一维随机微分方程的稳定性

华中科技大学 硕士学位论文 一维随机微分方程的稳定性 姓名：谢晶晶 申请学位级别：硕士 专业：应用数学 指导教师：雷冬霞 2011-05-15 r b '5?tq40!i##N �!�pV���u0j�yN�p��-yNW W��_��.?tq4�,V℄��F�$I_V�j_�V'5?tq4i�F� I_V��GW�{q�'5?tq4V℄VD0q"��q��`q:I_qWJ1� �GY�7b}��G�jV�l:�GVW��j�Yk7b}�N'5?tq 4 dx(t) = f(x(t), t)dt+ g(x(t), t)dB(t) VR� f � g t"2e�QO=4M�^q6 /4M��|'5?tq4V℄V�gqM�� ℄VD0q"��q��~.I_�> &�lG�I_�uG�I_WW�0Yk7V$�ZN�{q�h����_V'5 ?tq4R8��q<�u�,V℄AG�6/�53�� �,�A��kVR8�, V℄�ia 6/V�$�q<�u�,V℄��`V�3��� �,�A��kVR 8�,V℄AG�6/V�dj|C�3�D� Be'5?tq4VG�6/�� � K,VG�6/�0Yy7N{q�'5?tq4 dx(t) = f(x(t), t)dt+ g(x(t), t)dB(t) VI_q�TN��R� f � g 2es'�QO=4M� f 2eL�ia 6/4 M�Y g 2eia 6/4M��7NQ ?��V β � �|'5?tq4R8 D0��Vgs℄�#YD0H"8�D�ÆV.� Kp, �|q4V℄A�`V�g$ +Q.0e�GV q Fj |R8V℄2e>&�lG�I_q�g$+�9�_�V �beo ?� Æ�y�'5?tq4� I_q� G�6/� D0q� �`q� %�-b� 3�� I Abstract Stochastic differential equations have a very wide range of applications in many areas , such as economics, population ecology and so on. Given an ordinary differential equation, its solution may be unstable, while the corresponding stochastic differential equation might be stable.This paper studies the existence and uniqueness of the solutions, boundedness and stability issues of the stochastic differential equations. The first chapter introduces the background of this work and the main work of this paper. The second chapter describes the properties of the solutions of the stochastic differential equations dx(t) = f(x(t), t)dt+ g(x(t), t)dB(t), when the coefficient f and g of the stochastic differential equations satisfy the Lipschitz conditions and linear growth conditions, including the existence and uniqueness of the solutions, stability in probabil- ity, almost sure exponential stability and moment exponential stability and so on. In the last two sections of the second chapter, we have studied for a given stochastic differential system. On one hand, if the solution is for the exponential growth,then