局部时与分数次Brown运动的拟必然分析
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摘要
本文我们主要研究三个方面的内容,一是研究光滑半鞅的局部时的拟必然平方变差,二是讨论分数次Brown运动和分数次Brown单的拟必然性质及其拟必然p变差,三是研究由无穷个Brown运动驱动的具有分布值漂移系数的随机微分方程的解的存在唯一性。
1 几个有用的不等式
首先我们准备几个有用的不等式,它们在我们后面几部分结果的证明中起到重要的作用。
下面的定理来源于[72]。
定理1 (1) (?)n∈N,p>1,α>0,存在常数C(n,p,α)>0,使(?)F∈D_α~(np)(E),有
‖F~((?)n)‖_(p,α)≤C(n,p,α)‖F‖_(np,α)~n
(2) (?)p>1,0<α<1/2,0≤α<1/2-α,存在常数C(p,α,σ)>0,使(?)F∈D_(α+σ)~p(E),有
‖F‖_(p,α)≤C(p,α,σ)‖F‖_p~(σ/(α+σ)‖F‖_(p,α+σ)~(α/(α+σ)),且(?)n∈N,存在常数C(n,p,α,σ)>0,使
‖F~((?)n)‖_(p,α)≤C(n,p,α,σ)‖F‖_(np)~(nσ/(α+σ)‖F‖_(np,α+σ)~(nα/(α+σ))
下面的不等式其证明思想可参看Ren[32],只是我们的证明方法略有不同。具体可参看文章[58]。
定理2 (?)p>1,k∈N,n≥k,存在常数C=C(n,p,k)>0,使(?)F∈D_k~(np),有
‖F~n‖_(p,k)≤C‖F‖_(np)~(n-k)‖F‖_(np,k)~k‖≤C‖F‖_(np,k)~n
如下的不等式同样可参看文章[58]。
定理3 (1) 设p≥2,r≥1,F∈D_1~(rp),则|F|~r∈D_1~p,且存在常数C=C(p,r)>0,使(?)F∈D_1~(rp),有
‖F~r‖_(p,1)≤C‖F‖_(rp)~(r-1)‖F‖_(rp,1)≤C‖F‖_(rp,1)~r
In this paper, we mainly study the following three aspects: the first is the quasi sure quadratic variation of local times of smooth semimartingales, the second is the quasi sure property of fractional Brownian motion and fractional Brownian sheet and their quasi sure p-variation, and the last is the existence and pathwise uniqueness of the strong solution to stochastic differential equations driven by countably many Brownian motions with distributional drift coefficients.1 Several useful inequalitiesFirstly we prepare several useful inequalities, which play a crucial role in proving the main results of the following parts.The following theorem is taken from [72]. Theorem 1 (1) Vn ∈ N, p > 1, α > 0, there exists a constant C(n,p,α) > 0 such that(2) Vp > 1, 0 < α < 1/2, 0≤σ< 1/2 -α, there exists a constant C(p,α,σ) > 0 such thatand Vn ∈ N, there exists a constant C(n,p,α,σ) > 0 such thatThe proof of the following inequality is borrowed from Ren [32], and we improve it slightly in [58].Theorem 2 Vp > 1, k∈N, n≥k, there exists a constant C = C(n,p,k) > 0 such that VF ∈ D_k~np, andThe following inequalities also come from [58].
