Explicit form of global solution to stochastic logistic differential equation and related topics

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1 SAISICS, OPIMIZAION AND INFOMAION COMPUING Sa., Opim. Inf. Compu., Vol. 5, March 17, pp Publihed online in Inernaional Academic Pre ( Explici form of global oluion o ochaic logiic differenial equaion and relaed opic Dmyro Boryenko 1, Olekandr Boryenko, 1 Deparmen of Inegral and Differenial Equaion, ara Shevchenko Naional Univeriy of Kyiv, Kyiv, Ukraine Deparmen of Probabiliy heory, Saiic and Acuarial Mahemaic, ara Shevchenko Naional Univeriy of Kyiv, Kyiv, Ukraine (eceived: 5 November 16; Acceped: 5 February 17) Abrac hi paper preen he lici form of poiive global oluion o ochaically perurbed nonauonomou logiic equaion dn() = N() (a() b()n())d + α()dw() + γ(, z) ν(d, dz), N() = N, w() i he andard one-dimenional Wiener proce, ν(, A) = ν(, A) Π(A), ν(, A) i he Poion meaure, which i independen on w(), Eν(, A) = Π(A), Π(A) i a finie meaure on he Borel e in. If coefficien a(), b(), α() and γ(, z) are coninuou on, -periodic on funcion, a() >, b() > and a() α () γ (, z) 1 + γ(, z) Π(dz) d >, hen here exi unique, poiive -periodic oluion o equaion for E1/N(). Keyword Nonauonomou Sochaic Logiic Differenial Equaion, Explici Form, Global Soluion, Periodical Soluion AMS 1 ubjec claificaion. Primary: 9D5, 6H1 Secondary: 34C5 DOI: /oic.v5i Inroducion he conrucion of he logiic model and i properie are preened in M. Iannelli and A. Pugliee 1. A deerminiic nonauonomou logiic equaion ha a form dn() = N() (a() b()n()) d, N() = N >, and model he number N of a ingle pecie whoe member compee among hemelve for a limi amoun of food and living pace. Here a() i he rae of growh and a()/b() i he carrying capaciy a ime. In he paper by D. Jiang and N. Shi i i conidered he nonauonomou ochaic logiic differenial equaion dn() = N() (a() b()n())d + α()dw(), N() = N, (1) Correpondence o: Olekandr Boryenko ( odb@univ.kiev.ua). Deparmen of Probabiliy heory, Saiic and Acuarial Mahemaic, ara Shevchenko Naional Univeriy of Kyiv, Volodymyrka 64 Sr., Kyiv 161, Ukraine. ISSN (online) ISSN 311-4X (prin) Copyrigh c 17 Inernaional Academic Pre

2 D. BOYSENKO AND O. BOYSENKO 59 w() i he andard one-dimenional Wiener proce, N > i a random variable independen on w(), a(), b() and α() are bounded, coninuou funcion. he auhor prove, ha if a() >, b() >, hen here exi a unique coninuou, poiive global oluion N() o equaion (1). I i obained he lici repreenaion of hi global oluion. If b() <, hen he equaion (1) ha he only local oluion. In he cae, when a(), b() and α() are coninuou -periodic funcion, a() >, b() > and a() α ()d > he auhor prove ha equaion for funcion E1/N() ha a unique poiive -periodic oluion E1/N p () and lim + (E1/N() E1/N p ()) = N() i he oluion o he equaion (1) for any iniial value N() = N >. In hi paper, we conider he ochaic nonauonomou logiic differenial equaion of he form dn() = N() (a() b()n())d + α()dw() + γ(, z) ν(d, dz), N() = N, () w() i he andard one-dimenional Wiener proce, ν(, A) = ν(, A) Π(A), ν(, A) i he Poion meaure, which i independen on w() and N >, Eν(, A) = Π(A), Π(A) i a finie meaure on he Borel e A in. he coefficien of equaion () do no aify he linear growh condiion, bu hey are local Lipchiz coninuou, o (cf. I.I. Gikhman and A.V. Skorokhod 3) he local oluion o equaion () exi. We will obain he lici form of global oluion o equaion (). In he cae when coefficien a(), b(), α() and γ(, z) are coninuou on, -periodic on funcion, a() >, b() > and a() α () γ (, z) 1 + γ(, z) Π(dz) d >, we will how, ha he equaion for funcion E1/N() ha a unique poiive -periodic oluion. Mainly we ue he noaion and approache propoed in D. Jiang and N. Shi. If coefficien γ(, z), hen our reul are conien wih he correponding reul in D. Jiang and N. Shi. he re of hi paper i organized a follow. In Secion, we obain he lici repreenaion of he unique global poiive oluion o equaion (). In Secion 3, we derive he lici form of unique poiive -periodic oluion o he equaion for E1/N() in he cae of periodical coefficien, and in Secion 4, we conider he ochaic nonauonomou logiic differenial equaion of he form dn() = N() θ > i an odd ineger. (a() b()n θ ())d + α()dw() + γ(, z) ν(d, dz),. Explici form of global oluion Le (Ω, F, P ) be a probabiliy pace and w(), i a andard one-dimenional Wiener proce on (Ω, F, P ), N > i a random variable on (Ω, F, P ), which i independen on w(), and ν(, A) = ν(, A) Π(A) i a cenered Poion meaure defined on (Ω, F, P ) independen on w() and on N. Here Eν(, A) = Π(A), Π( ) i a finie meaure on he Borel e in. On he probabiliy pace (Ω, F, P ) we conider an increaing, righ coninuou family of complee ub-σ-algebra F, F = σn, w(), ν(, A),. he main reul of hi ecion i following heorem 1 Le a() >, b() > and α() be a bounded coninuou funcion defined on, + ). Aume ha Π() < and γ(, z) i coninuou on funcion and ln(1 + γ(, z)) K for ome conan K >. hen here exi a unique poiive oluion N() o equaion () for any iniial value N() = N >, which i global and ha a repreenaion a() β()d + α()dw() + ln(1 + γ(, z)) ν(d, dz) N() = 1/N + b() a(τ) β(τ)dτ + α(τ)dw(τ) + ln(1 + γ(τ, z)) ν(dτ, dz) d, Sa., Opim. Inf. Compu. Vol. 5, March 17

