order o prove ha X defined by 6 i he oluion of 5 we need o how ha X ε converge o X in L Ω a ε and ha boh X ε and i limi X belong o H. Pu Y ε = µ σ ε α d + σw H,ε,Y = µd + σw H hen Y ε Y in L 4 Ω a ε. Indeed, we have E Y ε Y 4 = E σ ε α + σw H,ε 8 W H 4 σ ε α 4 + σ p E W H,ε W H 4 8 σ ε α T 4 + σ 4 c 4,T ε 4H Cε 8α. An applicaion of Lagrange heorem yield where Aε, = up miny ε,y x maxy ε,y E X ε X X E Aε,Y ε Y e x aify he following eimae Aε, exp Y ε +Y + Y ε Y ne+ n Y ε + ne + n Y + e n+ Y ε Y n a.. n + The condiion 3 and Gauian propery of W H,ε,W H imply ha Aε, ha finie qh momen by chooing n large enough. Hence, by applying Hölder inequaliy we ee ha here exi a finie conan C no depending on ε uch ha E X ε X C E Y ε Y 4 Cε 4α 3 which mean ha X ε converge o X in L Ω a ε. We now are poiion o prove ha X ε belong o H. For < T and p [,4] we have E X ε X ε p CX E Y ε Y ε p p p E µ p d + p σ p E W H,ε W H,ε p 4 C p + p C p. noing ha E W H,ε W H,ε C, ee [5]. Pu Z ε = D W u µd + σ u + ε α. We have alo E Z ε Z ε E D W u µ 4 d + σ 4 4α C 4α 5 we ued a fundamenal inequaliy ha a + b α a α + b α a,b >,α,. Conequenly E D W u X ε D W u X ε E Z ε X ε X ε + E X ε Z ε Z ε E Z ε 4 E X ε X ε 4 + E X ε 4 E Z ε Z ε 4 C α. 6 Combining 4 and 6 we ge X ε L, β C up < < <T α β.

6 Thu, X ε aifie he condiion i in H wih any β uch ha H < β < α = H. In order o prove X ε H,ε aifie he condiion ii in H we oberve ha up eσw belong o L p Ω for any p >. Moreover, up X ε = X <<T up <<T e µ σ ε α d+σw H,ε X <<T up <<T e µd up e <<T H,ε σw. 7 Applying Hölder inequaliy o 7 and uing he condiion we ge deired reul by chooing p large enough. The fac X belonging o H can be proved imilarly wih noing ha The proof of he heorem i complee. D W u X = u D W u µd + σ u α X. Remark 3.. If we ue he Sraonovich inegral a in [9] hen o he be of our knowledge, here are no any lieraure abou he explici oluion of he equaion 5. In order o prove he exience and uniquene of he oluion of 5 we ue he mehod of ep a in he heory of claical delayed differenial equaion. We hall fir prove he reul for he inerval [, r], hen we ue hi oluion proce a he iniial condiion o olve he equaion wihin he inerval [r,r], and o on. We need he following echnical lemma. Corollary 3.. Le µ be a proce of bounded from above, i.e. here exi a conan M > uch ha µ M a.. Moreover, T µ,4 n d < for ome n. 8 Then he unique oluion X in H of he equaion 5 aifie T X,4 n d <. 9 Proof. The condiion µ M a.. and 8 imply, 3 and 4. So he equaion 5 ha a unique oluion in H. The inequaliy 9 i eay o check ince X = X e µd+σw H X e M+σW H, D W u X = u D W u µd + σ u α X X u D W u µd + σt α e M+σW H. Theorem 3.. The delayed logiic equaion driven by fracional Brownian moion 5 admi a unique oluion in H: X = φ, [ r,], X = φe a bx r d+σw H, [,T ].

7 Proof. For impliciy le u aume T = Nr, where N i a poiive ineger number. The heorem will be proved by inducion where our inducion hypohei, for n < N, i he following: H n The equaion X = φ + a bx r X d + wih X =, > nr, ha a unique oluion in H and hi oluion aifie σx dw H, [,nr], 3 T X,4 N nd <. 3 Check H. Le [,r]. Then X r = φ r and he equaion 3 become X = φ + µ X d + σx dw H, [,r], 3 where µ = a bφ r. I i obviou ha µ aify he condiion in Corollary 3. ince φ i a coninuou deerminiic funcion. So H i rue. Inducion. Aume ha H i i rue for i n, wih n < N. We wih o prove ha H n+ i rue alo. Conider he ochaic proce defined a φ r if r, V = X r if r < n + r, if > n + r where X i he oluion obained in H n. Pu µ n = a bv. Thu, for [,n + r], our problem ha become he equaion X = φ + µ n X d + σx dw H, [,n + r], 33 Since φ i a coninuou deerminiic funcion and X r i poiive a.. we have µ n i bounded from above. Moreover, µ n aifie he condiion 8 wih n = N n by inducion hypohei. Applying Corollary 3. one again yield H n+ i rue. The proof of he heorem i complee. 4 Some oher model in Phyiology From pracical poin of view, i i imporan o find he explici expreion for he oluion of each pecific model. In hi ecion we wan o menion wo model in phyiology propoed by Mackey and Gla [6] o ee more clearly he advanage of emimaringale approximae mehod. If we aume ha he noie i in proporion wih he ize of populaion hen we ge he following wo model driven by fracional Brownian moion dx = λ αv mxx n r θ n + x n d + σxdw H, 34 r β θ n d p = θ n + p n r γp d + σpdw H, 35