the noise can turn it into a new system, its solution is for the polynomial growth. On the other hand, if its solution is bounded, the noise can also turn it into a new system,its solution is for the exponential growth. All in all, the noise can not only promote the exponential growth of stochastic differential equations but also suppression the exponential growth of the equations. In the third chapter, we have studied the stability of the stochastic differential equations dx(t) = f(x(t), t)dt+ g(x(t), t)dB(t),both of the coefficients f and g satisfy Lipschitz conditions, f satisfies the unilateral polynomial growth condition, and g satisfy the polynomial growth condition. This chapter will explain that appro- priate β can guarantee the existence and uniqueness of the global solution of stochastic differential equations,and there is a constant Kp dependent only on the initial value, such that the solution of the equations are bounded.Finally, we will also discuss the q large enough to ensure that the solution of the system to meet the almost sure exponential stability. Finally, corresponding examples are given for demonstration. Key words: Stochastic differential equations; Stability; Exponential growth; Existence; Boundedness; Brownian motion; Noise. II �xZ5/ �p�?*2TVyD0G�K�p0Q�GQVeoV{q�j: TV{q 1��gK*B�:GN j ?��VEt:��0G$��q�T+�p282 jl!2℄h�V{q1��h�GV{qh9 [V�p�82�z 0GN ?j q ?��p;g R���?Vm,[���p5K� yD0Gj:V�� sS� H ) s _I*Jm26h92: j:;g�℄yf�Æ &���yD0GV�_�;�yf�f &#b�E� Æ'825�$T0GVw�M�\b �,s0G�,*�a*��p�f(N? Gy� Q�yD0GVg'2'tEt�v�Æ�w�eoI)�� (����( �2z=WwK�f D�0��yD0G� ; 2 , 0 H℄;$����f�� �0G�� $ ; 2. (^0 |q�EF ” √ ”) yD0Gj:V�� GQV�V�� sS� H ) s sS� H ) s 1 ^ + 1.1 oJ`"|n!�Qy h���.?tq4R8 dx(t) = f(x(t), t)dt ,V℄��F�I_V����F�$I_V� K9GE BS�3�n� ���$I_VR8�TI_��� ���I_ VR8�T�FI_ [1]. h�'5?tq4R8 dx(t) = f(x(t), t)dt+ g(x(t), t)dB(t) VI_qV{q�r��0GVW�Et� Æ�;��01�;gH�!iy: h�!i{q�1981 H�Hasminskii[2]g1 6������3�a&����kBV^qR8�TI_�1983 H�Arnold, Crauel, � Wihstutz[3] �?NYrNu< A V7d� 0 ��S>DA 0 VOI*�3�� �^qR8�TI_� h����BVr^qR8 x˙(t) = f(x(t), t) t > 0; TN f(0, t) = 0, x(0) = x0 ∈ R, VI_qVV{q�03 �V� [1] N�!