EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE BACKWARD STOCHASTIC LORENZ SYSTEM

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1 Communicaions on Sochasic Analysis Vol. 1, No. 3 (27) EXISTENCE AND UNIQUENESS OF SOLUTIONS TO THE BACKWARD STOCHASTIC LORENZ SYSTEM P. SUNDAR AND HONG YIN Absrac. The backward sochasic Lorenz sysem is sudied in his paper. Suiable a priori esimaes for adaped soluions of he backward sochasic Lorenz sysem are obained. The exisence and uniqueness of soluions is shown by he use of suiable runcaions and approximaions. The coninuiy of he adaped soluions wih respec o he erminal daa is also esablished. 1. Inroducion In a celebraed work, Edward N. Lorenz inroduced a nonlinear sysem of ordinary differenial equaions describing fluid convecion of nonperiodic flows (Lorenz [9]). The derivaion of hese equaions is from a model of fluid flow wihin a region of uniform deph and wih higher emperaure a he boom (Rayleigh [14]). Lorenz inroduced hree ime-dependen variables. The variable X is proporional o he inensiy of he convecive moion, Y is proporional o he emperaure difference beween ascending and descending currens, and Z is proporional o disorion of he verical emperaure profile from lineariy. The model consiss of he following hree equaions: Ẋ = ax + ay Ẏ = XZ + bx Y (1.1) Ż = XY cz where a is he Prandl number, b is he emperaure difference of he heaed layer and c is relaed o he size of he fluid cell. The numbers a, b, and c are all posiive. In he pas 4 years, ranging from physics (Sparrow [17]) o physiology of he human brain (Weiss [19]), Lorenz sysems have been widely sudied in many areas for a variey of parameer values. Randomness has also been inroduced ino Lorenz sysem and some properies of he forward sysem have been sudied (Schmalfuß[16] and Keller [7]). Le (Ω, F, {F }, P ) be a complee probabiliy space on which a 3-dimensional Wiener process {W } is defined, such ha {F } is he naural filraion of {W }, 2 Mahemaics Subjec Classificaion. 6H1, 34F5 76F35. Key words and phrases. Backward sochasic Lorenz sysem, he Iô formula, runcaed sysem, Gronwall inequaliy. 473

2 474 P. SUNDAR AND HONG YIN augmened by all he P -null subses of F. Define a marix A as follows: a a A = b 1 c For any y = ( ) ( y1 ȳ 1 ) y 2 y and ȳ= ȳ 2 in R 3, we define an operaor B as follows: 3 ȳ 3 B(y, ȳ) = y 1 ȳ 3 y 1 ȳ 2 Then he backward sochasic Lorenz sysem corresponding o equaion (1.1) is given by he following erminal value problem: { dy () = (AY () + B(Y (), Y ()))d + Z()dW () (1.2) Y (T ) = ξ for [, T ] and ξ L 2 F T (Ω; R 3 ). The inegral form of he backward sochasic Lorenz sysem is as follows: Y () = ξ + (AY (s) + B(Y (s), Y (s)))ds Z(s)dW (s). The problem consiss in finding a pair of adaped soluions {(Y (), Z())} [,T ]. Definiion 1.1. A pair of processes (Y (), Z()) M[, T ] is called an adaped soluion of (1.2) if he following holds: Y () = ξ + (AY (s) + B(Y (s), Y (s)))ds Z(s)dW (s) [, T ], P-a.s. Here M[, T ]= L 2 F (Ω; C([, T ]; R3 )) L 2 F (Ω; L2 (, T ; R 3 3 )) and i is equipped wih he norm Y ( ), Z( ) M[,T ] = {E( sup Y () 2 ) + E T Z() 2 d} 1 2. I is worhwhile o emphasize ha he soluion pair {(Y (), Z()) : [, T ]} is required o be adaped o he forward filraion {F : [, T ]}, and Y (T ) is specified as an F T -measurable random variable where T is he erminal ime. The sochasic inegrals ha appear hroughou his aricle are herefore forward Iô inegrals hough he ime parameer appears as he lower limi of he inegrals. This is in conras o he backward Iô inegrals ha are employed in he book by Kunia [8]. Linear backward sochasic differenial equaions were inroduced by Bismu in 1973 [1], and he sysemaic sudy of general backward sochasic differenial equaions (BSDEs for shor) were pu forward firs by Pardoux and Peng in 199 [13]. Since he heory of BSDEs is well conneced wih nonlinear parial differenial equaions, nonlinear semigroups and sochasic conrols, i has been inensively sudied in he pas wo decades. There are also various applicaions of BSDEs in