Theorem 3 (1) If p>2, r>l, F G Dj", then |F|r G Df, and there exists a constant C = C(p,r) > 0, such that VF G DJ",(2) If p>2, r>l, /c<[r], F G Drkp, then |F|r G ￡>￡, and there exists a constant C = C(p, r, k) > 0, such that VF G Z)^,||||II \\p,k2 Quasi sure quadratic variation of local times of smooth semi-martingalesThis part of work is published in Bull.Sci.Math.(see [72]).Suppose that X is a smooth semimartingale, L? is the local time of X at x, An = {a^ < a" < ? ? ? < d%n} is a sequence of partition of [a,b], where a" = i(b — a)/2n + a,We prove the following main theorem:Theorem 4Vp>l,0 1, n G N, the sample paths of which are Holder continuous of order 7 G [0, H), (p,n)-q.s..Suppose that An = {t$ < ￡" < ? ? ? < ??} is a sequence of partition of [0,1], where tfl = i2~n, i = 0,1, ? ? ? 2n. We prove the following main theorems:Theorem 6 Vp > —, q > 1, H2?-lV^ p = 0lim y Btn At - BtnAti=0
holds uniformly in t G [0,1], (q, [p])-q.s.. Thus, in particular, it holds uniformly in t[0,l],(2,oo)-q.s..Theorem 7 Vp = 1m > —, m ￡ N,f2n-l ?2mlimn—>i=0= 0holds uniformly in t ￡ [0,1] q.s..4 Quasi sure p-variation of fractional Brownian sheetLet a,/3 ￡ (0,1), {Bz,z ￡ K2} is the fractional Brownian sheet with Hurst parameter a, f3. We construct the abstract Wiener space by the properties of B, and obtain the following results. This part of work is taken from [59].Theorem 8 Bz admits oo-modifications, and Vp > 1, n e N, the sample paths of which are Holder continuous of order 76 [0,a A 0), (p,n)-q.s..Suppose that A^(z) = (zfj A z, z?+1J+1 A z], where z = (s, t), z?j = (s?, q), sn{ = i2~n, tj = j2~n, i = 0,1, ? ? ? 2n, j = 0,1, ? ? ? 2n. We prove the following main theorems:2 Theorem 9 If a + (3 > 1, then Vp >-------, q>l,ol + purn yyV= 0 i=o i=oholds uniformly in z e [0,1]2, (g, [p])-q.s.. Thus, in particular, it holds uniformly in z E [0,1]2, (2,oo)-q.s..2 Theorem 10 If a + (3 > 1, then Vp = 2m >-------, m ￡ N,2?-12n-l x—^ x—\ Plim 2^ 2^ B(A^-(z)) =0i=0 j'=0holds uniformly in z ￡ [0,1]2 q.s..5 SDEs driven by countably many Brownian motionsConsider the following time homogeneous Markovian stochastic differential equations:a(Xs)dWs+ [ b(Xs)ds, (1)o Jowhere W = (W1, W2, ? ? ?), {W\j = 1,2,---} is an infinite sequence of independent Brownian motions on (Q, JT,P; JF^), which is a complete filtered probability space.
Similar to the finite dimensional case, we define the conceptions of the weak solution, the strong solution and the unique strong solution, then develop the results which parallel classical theories, and give some conditions on existence of the unique strong solution to the equation with non-Lipschitz coefficients. This part of work is published in J.Funct.Anal.(see [55]).Theorem 11 Eq.(l) has a unique strong solution if and only if there is a solution to (1) and the pathwise uniqueness holds.Theorem 12 Suppose that a and b in Eq.(l) are bounded, and the following conditions are satisfied:(1) there exists a continuous non-decreasing function p on R+ such that p(0) = 0,I p~2(u) du = +oo, Jo+and Vx,y G E,\\a(x)-a(y)\\^p(\x-y\),(2) there exists a concave non-decreasing function 7 on E+ such that 7(0) = 0,rI 7~1(m) du = +00, Jo+and \/x,y e R\b(x)-b(y)\^(\x-y\), then Eq.(l) has a unique strong solution.The following work is taken from [73]. Firstly we discuss Eq.(5.6) with measurable drift by Zvonkin's method (see [26]) and give the following results:Theorem 13 Suppose that a and b in Eq.(l) are bounded, and the following conditions are satisfied:(1) there exists a continuous non-decreasing function p on R+ such that p(0) = 0,/ 2/ \ 1—jd? = +oo, Jo+ pz{u) + u2and \/x,y G E,\\a(x) -a(y)\\^p(\x-y\).(2) the function / = -—^ is bounded and / G C1\\cr\\then Eq.(l) has a unique strong solution.
Theorem 14 Keep the same assumption as Theorem 13, where p satisfies that0 (u = 0)p(u) = (2)Denote by X(x, ?) the unique strong solution to Eq.(l) with initial value x, then X admits a coutinuous modifications X(x,t) such that W > 0 and19xh X(x, ?) G C([0, t]) is /3-Holder continuous, and for a.a.w, W G R+, X(-, J):1h1 is a homomorphism.Secondly, let G = {p : p is a continuous non-decreasing function on M+,p(0) = 0, Jo+ p~2(x) dx = +oo}. DefineCp = {/ : / is a Borel measurable function on R, 3p e G, such that Va;,?/ e R, \f(x) - f(y)\ I2 such that a E B(R)/B(R 0 I2), A is & continuous process with zero energy, formally we write A asctAt= I b'{Xs)\\a(Xs)\\2ds, Jowhere bf is the distributional derivative of some function b G Cp.We define the conception of the solution to Eq.(3), then prove the following results: Theorem 15 Suppose that a and b in Eq.(3) are bounded, and Vx,y G R\\a(x) - a(y)\\ < p(\x-y\), \b(x)-b(y)\ < 7(|z-0l),where p,j G G such that / ------—— da; = +oo, then Eq.(3) has a pathwise uniqueJ o+P {x) +7 Wsolution.