8 where λ,α,v m,n,r,θ,β and γ are poiive conan. The fir equaion i ued o udy a dynamic dieae involving repiraory diorder, where x denoe he arerial CO concenraion of a mammal, λ i he CO producion rae, V m, denoe he maximum venilaion rae of CO, and r i he ime beween oxygenaion of blood in he lung and imulaion of chemorecepor in he brainem. The la equaion i propoed a model of hemaopoiei, p denoe he deniy of maure cell in blood circulaion, and r i he ime delay beween he producion of immaure cell in he bone marrow and heir mauraion for releae in he circulaing bloodream. A we have een in Secion 3, he key problem i o udy ochaic differenial equaion driven by fracional Brownian moion of he following form dx = λ + µx d + σx dw H, X = x >, 36 where µ and λ are bounded ochaic procee ince he ize of populaion i poiive and µ = αv mx n r θ n +x n r,λ = λ for equaion 34, µ = γ,λ = β θ n θ n +p n r for equaion 35, Under he bounded condiion of µ and λ, by uing emimaringale approximae mehod we can find he explici oluion of 36 which i given X = Φ X + λφ d, where Φ = exp µd + σw H. Then, for example, he oluion of 34 i given by x = exp αv m x n r H θ n + x n d + σw r φ + αv m x n u r H λexp θ n + x n du σw d. u r 5 Concluion In hi paper, we udied he delayed logiic model driven by fracional Brownian moion. Our main conribuion i o inroduce a mehod of approximaion for finding an explici expreion of he oluion. We alo howed ha hi mehod can be ued o udy ome oher model in phyiology. Acknowledgmen. The auhor would like o hank he anonymou referee for heir valuable commen for improving he paper. Reearch wa parially compleed while he auhor wa viiing he Iniue for Mahemaical Science, Naional Univeriy of Singapore in. Reference [] Carmona, P., Couin, L. and Moneny, G., 3. Sochaic inegraion wih repec o fracional Brownian moion. Ann. In. H. Poincaré Probab. Sai., 39, pp [] Verhul, P.F., 838. Noice ur la loi que la populaion ui dan on accroiemen. Correpondence Mah. Phy.,, pp. 3-. [3] Huchinon, G.E., 948. Circular caual yem in ecology. Ann. NY Acad. Sci., 5, pp [4] Alvarez, L. and Shepp, L., 998. Opimal harveing of ochaically flucuaing populaion. J. Mah. Biol., 37, pp [5] Guillouzic, S., L Heureux, I., Longin, A., 999. Small delay approximaion of ochaic delay differenial equaion. Phy. Rev. E, 59, pp [6] Bainghwaighe, J.B., Liebovich, L.S., and We, B.J., 994. Fracal Phyiology Mehod in Phyiology, Vol.. American Phyiological Sociey / Oxford. [7] Loa, G.A., Merlini, D., Nonnenmacher,T.F., and Weibel, E.R., 5. Fracal in Biology and Medicine: Volume IV Mahemaic and Biocience in Ineracion. Birkhäuer Verlag. [8] Dung, N.T., 8. A cla of fracional ochaic differenial equaion. Vienam Journal of Mahemaic, 363, pp

9 [9] Ferrane, M., Rovira, C., 6. Sochaic delay differenial equaion driven by fracional Brownian moion wih Hur parameer H >. Bernoulli,, pp [] Ferrane, M., Rovira, C.,. Convergence of delay differenial equaion driven by fracional Brownian moion. Journal of Evoluion Equaion, 4, pp [] Dung, N.T.,. Semimaringale approximaion of Fracional Brownian moion and i applicaion. Compuer and Mahemaic wih Applicaion, 6, pp [] Thao, T.H., 6. An Approximae Approach o Fracional Analyi for Finance. Nonlinear Analyi, 7, pp [3] Couin, L., 7. An Inroducion o Sochaic Calculu wih Repec o Fracional Brownian moion. In Séminaire de Probabilié XL. Springer, Verlag Berlin Heidelberg, pp [4] Nualar, D., 6. The Malliavin Calculu and Relaed Topic. nd ediion, Springer. [5] Dung, N.T., Thao, T.H.,. An Approximae Approach o Fracional Sochaic Inegraion and I Applicaion. Braz. J. Probab. Sa., 4, pp [6] Mackey, M.C., Gla, L., 977. Ocillaion and chao in phyiological conrol yem. Science, 97, pp