3 BACKWARD STOCHASTIC LORENZ SYSTEM 475 he heory of mahemaical finance. For insance, he hedging and pricing of a coningen claim can be described as linear BSDEs. In he presen work, since he coefficien B is nonlinear and unbounded, he exising heory of BSDEs does no apply. To overcome his difficuly, a runcaion of he coefficien and an approximaion scheme have been used. The Lorenz sysem shares cerain feaures wih he Navier-Sokes equaions. Adaped soluions of he wo dimensional backward sochasic Navier-Sokes equaions have also been sudied recenly [18]. The organizaion of he paper is as follows. In secion 2, a priori esimaes for he soluions of sysems wih uniformly bounded erminal values are obained and a runcaion of he sysem is inroduced. In secion 3, we prove he exisence and uniqueness of he soluion o he Lorenz sysem using an approximaion scheme. Secion 4 is devoed o he coninuiy of he soluions wih respec o erminal daa. 2. A Priori Esimaes Le us lis wo frequenly used resuls. The firs one is a simple propery of B, and he second resul is he Gronwall inequaliy for backward differenial equaions. Proposiion 2.1. If y and ȳ R 3, hen B(y, ȳ), ȳ = and B(y, y) B(ȳ, ȳ) 2 y 2 + ȳ 2 ) y ȳ 2 Proposiion 2.2. Suppose ha g(), α(), β() and γ() are inegrable funcions, and β(), γ(). For T, if g() α() + β() γ(ρ)g(ρ)dρ (2.1) hen g() α() + β() α(η)γ(η)er η β(ρ)γ(ρ)dρ dη. In paricular, if α() α, β() β and γ() 1, hen g() α(2 e β(t ) ) Le E F X o be he condiional expecaion E(X F ), and le us lis wo assumpions: 1 ξ 2 K for some consan K, P-a.s. 2 ξ L 2 F T (Ω; R n ) Proposiion 2.3. Under Assumpion 1, if (Y (), Z()) is an adaped soluion for Lorenz sysem (1.2), hen here exiss a consan N, such ha Y () N for all [, T ], P-a.s. Proof. Applying he Iô formula o Y () 2, and by Proposiion 2.1, Y () 2 + Z(s) 2 ds = ξ Y (s), AY (s) ds 2 Y (s), Z(s) dw (s).

4 476 P. SUNDAR AND HONG YIN For all r T, we have: E Fr Y () 2 + E Fr Z(s) 2 ds = E Fr ξ 2 + 2E Fr Y (s), AY (s) ds By Gronwall s inequaliy (2.1), E Fr ξ A E Fr Y (s) 2 ds. E Fr Y () 2 + E Fr Z(s) 2 ds E Fr ξ A =E Fr ξ 2 (2 e 2 A (T ) ). E Fr ξ 2 e R s 2 A dv ds Leing r o be, and by Assumpion 1, Y () 2 N for some consan N > which is only relaed o K. Definiion 2.4. Le b(y) = Ay + B(y, y), and for all N N and y R 3, we define { b(y) if y N b N (y) = b( y y N) if y > N, and he runcaed Lorenz sysem is he following BSDE: { dy N () = b N (Y N ())d + Z N ()dw () Y N (T ) = ξ (2.2) where W () is he 3-dimensional Wiener process defined on a complee probabiliy space (Ω, F, P ) and ξ L 2 F T (Ω; R 3 ). Corollary 2.5. Under Assumpion 1, if (Y N (), Z N ()) is an adaped soluion for runcaed Lorenz sysem (2.2), hen here exiss a consan N, such ha Y N () N for all N N and [, T ], P-a.s. Proof. If Y N () N, hen Y N (), B(Y N (), Y N ()) =. If Y N () > N, hen we also have Y N (), B( Y N () Y N () N, Y N () N) =. Le Y N () { Ay if y N a N (y) = A y y N if y > N. Then Y N (), b N (Y N ()) = Y N (), a N (Y N ()). An applicaion of Iô formula o Y N () 2 and he above equaliy yields Y N () 2 + Z N (s) 2 ds = ξ Y N (s), a N (Y N (s)) ds Y N (s), Z N (s) dw (s).