`S Vb}� 2005 H� Appleby � Mao[5],2008 H� Appleby,Mao � Rodkina[6] I~� N f 2eL�^q6/4M��'5?tq4R8VI_qM� 2002 H�Mao,Marion,Renshaw[7] � 2004 H Bahar,Mao[8] �?���r.S �V� �r�*n3�� �Kp��-yR80�℄�HE�P� 2009 H�M � [11] {q�r^qR8VI_q��_��$I_V?tq4 x˙(t) = f(x(t), t) 1 TN f 2eL�ia 6/4M��e��%�3�n �'5Xnb|&�lG�I_�uG�I_WW�Yk7V$�ZW�b}�3�� �K'5?tq4VG�6/��r� h���G�6/V?tq4�0�_V4 MV�3�� �,�Aia 6/�T?�h����`V?tq4R8�3��� Be_�V'5?tq4R8VG�6/� Yy7�G7VW�Et�W�b}�N'5?tq4VR�2eia 6/4M ��; f 2eL�ia 6/4M xf(x(t), t) ≤ δxα+2 + γx2 � g 2eia 6/4M g2(x, t) ≤ σxβ+2 + qx2 V�#�|'5?tq4V℄VD0q"��q��`q�I_qWW��7EttZ b}�;gqM��Wb}�0R� β > α > 0 \�V�|'5?tq4R8V℄�D 2 0V#Y���V�T?���0 β > α > 0 2eV\�V�e�GVR� q �� �|'5?tq4V℄�>&�lG�I_V� |r��"�0GVW��j�TN0_�V$<�v9��g�b���X ?_�V1�q� 1.3 �o��j�1 �_uV'5?tq4 dx(t) = f(x(t), t)dt+ g(x(t), t)dB(t) (1.1) TN f : R× R+ → R " g : R× R+ → R � Borel �+��� 0;MG7N�V�To��x A A��u<�5,V7o�A |A| = √ trace(ATA) 〈X, Y 〉 !Æb� X, Y VE6�hq V a, b ∈ R � a ∨ b !Æ a, b NVWGD� a ∧ b !Æ a, b NVWdD� % (Ω,F ,P) A��;�~.�H�Y���2e� 4MV( {F}t≥0 �* N F0 ��*�V�8�� {F}t≥0 �����uYL℄Vt!� B(t) �_Æ0;�~. �H|V��d�V �%�-b� LpFt(Ω, R) !Æ R DV Ft �+V'5����Y '5�� ξ 2e E|ξ|P <∞ �j LpFt(Ω, R) !Æ Ft �+�`V R D'5���� 0V8�Dj*TVq4V℄CA x(t, ξ) 2JCA x(t) �TN t ≥ 0. g& 1.1 (Borel − Cantelli′s ��)[1] (1) u�D0��t! {Ak} ⊂ F �Y� ∞∑ k=1 P (Ak) <∞, 5K9� 99 P (lim sup k→∞ Ak) = 0 3 ;�r� �D0��8� Ω0 ∈ F , #Y,2e P (Ω0) = 1, YD0��>�DV'5 �� k0 ��Th�q V ω ∈ Ω0 �N k ≥ k0(ω) �� � ω /∈ Ak 1�� (2) u�t! {Ak} ⊂ F �d�V�Y ∞∑ k=1 P (Ak) =∞, 5 P (lim sup k→∞ Ak) = 1 �r� �D0��8� Ωθ ∈ F �2e P (Ωθ) = 1 �Yh�q V ω ∈ Ωθ � D0 ��bt! {Aki} ��T6�� Aki ��$% ω. g& 1.2 (Y oung $W ) u�D0.� x, y, α, β ≥ 0, Y2e α + β ≥ 0, ε > 0, 5V 0, D06�V?.