3 6 EXPLICI FOM OF GLOBAL SOLUION O SOCHASIC LOGISIC EQUAION β() = α () + γ(, z) ln(1 + γ(, z))π(dz),. Proof. he coefficien of he equaion () are local Lipchiz coninuou. herefore for any iniial random value N >, which i independen on w() and ν(, A), here exi a unique local oluion N() on, τ e ), up <τe N() = + (cf. heorem 6, p.46,3). We will derive he lici form of hi oluion and we will ee, ha hi oluion i global. Le ) x() = e (1/N η() + b()e η() d, (3) η() = a() β()d + α()dw() + ln(1 + γ(, z)) ν(d, dz). Uing he Io formula, we derive he ochaic differenial equaion for he proce x(): dx() = ( ) b() x() a() α γ (, z) () 1 + γ(, z) Π(dz) d γ(, z) x() α()dw() + ν(d, dz), x() = 1/N. (4) 1 + γ(, z) Le N() = 1/x(), hen N() > and N() i ochaically coninuou proce wih rajecorie, which are righ coninuou and have a lef limi a every almo urely. Under condiion of he heorem, N() will no lode in a finie ime. By Io formula, we obain from (4): dn() = N() (a() N()b()) d + α()dw() + γ(, z) ν(d, dz), N() = N. hu N() i a unique poiive oluion o equaion () and N() i global on, + ), i.e. τ e = + almo urely. emark 1 If b() <, hen equaion () ha only he local oluion N() = a() β()d + α()dw() + ln(1 + γ(, z)) ν(d, dz) 1/N b() a(τ) β(τ)dτ + α(τ)dw(τ) + ln(1 + γ(τ, z)) ν(dτ, dz) d, < τ e, loion ime defined by τ e = inf : b() a(τ) β(τ)dτ + α(τ)dw(τ) + ln(1 + γ(τ, z)) ν(dτ, dz) d = 1/N. 3. Periodic oluion o equaion for E1/N() From he Lemma (p.465,3) we have he following reul. Sa., Opim. Inf. Compu. Vol. 5, March 17

4 D. BOYSENKO AND O. BOYSENKO 61 Lemma 1 Le Π() <, funcion α(), γ(, z) be coninuou on and bounded, hen E α()dw() + γ(, z) ν(d, dz) = σ()d,, σ() = α () + e γ(,z) 1 γ(, z) Π(dz). Le u conider equaion () wih -periodic on coefficien. heorem Le coefficien a(), b(), α() and γ(, z) be coninuou on and -periodic on funcion, a() >, b() > and a() δ ()d >, δ () = α γ (, z) () γ(, z) Π(dz), Π() <, ln(1 + γ(, z)) K for ome conan K >. hen he equaion d d E1/N() = δ () a()e1/n() + b(), (5) N() i a oluion o equaion (), ha a unique poiive -periodic oluion E1/N p () = + a(τ) δ (τ)dτ b()d, (6) a(τ) δ (τ)dτ 1 and for oluion N() o equaion () wih any iniial value N() = N >, E1/N < we have Proof. From (3) we have x() = β() a()d + By Lemma 1 we obain b() β(τ) a(τ)dτ E α () = d + lim (E1/N() E1/N p()) =. (7) α()dw() α()dw() α(τ)dw(τ) herefore, uing he independence of w(), ν(, A) and N, we derive Ex() = E1/N() = α () + + b() 1 ln(1 + γ(, z)) ν(d, dz) N ln(1 + γ(τ, z)) ν(dτ, dz) d. ln(1 + γ(, z)) ν(d, dz) ln(1 + γ(, z)) Π(dz)d, <. 1 + γ(, z) α (τ) + γ (, z) Π(dz) a() 1 + γ(, z) γ (τ, z) Π(dz) a(τ) 1 + γ(τ, z) d E1/N dτ d. Sa., Opim. Inf. Compu. Vol. 5, March 17