5 BACKWARD STOCHASTIC LORENZ SYSTEM 477 Similar o he proof of Proposiion 2.3, we ge Y N () 2 N for some consan N > which is only relaed o K. 3. Exisence and Uniqueness of Soluions Proposiion 3.1. The funcion b N is Lipschiz coninuous on R 3. Proof. For any y and ȳ R 3, le us assume ha y = ( ) ( y1 ȳ 1 ) y 2 y and ȳ= ȳ 2. 3 ȳ 3 Case I: y, ȳ N. Then we have b N (y) b N (ȳ) = b(y) b(ȳ) = Ay + B(y, y) Aȳ B(ȳ, ȳ) A y ȳ + (y 1 y 3 ȳ 1 ȳ 3 ) 2 + (y 1 y 2 ȳ 1 ȳ 2 ) 2 ac abc y ȳ + N y ȳ. Le L N = ac abc + N. Thus b N (y) b N (ȳ) L N y ȳ. Case II: y N, bu ȳ > N. Then by Case I, we have b N (y) b N (ȳ) = b(y) b( ȳ ȳ N) L N y ȳ ȳ N. Le us prove ha y ȳ ȳ N y ȳ. By carefully choosing a coordinae sysem, I is possible o make ȳ = (ȳ 1,, ). Under such coordinae sysem, we have y ȳ ȳ N = [(y 1 sign(ȳ 1 )N) 2 + y y 2 3] 1 2 y ȳ = [(y 1 ȳ 1 ) 2 + y y 2 3] 1 2. Since y 1 N < ȳ 1, i is clear ha y 1 sign(ȳ 1 )N y 1 ȳ 1. So y ȳ ȳ N y ȳ and hus b N (y) b N (ȳ) L N y ȳ. Case III: y > N and ȳ > N. Then by Case I, we have and b N (y) b N (ȳ) = b( y ȳ N) b( y ȳ N) L N y y N ȳ ȳ N. Wihou lose of generaliy, le us assume ha y ȳ. Consider y as N in Case II. I is clear ha Thus we have shown ha y y N ȳ ȳ N = N y y y ȳ ȳ N y ȳ, y b N (y) b N (ȳ) L N y ȳ for Case III and he proof is complee. Theorem 3.2. Under Assumpion 1, he Lorenz sysem (1.2) has a unique soluion.