� cp � Cp(z" p �Æ) �2e cpE|A(t)|P2 ≤ E( sup 0≤s≤t |x(s)|p) ≤ CpE|A(t)|P2 h*�V t ≥ 0, K9+� uVV/�D� 5 cp = (p/2) p, Cp = (32/p) p/2, N 0 < p < 2 �� cp = 1, Cp = 4, N p = 2 �� cp = (2p) −p/2, Cp = [p p+1/2(p− 1)p−1]p/2, N p > 2 �� �& 1.6 (G�l$W )[1] % g = (g1, g2, · · · gm) ∈ L2(R+;R1×m), Y% T, α, β �?��5� P{ sup 0≤t≤T [ ∫ T 0 g(s)dB(s)− α 2 ∫ t 0 |g(s)|2ds] > β} ≤ e−αβ . �& 1.7 (Gronwall $W )[1] % T > 0, C ≥ 0, % µ(·) ���_Æ0 [0, T ] |V Borel �+�`Vrz���j υ(·) 5���_Æ0 [0, T ] |Vrz�6���u� µ(t) ≤ C + ∫ T 0 υ(s)µ(s)ds, 0 ≤ t ≤ T 5� µ(t) ≤ C exp( ∫ T 0 υ(s)ds), 0 ≤ t ≤ T �& 1.8 (℄VD0q"��q_�)[1] GD0��?.� K � K¯ 2e (1)(�QO=4M) h�*�V x, y ∈ R � t ∈ [t0, T ], �V 0, p > 0 Y X ∈ Lp, 5�$W P{ω : |X(ω)| ≥ c} ≤ c−pE|X|p 1�� 7 2 P>.{U\ptC�~A�G��w~N�\ 0;�7N�K9Qb}N'5?tq4VR�2e�QO=4M�^q6/4M ��,V�gI_qM�K9�_uV'5?tq4 dx(t) = f(x(t), t)dt+ g(x(t), t)dB(t) (1.4) TN t ≥ t0, Y f : R × R+ → R " g : R × R+ → R � Borel �+���#Y2e f(x(0), 0) ≡ 0, g(x(0), 0) ≡ 0. 0;��K9G*�9Vq4 �2e℄VD0q"� �q_�V��r�2e_� 1.8 V�;D0��?.� K � K¯ 2e (1)(�QO=4M) h�*�V x, y ∈ R � t ∈ [t0, T ], � |f(x, t)− f(y, t)|2 ∨ |g(x, t)− g(y, t)|2 ≤ K¯|x− y|2 1�� (2)(^q6/4M) h�*�V (x, t) ∈ R× [t0, T ), � |f(x, t)|2 ∨ |g(x, t))|2 ≤ K(1 + |x|2) 1�� TN8�DA x(t0) = x0 ∈ R, 5q4 (1.4) D0��V��℄�K9�,CA x(t; t0, x0), JCA x(t). K9W�9;�7N*r�V�g_Æ�C K A*�V�urK���� µ : R+ → µ : R+, �> µ(0) = 0, Yu� r > 0, 5 µ(r) > 0. hq V h > 0, % Sh = {x ∈ R : |x| < h}. ?_��V_Æ���_Æ0 Sh× [t0,∞) |V��0A?_V�u� V (0, t) ≡ 0, Yh� µ ∈ K � V (x, t) ≥ µ(|x|), h*�V (x, t) ∈ Sh × [t0,∞) 1�� ���� V �0A�z_V�u� −V �?_V� 8 ���uVrz�� V (x, t) �0A��KV�u�h�g µ ∈ K � V (x, t) ≤ µ(|x|), h*�V (x, t) ∈ Sh × [t0,∞) 1�� ��_Æ0 R× [t0,∞) |V�� V (x, t), u�� lim inf |x|→∞ t≥t0 V (x, t) =∞, 5;���0A�obL`V� C C1,1(Sh× [t0,∞);R+) A*�� Sh× [t0,∞) R R+ |D0�Y�uNQ�V� u�� V (x, t). 2.1 d�)L 0;�ZN�K9Q.0�~.I_;�J1�K9%8�D x0 ���.