5 6 EXPLICI FOM OF GLOBAL SOLUION O SOCHASIC LOGISIC EQUAION I i eay o ee ha funcion Ex() = E1/N() aifie he equaion (5). By he condiion he equaion (5) ha a unique poiive -periodic oluion (6)(cf. p.8, 4). Le u denoe p() = a() δ (), p = p() d. If N() i he oluion o equaion () for any iniial value N() = N >, hen + From (8) we have Beide = For F () we have E1/N() E1/N p () = b() b() F () = b() a() δ () d > (8) p() d E1/N p(τ) dτ d 1 + b() p 1 lim E1/N + p(τ) dτ b() p(τ) dτ d 1 + b() p 1 = p(τ) dτ d. p() d =. (9) p(τ) dτ d 1 + b() p 1 p(τ) dτ d p(τ) dτ d p(τ) dτ F (). (1) p(τ) dτ 1 + b( + ) p(τ) dτ b() p 1 = b() p(τ) dτ 1 becaue from -periodiciy of p() we can derive, ha + p(τ) dτ p 1 + p(τ) dτ 1 = 1. p(τ) dτ 1 Hence F () doen depend on and from (9) and (1) we deduce (7). 1 =, p(τ) dτ Sa., Opim. Inf. Compu. Vol. 5, March 17

6 D. BOYSENKO AND O. BOYSENKO Some relaed logiic equaion Le u conider ochaic differenial equaion of he form (a() dn() = N() b()n θ () ) d + α()dw() + γ(, z) ν(d, dz), (11) θ > i an odd ineger; N() = N > i he random variable independen on andard one-dimenional Wiener proce and cenered Poion meaure ν(, A) = ν(, A) Π(A), Π() < ; w() and ν(, A) are independen. Le non-random funcion a(), b(), α(), γ(, z) be bounded, coninuou on, ), a() >, b() >. Le N() be a oluion o equaion (11), hen by Io formula applied o ochaic proce N θ () we derive dn θ () = N θ () ( θa() + θ(θ 1) α () θb()n θ () + (1 γ(, z)) θ 1 θγ(, z) ) Π(dz) d +θα()dw() + (1 + γ(, z)) θ 1 ν(d, dz), N θ () = 1/N θ. For ochaic proce M() = N θ () we have an ()-ype equaion dm() = M() (â() ˆb()M())d + ˆα()dw() + ˆγ(, z) ν(d, dz), M() = 1/N θ, â() = θa() + θ(θ 1) α () + (1 + γ(, z)) θ 1 θγ(, z) Π(dz), ˆb() = θb(), ˆα() = θα(), ˆγ(, z) = (1 + γ(, z)) θ 1. I i eay o ee ha under condiion of heorem 1 on coefficien a(), b(), α(), γ(, z) he ame condiion are fulfilled for coefficien â(), ˆb(), ˆα(), ˆγ(, z). herefore we obain he following reul. heorem 3 Le condiion of heorem 1 are fulfilled. hen here exi a unique poiive oluion N() o equaion (11) wih iniial condiion N() = N >, which i global and ha a repreenaion N() = ( θ a() β()d + α()dw() + ) ln(1 + γ(, z)) ν(d, dz) 1/N θ + θ b() θ ( a(τ) β(τ)dτ + α(τ)dw(τ) + β() = α () + γ(, x) ln(1 + γ(, z)) Π(dz). ln(1 + γ(τ, z)) ν(dτ, dz)) d heorem 4 Le a(), b(), α() and γ(, z) be coninuou on and -periodic on funcion, a() >, b() > and θa() σ ()d >. σ θ(θ + 1) () = α () + (1 + γ(, z)) θ 1 + θγ(, z) Π(dz), Π() <, ln(1 + γ(, z)) K for ome K >. hen he equaion d d E1/N θ () = σ () θa()e1/n θ () + θb() Sa., Opim. Inf. Compu. Vol. 5, March 17 1/θ

7 64 EXPLICI FOM OF GLOBAL SOLUION O SOCHASIC LOGISIC EQUAION ha a unique poiive -periodic oluion E1/Np θ = θ + θa(τ) σ (τ) dτ b() d, θa(τ) σ (τ) dτ 1 and for oluion N() o equaion (11) wih any iniial value N() = N >, E1/N <, we have ( E1/N θ () E1/Np θ () ) =. lim + EFEENCES 1. M.Iannelli, and A. Pugliee, An Inroducion o Mahemaical Populaion Dynamic, Springer, 14.. D. Jiang, and N. Shi, A noe on nonauonomou logiic equaion wih random perurbaion, J. Mah. Anal. Appl., 33, pp , I.I Gikhman, and A.V. Skorokhod, Sochaic Differenial Equaion and i Applicaion, Kiev, Naukova Dumka, 198.(In uian) 4. G. Sanone, Ordinary Differenial Equaion, v.1, Mocow, Inorannaya Lieraura, 1953.(In uian) Sa., Opim. Inf. Compu. Vol. 5, March 17