6 478 P. SUNDAR AND HONG YIN Proof. Firs le us prove he exisence of he soluion of Lorenz sysem (1.2). By Proposiion 3.1, b N is Lipschiz. Thus here exiss a unique soluion (Y N (), Z N ()) of runcaed Lorenz sysem (2.2) wih such b N for each N N(see Yong and Zhou [2]). Because of Assumpion 1, by Corollary 2.5, here exiss a naural number N, such ha Y N () N for all N N. By aking N = N, i follows ha Y N () N = b N (Y N ()) = b(y N ()) by he definiion of b N (y). Thus for N, runcaed Lorenz sysem (2.2) is he same as Lorenz sysem (1.2). Hence (Y N (), Z N ()) is also soluion of Lorenz sysem (1.2). Le (Y (), Z()) and (Ȳ (), Z()) be wo pairs of soluions of Lorenz sysem (1.2). By Proposiion 2.3, here exiss a naural number N, such ha Y () N and Ȳ () N. Since Lorenz sysem (1.2) and runcaed Lorenz sysem (2.2) for N = N are he same, (Y (), Z()) and (Ȳ (), Z()) are also soluions of runcaed Lorenz sysem (2.2) for N = N. Since he runcaed Lorenz sysem (2.2) has a unique soluion for N = N, we know ha (Y (), Z()) = (Ȳ (), Z()) P-a.s. Thus he uniqueness of he soluion has been shown. Definiion 3.3. For any ξ saisfies Assumpion 2 and n N, we define ξ n = ξ ( n) n, and he n-lorenz sysem is he following BSDE: { dy n () = b(y n ())d + Z n ()dw () Y n (T ) = ξ n (3.1) where W () is he 3-dimensional Wiener process defined on a complee probabiliy space (Ω, F, P ). Proposiion 3.4. Under Assumpion 2,he soluions of n-lorenz sysems are Cauchy in M[, T ]. Proof. Since ξ n is bounded by n, he exisence and uniqueness of he soluion of n-lorenz sysem is guaraneed by Proposiion 2.3. For all n and m N, le (Y n (), Z n ()) and (Y m (), Z m ()) be he unique soluions of n-lorenz sysem and m-lorenz sysem, respecively. Define Ỹ () = Y n () Y m (), Z() = Z n () Z m () and ξ = ξ n ξ m. Apply Iô formula o Ỹ () 2 o ge +2 2 Ỹ () 2 + Z(s) 2 ds ξ A Ỹ (s) 2 ds Ỹ (s), B(Y n (s), Y n (s)) B(Y m (s), Y m (s)) ds (3.2) Ỹ (s), Z(s) dw (s)

7 BACKWARD STOCHASTIC LORENZ SYSTEM 479 I is easy o show ha Ỹ (s), B(Y n (s), Y n (s)) B(Y m (s), Y m (s)) = Y n (s) Y m (s), B(Y n (s), Y n (s)) B(Y m (s), Y m (s)) = Y n (s), B(Y m (s), Y m (s)) Y m (s), B(Y n (s), Y n (s)) = Y m (s), B(Y m (s), Y n (s)) Y m (s), B(Y n (s), Y n (s)) = Y m (s), B(Y m (s) Y n (s), Y n (s)) = Y m (s) Y n (s), B(Y m (s) Y n (s), Y n (s)) = Ỹ (s), B(Ỹ (s), Y n (s)) Ỹ (s) 2 Y n (s) I follows from Proposiion 2.3 ha So Y n () 2n. Thus i has been shown ha From (3.2) and (3.3), one ges Ỹ () 2 + Y n () 2 (2 e 2 A (T ) )E F ξ n 2 2n 2. Ỹ (s), B(Y n (s), Y n (s)) B(Y m (s), Y m (s)) Ỹ (s) 2 Y n (s) 2n Ỹ (s) 2 (3.3) Z(s) 2 ds ξ 2 + (2 A + 2 2n) 2 Taking expecaion on boh sides of (3.4), i follows ha E Ỹ () 2 + E Hence by Gronwall s inequaliy, we ge Z(s) 2 ds E ξ 2 + (2 A + 2 2n) E Ỹ () 2 + E Z(s) 2 ds E ξ 2 + (2 A + 2 2n) 2E ξ 2. Ỹ (s) 2 ds Ỹ (s), Z(s) dw (s). (3.4) E Ỹ (s) 2 ds. (3.5) E ξ 2 e (2 A +2 2n)( s) ds (3.6) Since E ξ 2 < and ξ 2 2 ξ 2, an applicaion of he Lebesgue dominaed convergence heorem yeilds So i follows from (3.6) ha lim E ξ 2 = E lim ξ 2 = (3.7) m, m, lim E Z(s) 2 ds = and m, lim E Ỹ m, () 2 = (3.8)