�j$ ���'5��� �f 2.1[1] (1) q4 (1.4) V℄0A�'5I_V2��~.I_V�u�h�q V ε ∈ (0, 1) �"C_�V r > 0, D0�� δ = δ(ε, r, t0) > 0 2e P{|x(t; t0, x0)| < 0, ∀ t ≥ t0} ≥ 1− ε ;� |x0| < δ. v50q4 (1.4) V℄�'5$I_V� (2) q4V℄0A�'5NfI_V�u�,�'5I_V�#Yhq V ε ∈ (0, 1) �D0�� δ0 = δ0(ε, t0) > 0 2e P{ lim t→∞ |x(t; t0, x0)| = 0} ≥ 1− ε ;� |x0| < δ0. (3) q4V℄0A�gs'5NfI_V�u�,�'5I_V�#Yh*�V x0 ∈ R, � P{ lim t→∞ |x(t; t0, x0)| = 0} = 1 1�� �& 2.1[1] u�D0��?_�� V (x, t) ∈ C2,1(Sh × [t0,∞);R+), 9 h*�V (x, t) ∈ Sh × (t0,∞), � LV (x, t) ≤ 0 1��50q4 (1.4) V℄�'5I _V� �& 2.2[1] u�D0��?_�KV�� V (x, t) ∈ C2,1(Sh × [t0,∞);R+), #Y LV (x, t) �z_V�50q4 (1.4) V℄�'5NfI_V� �& 2.3[1] u�D0��?_�K#YobL`V�� V (x, t) ∈ C2,1(R× [t0,∞);R+), �T LV (x, t) �z_V�50q4 (1.4) V℄�gs'5NfI_V� 2.2 ��p3t=L �f 2.2[1] q4 (1.4) V℄�0A�>&�lG�I_V�u�h*�V x0 ∈ R, � lim sup t→∞ 1 t log |x(t; t0, x0)| < 0 a.s. 1�� g& 2.1[1] h�*�0 R NV8�D x0 6= 0, � P{|x(t; t0, x0)| 6= 0, ∀ t ≥ t0} = 1 �r� �℄Vq��r"_-{�V�)o $/..R%[� �& 2.4[1] GD0���� V (x, t) ∈ C2,1(R× [t0,∞);R+), �.� p > 0, c1 > 0, c2 ∈ R, c3 ≥ 0, h*�V x 6= 0 � t ≥ t0 � (1) c1|x|p ≤ V (x, t), (2) LV (x, t) ≤ c2V (x, t), (3) |Vx(x, t)g(x, t)|2 ≥ c3V 2(x, t). 5 lim sup t→∞ 1 t sup |x(t; t0, x0)| ≤ −c3 − 2c2 2p a.s. h*�V x0 ∈ R z1�� /�X�u� c3 > 2c2, 5q4 (1.4) V℄r0A�>&�lG�I_V� 10 D, 2.1[1] GD0���� V (x, t) ∈ C2,1(R × [t0,∞);R+), �?� p, α, λ, � Th*�V x 6= 0, t ≥ t0 �� α|x|p ≤ V (x, t), Y LV (x, t) ≤ −λV (x, t). 5 lim sup t→∞ 1 t log|x(t; t0, x0)| ≤ −λ p a.s. h*�V x0 ∈ R z1�� ,y) �q4 (1.4) V℄r0A�>&�lG�I_V� �& 2.5[1] GD0���� V (x, t) ∈ C2,1(R× [t0,∞);R+), �.� p > 0, c1 > 0, c2 ∈ R, c3 > 0, �Th�*�V x 6= 0 � t ≥ t0, � (1) c1|x|p ≥ V (x, t) > 0, (2) LV (x, t) ≥ c2V (x, t), (3) |Vx(x, t)g(x, t)|2 ≤ c3V 2(x, t). 5 lim inf t→∞ 1 t log |x(t; t0, x0)| ≥ −2c2 − c3 2p a.s. h*�0 R NV x0 6= 0 z1�� /�X�u� 2c2 > c3, 5 |x(t; t0, x0)| V>&*�V�)oQ/bbL_G�K 900;R\�V�q4 (1.4) V℄�>&�lG�$I_V� 2.3 #t=L �f 2.3[1] u�D0�h?� λ � C, h*�V x0 ∈ R 2e E|x(t; t0, x0)|p ≤ C|x0|pe−λ(t−t0), t ≥ t0 50q4 (1.4) V℄� p YuG�I_V� N p = 2 ��6.0q4V℄�zqG�I_V� �& 2.6[1] GD0��?.� K, 2e x⊤f(x, t) ∨ |g(x, t)|2 ≤ K|x|2 11 h*�V (x, t) ∈ R× [t0,∞) z1��5�q4 (1.4) V℄� p YuG�I_V� 9 9,V℄�>&�lG�I_V� �& 2.7[1] GD0���� V (x, t) ∈ C2,1(R × [t0,∞);R+), �?� c1, c2, c3, 5� c1|x|p ≤ V (x, t) ≤ c2|x|p � LV (x, t) ≤ −c3V (x, t), h*�V (x, t) ∈ R× [t0,∞) z1��Y E|x(t; t0, x0)|p ≤ c2 c1 |x0|pe−c3(t−t0), t ≥ t0 h*�V x0 ∈ R z1���r� q4 (1.4) V℄� p YuG�I_V� �& 2.8[1] % q > 0, GD0���� V (x, t) ∈ C2,1(R × [t0,∞);R+), �?� c1, c2, c3, 5� c1|x|q ≤ V (x, t) ≤ c2|x|q � LV (x, t) ≥ c2V (x, t), h*�V (x, t) ∈ R× [t0,∞) z1��Y E|x(t; t0, x0)|q ≥ c1 c2 |x0|pec3(t−t0), t ≥ t0 h*�V x0 ∈ R z1��5K9 q4 (1.4) V℄� p YuG�$I_V� 2.4 n5ewt=L 0;�ZN�K9Qb}3�n�� �K�_R8VG�6/��,�Aia 6/V� �4 2.1 G��R� f � g 2es'�QO=4M��r� �h�6�� k = 1, 2, · · · , D0��?� Hk, �T |f(x, t)− f(y, t)| ∨ |g(x, t)− g(y, t)| ≤ Hk|x− y| h*�V t ≥ 0 z1��#Y x, y ∈ R, |x| ∨ |y| ≤ k. 12 �4 2.2 GD0rz.� α, β, η � γ, 2e 〈x, f(x, t)〉 ≤ α + β|x|2 � |g(x, t)|2 ≤ η + γ|x|2 h*�V (x, t) ∈ R× R+ z1��5D0.� T , �T x⊤f(x, t) + 1 2 |g(x, t)|2 ≤ T (1 + |x|2), (TN T = max(α + 1 2 η, β + 1 2 γ)). �_� (1.8) �B�0G 2.1, 2.2 2eV4MV�'5?tq4 (1.4) 0 t ∈ R+ |�����Vgs℄ x(t), j�_� (1.9) K9� BS�q4 (1.4) V℄2e lim sup t→∞ 1 t log |x(t)| ≤ T a.s. ;�r� �q4 (1.4) V℄Q/�~.G�6/�V β + 1 2 γ 1��5q4 (1.4) V℄Q/2e lim sup t→∂ log(|x(t)|) log t ≤ δ 2δ − 2β − γ a.s. s v ��V θ �,2e 0 < θ < 2δ − 2β − γ δ % V = (1 + |x|2)θ, 5� Itoˆ K9�T� dV = d[(1 + |x|2)θ] = LV dt+M(t) (2.5) 13 TN LV = θ(1 + |x|2)θ−1(2x⊤f + |g|2) + 2θ(θ − 1)(1 + |x|2)θ−2|x⊤g|2, M(t) = 2θ(1 + |x|2)θ−1x⊤gdB(t). * �G (2.2) � TR LV ≤ θ(1 + |x|2)θ−2 × [(1 + |x|2)[2α + η + (2β + γ)|x|2] − 2(1− θ)(δ|x|4 − ρ)] = θ(1 + |x|2)θ−2[2α + η + 2ρ+ (2α+ η + 2β + γ)|x|2 − [2δ(1− θ)− 2β − γ]|x|4] .% V¯ = eεt(1 + |x|2)θ, 5� Itoˆ �T dV¯ = LV¯ dt+ L¯(t), � M¯(t) = 2θeεt(1 + |x|2)θ−1x⊤gdB(t) Y LV¯ ≤ εeεt(1 + |x|2)θ + eεtLV = θeεt(1 + |x|2)θ−2[ε θ (1 + |x|2)2] + eεtLV ≤ θeεt(1 + |x|2)θ−2[ε θ (1 + |x|2)2 + 2α + η + 2ρ + (2α+ η + 2β + γ)|x|2 − (2δ(1− θ)− 2β − γ)|x|4] v e�dV ε > 0, #�,2e ε θ < 2δ(1− θ)− 2β − γ, 5�ia ��V�`q�B�D0?