8 48 P. SUNDAR AND HONG YIN On he oher hand, by means of he Bürkholder-Davis-Gundy inequaliy, one ges E{ sup ρ T 2E sup ρ ρ ρ T 4 2E{ Ỹ (s), Z(s) dw (s) } Ỹ (s), Z(s) dw (s) Ỹ (s) 2 Z(s) 2 ds} 1 2 (3.9) 1 E( sup Ỹ 4 (ρ) 2 ) + 32E ρ T Z(s) 2 ds Thus from (3.2),(3.3) and (3.9), one ges E( sup ρ T Ỹ (ρ) 2 ) + E Z(s) 2 ds E ξ 2 + E (2 A + 2 Y n (s) ) Ỹ (s) 2 ds + 1 E( sup Ỹ 2 (ρ) 2 ) + 64E ρ T Hence i follows from (3.6) and he above inequaliy ha 1 E( sup Ỹ 2 (ρ) 2 ) 127E ξ 2 + E ρ T From Proposiion 2.3, one has Z(s) 2 ds (2 A + 2 Y n (s) ) Ỹ (s) 2 ds (3.1) Y n (s) 2 E Fs ξ n 2 (2 e 2 A (T ) ) 2E Fs ξ 2 (3.11) Clearly {E Fs ξ 2 } s [,T ] is a F-adaped maringale. By Doob s submaringale inequaliy, for any λ >, P { sup E Fs ξ 2 λ} 1 ξ 2 = 1 s T λ EEFT λ E ξ 2 as λ Le τ R = inf{ : E F ξ 2 > R} T for R >. I is easy o show ha τ R T a.s. as R. From (3.1) and Gronwall s inequaliy, one ges E sup Ỹ (ρ τ R) 2 ρ T 254E ξ 2 + 2E 254E ξ 2 + 4( A + R) (2 A + 2 Y n (s τ R ) ) Ỹ (s τ R) 2 ds 58E ξ 2 for all [, T ] E sup s ρ T Ỹ (ρ τ R) 2 ds

9 BACKWARD STOCHASTIC LORENZ SYSTEM 481 An applicaion of monoone convergence heorem yields lim E( sup Ỹ m, (ρ) 2 ) ρ T = lim E( lim m, = lim m, R sup R ρ T lim Ỹ (ρ τ R) 2 ) E( sup Ỹ (ρ τ R) 2 ) ρ T lim 58E ξ 2 = (3.12) m, From (3.8), (3.12), and he definiion of he norm of M[, T ], we know ha he soluions of n-lorenz sysems are Cauchy. Since M[, T ] is a Banach space, we know ha here exiss (Y, Z) M[, T ], such ha lim {E( sup Y n () Y () 2 ) + E T Z n () Z() 2 d} = We wan o show ha (Y, Z) is acually he soluion of Lorenz sysem (1.2). Theorem 3.5. Under Assumpion A2, Lorenz sysem (1.2) has a unique soluion. Proof. By Proposiion 3.4, we know ha Y n () converge o Y () uniformly. Since Y n () and Y () are all bounded, i is easy o see ha By Iô isomery, we have lim E b(y n (s)) b(y (s)) 2 ds = lim E( (Z n (s) Z(s))dW (s)) 2 = lim E Z n (s) Z(s) 2 ds = Thus we have shown ha (Y (), Z()) saisfies Lorenz sysem (1.2), i.e. (Y (), Z()) is a soluion of Lorenz sysem (1.2). Now assume ha (Y (), Z()) and (Y (), Z ()) are wo soluions of Lorenz sysem (1.2). Similar o he proof of Proposiion 3.4, we can ge E( sup Y () Y () 2 ) = and E T Z() Z () 2 d =, Tha is, (Y (), Z()) (Y (), Z ()) M[,T ] =. Thus we have proved he uniqueness of he soluion. 4. Coninuiy wih respec o Terminal Daa Theorem 4.1. Assume ha ξ L 2 F T (Ω; R n ). Then he soluion of (1.2) is coninuous wih respec o he erminal daa. Proof. For any ξ, ζ L 2 F T (Ω; R n ), le (Y (), Z()) and (X(), V ()) be soluions of (1.2) under erminal values ξ and ζ, respecively. Le (Y n (), Z n ()) and