� C1, �Thq V x ∈ R, � θ(1 + |x|2)θ−2(ε θ (1 + |x|2)2 + 2α + η + 2ρ +(2α+ η + 2β + γ)|x|2 − (2δ(1− θ)− 2β − γ)|x|4) ≤ C1 14 5 dV¯ ≤ C1eεtdt+ M¯(t), ; dV¯ ≤ C1eεtdt+ 2θeεt(1 + |x|2)θ−1x⊤gdB(t) * h| ��WaS>.a6t�T ∫ t 0 Ed[eεs(1 + |x(s)|2)θ] ≤ ∫ t 0 EC1e εsds ; E[eεt(1 + |x(t)|2)θ] ≤ (1 + |x0|2)θ] + C1 ε eεt, * lim sup t→∞ E[(1 + |x(t)|2)θ] ≤ C1 ε . (2.6) e�&K9� C1 HR (2.5) N� TR d[(1 + |x(t)|2)θ] ≤ C1dt+ 2θ(1 + |x(t)|2)θ−1x⊤(t)g(x(t), t)dB(t) ��6t�T E( sup t≤u≤t+1 (1 + |x(u)|2)θ) ≤ E[(1 + |x(t)|2)θ] + C1 +2θE( sup t≤u≤t+1 | ∫ u t (1 + |x(s)|2)θ−1x⊤(s)g(x(s), s)dB(s)|) M���X�V BDG $W �K9� TR 2 θE( sup t≤u≤t+1 | ∫ u t (1 + |x|2)θ−1x⊤gdB(s)|) ≤ 2θE( ∫ t+1 t (1 + |x|2)2θ−2|x⊤g|2ds) 12 ≤ 2θE( ∫ t+1 t (1 + |x|2)2θ−2|x|2 × (η + γ|x|2)ds) 12 ≤ 2θ√η ∨ γE([ sup t≤s≤t+1 (1 + |x|2)θ]× ∫ t+1 t (1 + |x|2)θ) 12 ≤ 1 2 E[ sup t≤s≤t+1 (1 + |x|2)θ] + 2θ2(η ∨ γ)E ∫ t+1 t (1 + |x|2)θds �| HRW��_D0��?� C2 2e E( sup k≤u≤k+1 |x(u)|2θ) ≤ C2, k = 1, 2, · · · .% ε¯ > 0 Aq V�5� Chebyshev $W ��T P{ sup k≤u≤k+1 |x(u)|2θ > k1+ε¯} ≤ C2 k1+ε¯ , k = 1, 2, · · · .�� Borel − Cantelli ���K9� TR�h>&*�V ω ∈ Ω, � sup k≤t≤k+1 |x(t)|2θ ≤ k1+ε¯. :��℄V k :�T+ 1�� �>�D0�� k0(ω), h>&*�V ω ∈ Ω, N k ≥ k0 ��| �1�V�� >�h�>&*�V ω ∈ Ω, u�� k ≥ k0 Y k ≤ t ≤ k + 1, 5� log(|x(t)|2θ) log t ≤ (1 + ε¯) log k log k = 1 + ε¯ �> lim sup t→∞ log |x(t)| log t ≤ 1 + ε¯ 2θ a.s. % ε¯→ 0, 5� lim sup t→∞ log |x(t)| log t ≤ 1 2θ a.s. �>;�V θ 2e; 0 < θ < 2δ − 2β − γ 2δ * lim sup t→∞ log(|x(t)|) log t ≤ δ 2δ − 2β − γ a.s. ' 1 h�'5?tq4 dx = f(x, t)dt+ g(x, t)dB(t) 16 u�K9_Æ f(x, t) = a + bx, � g(x, t) = σx TN (x, t) ∈ R× R+, B(t) ��� �%�-b� 50;R\�V�K9� TR |g(x, t)|2 = σ2x2, � |xg(x, t)|2 = σ2x4, #Yh�q dV ε > 0, � x⊤f(x, t) = ax+ bx2 ≤ 2a 2 ε + (b+ ε)x2 5�_� (2.9), K9� TR|q4V℄2e lim sup t→∞ log(|x(t)|) log t ≤ σ 2 σ2 − 2(b+ ε) a.s. % ε→ 0 K9� TR lim sup t→∞ log(|x(t)|) log t ≤ σ 2 σ2 − 2b a.s. �r� q4V℄�ia 6/� ' 2 K9W}+��JL�^q �.?tq4 