10 482 P. SUNDAR AND HONG YIN (X n (), V n ()) be soluions of corresponding n-lorenz sysem. By Proposiion 3.4, we know ha lim E sup Y () Y n () 2 = lim E Z() Z n () 2 d = and T lim E sup X() X n () 2 = lim E V () V n () 2 d = T Similar o he proof of Proposiion 3.4, one can show ha and Hence Thus E Z n () V n () 2 d 2E ξ n ζ n 2 E sup Y n () X n () 2 58E ξ n ζ n 2 T E sup Y n () X n () 2 + E T E sup Y () X() 2 + E T Z n () V n () 2 d 51E ξ n ζ n 2 Z() V () 2 d lim 3E sup ( Y () Y n () 2 + Y n () X n () 2 + X() X n () 2 ) T + lim 3E ( Z() Z n () 2 + Z n () V n () 2 + V () V n () 2 )d lim 153E ξn ζ n 2 = 153E ξ ζ 2 References 1. Bismu, J. M.: Conjugae Convex Funcions in Opimal Sochasic Conrol, J. Mah. Anal. Apl., 44 (1973) Ehier, S. N. and Kurz, T. G.: Markov Processes: Characerizaion and Convergence, John Willey & Sons, New York, Flandoli, F.: Lecure Noes on SPDEs, IMA/RMMC Summer Conference, Hu, Y., Ma, J. and Yong, J.: On semi-linear, degenerae backward sochasic parial differenial equaions, Probab. Theory Rela. Fields 123 (22) Karazas, I. and Shreve, S. E.: Brownian Moion and Sochasic Calculus, Springer-Verlag, New York, Inc., Karoui, N. E. and Mazliak, L.: Backward sochasic differenial equaions, Piman Research Noes in Mahemaics Series 364, Addison Wesley Longman Inc., Keller, H.: Aracors and bifurcaions of he sochasic Lorenz sysem, Tech. Rep. 389, Insiu für Dynamische Syseme, Universiä Bremen, Kunia, H.: Sochasic flows and sochasic differenial equaions, Cambridge Universiy Press, Lorenz, E. N.: Deerminisic Nonperiodic Flow, J. Amos. Sci. 2 (1963) Ma, J., Proer, P. and Yong, J.: Solving Forward-Backward Sochasic Differenial Equaions Explicily A Four Sep Scheme, Prob. Th. & Rel. Fields 98 (1994)

11 BACKWARD STOCHASTIC LORENZ SYSTEM Ma, J. and Yong, J.: On linear, degenerae backward sochasic parial differenial equaions, Probab. Theory Rela. Fields, 113 (1999) Øksendal, B.: Sochasic Differenial Equaions: An Inroducion wih Applicaions, 6h Ed., Springer-Verlag, New York, Inc., Pardoux, E. and Peng, S.: Adaped Soluion of a Backward Sochasic Differenial Equaion, Sysems and Conrol Leers 14 (199) Rayleigh, Lord: On convecive currens in a horizonal layer of fluid when he higher emperaure is on he under side, Phil. Mag. 32 (1916) Rong, S.: On soluions of backward sochasic differenial equaions wih jumps and applicaions, Sochasic Processes and heir Applicaions, 66 (1997) Schmalfuß, B.: The Random Aracor of he Sochasic Lorenz Sysem, Z. angew. Mah. Phys. 48 (1997) Sparrow, C.: The Lorenz equaions: bifurcaions, chaos and srange aracors, Springer- Verlag, New York, Sundar, P. and Yin, H.: Exisence and Uniqueness of Soluions o he Backward 2D Sochasic Navier-Sokes Equaions, (27) preprin. 19. Weiss, H. and Weiss, V.: The golden mean as clock cycle of brain waves, Chaos, Solions and Fracals 18 (23) Yong, J. and Zhou, X.: Sochasic Conrols: Hamilonian sysems and HJB equaions, Springer-Verlag, New York, P. Sundar: Deparmen of Mahemaics, Louisiana Sae Universiy, Baon Rouge, Louisiana address: sundar@mah.lsu.edu Hong Yin: Deparmen of Mahemaics, Louisiana Sae Universiy, Baon Rouge, Louisiana address: yinhong@mah.lsu.edu