y˙(t) = a+ by(t), TN t ≥ 0, Y8�D y(0) = y0 > 0, ;� a, b > 0. ℄;�.?tq4�K9� TR,V℄A y(t) = (y0 + a b )ebt − a b . �>r� lim t→∞ 1 t log(y(t)) = b. 17 �r� �|�.?tq4V℄bb�L_G�6/� u�K9'5Xnb;�q4��,�1��^qV'5?tq4� dx(t) = [a+ bx(t)]dt+ σx(t)dB(t). TN t ≥ 0, Y σ > 0, B(t) ��� �%�-b�Y8�DA x(0) = x0 > 0. 5;�'5?tq4�X℄ x(t) = exp[(b− 1 2 σ2)t+ σB(t)] × (x0 + a ∫ t 0 exp[(b− 1 2 σ2)s+ σB(t)]ds) �| 2b ��5|�'5?tq4V℄2e lim sup t→∞ log(x(t)) log t ≤ σ 2 σ2 − 2b a.s. ;�r� �hq V ε > 0, D0��?V'5�� Tε �~.2e x(t) ≤ tε+σ2/(σ2−2b), hq V t ≥ Tε z1��;�r� �'5?tq4V℄Q/�ia 6/� 2.5 n5z t=L 0;�ZN�K9Qb}3�n� Be�_R8VG�6/�,� ���� `VR8�AG�6/� �& 2.10[12] GD0rz� c1 − c6 2e c5 > c1, c6 > c2 + 2c4, −2〈x⊤f(x, t)〉 ≤ c1 + c2|x|2, |x⊤g(x, t)|2 ≤ c3|x|2 + c4|x|4. (2.7) #Y |g(x, t)|2 ≥ c5 + c6|x|2. (2.8) h*�V (x, t) ∈ R× R+ z1�� 18 % a = c5 − c1, b = c6 − c2 − 2c4 � c = c5 − c1 + c6 − c2 − 2c3. (1) u� c ≥ 2(a ∧ b), 5q4 (1.4) V℄2e lim inf t→∞ 1 t log |x(t)| ≥ (a ∧ b) a.s. (2) u� c < 2(a ∧ b), ; ab > 1 4 c2, 5q4 (1.4) V℄2e lim inf t→∞ 1 t log |x(t)| ≥ 1 2 min{a, b, ab− 0.25c 2 a + b− c } a.s. s � Itoˆ � (2.7) � (2.8) �K9� TR d[log(1 + |x(t)|2)] = (2〈x(t), f(x(t), t)〉+ |g(x(t), t)| 2 1 + |x(t)|2 − 2|x⊤(t)g(x(t), t)|2 1 + |x(t)|2 )dt + 2x⊤(t)g(x(t), t) 1 + |x(t)|2 dB(t) ≥ ((c5 − c1) + (c6 − c2)|x(t)| 2 1 + |x(t)|2 − 2c3|x(t)|2 + 2c4|x(t)|4 1 + |x(t)|2 )dt + 2x⊤(t)g(x(t), t) 1 + |x(t)|2 dB(t) = H(|x(t)|2) (1 + |x(t)|2)2dt+ 2x⊤(t)g(x(t), t) 1 + |x(t)|2 dB(t) ;� H(u) = a+ cu+ bu2 Y H : R+ → R, TN��V a, b, c _Æu_�N*Æ� Vr� H(|x(t)|2) (1 + |x(t)|2)2 ≥ (a ∧ b) * �| 1 4 c2 V\�V�K9� 9R��?� λ, �,2e H(u) ≥ λ(1 + u)2, hq V u ≥ 0 1��hj H(u)− λ(1 + u)2 = a− λ+ (c− 2λ)u+ (b− λ)u2 = ( 1 u )( a− λ 0.25(c− 2λ) 0.25(c− 2λ) b− λ )( 1 u ) �>| 1�V4M� λ ≤ a ∧ b, Y (a− λ)(b− λ) ≥ 0.25(c− 2λ)2. �r� λ ≤ a ∧ b, Y (a+ b− c)λ ≤ ab− 0.25c2. �A c < 2(a ∧ b), * c < 2(a ∧ b) ≤ a+ b, �>K9� v4?� λ = min{a, b, ab− 0.25c 2 a+ b− c }. h| 1���> H(|x(t)|2) (1 + |x(t)|2)2 ≥ λ. * �W