Theorem 16 Keep the same assumption as Theorem 15, where p is given by (2). Denote by X(x,-) the pathwise unique solution to Eq.(3) with initial value x, then X admits a coutinuous modifications X(x,t) such that Vt > 0 andl + \/l-e-T* , ?) e C([0, t]) is /3-Holder continuous, and for a.a.w, V￡ e R+, X(-, {):Ih1 is a homomorphism.
1 几个有用的不等式
首先我们准备几个有用的不等式,它们在我们后面几部分结果的证明中起到重要的作用。
下面的定理来源于[72]。
定理1 (1) (?)n∈N,p>1,α>0,存在常数C(n,p,α)>0,使(?)F∈D_α~(np)(E),有
‖F~((?)n)‖_(p,α)≤C(n,p,α)‖F‖_(np,α)~n
(2) (?)p>1,0<α<1/2,0≤α<1/2-α,存在常数C(p,α,σ)>0,使(?)F∈D_(α+σ)~p(E),有
‖F‖_(p,α)≤C(p,α,σ)‖F‖_p~(σ/(α+σ)‖F‖_(p,α+σ)~(α/(α+σ)),且(?)n∈N,存在常数C(n,p,α,σ)>0,使
‖F~((?)n)‖_(p,α)≤C(n,p,α,σ)‖F‖_(np)~(nσ/(α+σ)‖F‖_(np,α+σ)~(nα/(α+σ))
下面的不等式其证明思想可参看Ren[32],只是我们的证明方法略有不同。具体可参看文章[58]。
定理2 (?)p>1,k∈N,n≥k,存在常数C=C(n,p,k)>0,使(?)F∈D_k~(np),有
‖F~n‖_(p,k)≤C‖F‖_(np)~(n-k)‖F‖_(np,k)~k‖≤C‖F‖_(np,k)~n
如下的不等式同样可参看文章[58]。
定理3 (1) 设p≥2,r≥1,F∈D_1~(rp),则|F|~r∈D_1~p,且存在常数C=C(p,r)>0,使(?)F∈D_1~(rp),有
‖F~r‖_(p,1)≤C‖F‖_(rp)~(r-1)‖F‖_(rp,1)≤C‖F‖_(rp,1)~r
In this paper, we mainly study the following three aspects: the first is the quasi sure quadratic variation of local times of smooth semimartingales, the second is the quasi sure property of fractional Brownian motion and fractional Brownian sheet and their quasi sure p-variation, and the last is the existence and pathwise uniqueness of the strong solution to stochastic differential equations driven by countably many Brownian motions with distributional drift coefficients.1 Several useful inequalitiesFirstly we prepare several useful inequalities, which play a crucial role in proving the main results of the following parts.The following theorem is taken from [72]. Theorem 1 (1) Vn ∈ N, p > 1, α > 0, there exists a constant C(n,p,α) > 0 such that(2) Vp > 1, 0 < α < 1/2, 0≤σ< 1/2 -α, there exists a constant C(p,α,σ) > 0 such thatand Vn ∈ N, there exists a constant C(n,p,α,σ) > 0 such thatThe proof of the following inequality is borrowed from Ren [32], and we improve it slightly in [58].Theorem 2 Vp > 1, k∈N, n≥k, there exists a constant C = C(n,p,k) > 0 such that VF ∈ D_k~np, andThe following inequalities also come from [58].
Theorem 3 (1) If p>2, r>l, F G Dj", then |F|r G Df, and there exists a constant C = C(p,r) > 0, such that VF G DJ",(2) If p>2, r>l, /c<[r], F G Drkp, then |F|r G ￡>￡, and there exists a constant C = C(p, r, k) > 0, such that VF G Z)^,||||II \\p,k2 Quasi sure quadratic variation of local times of smooth semi-martingalesThis part of work is published in Bull.Sci.Math.(see [72]).Suppose that X is a smooth semimartingale, L? is the local time of X at x, An = {a^ < a" < ? ? ? < d%n} is a sequence of partition of [a,b], where a" = i(b — a)/2n + a,We prove the following main theorem:Theorem 4Vp>l,0
holds uniformly in t G [0,1], (q, [p])-q.s.. Thus, in particular, it holds uniformly in t[0,l],(2,oo)-q.s..Theorem 7 Vp = 1m > —, m ￡ N,f2n-l ?2mlimn—>
Similar to the finite dimensional case, we define the conceptions of the weak solution, the strong solution and the unique strong solution, then develop the results which parallel classical theories, and give some conditions on existence of the unique strong solution to the equation with non-Lipschitz coefficients. This part of work is published in J.Funct.Anal.(see [55]).Theorem 11 Eq.(l) has a unique strong solution if and only if there is a solution to (1) and the pathwise uniqueness holds.Theorem 12 Suppose that a and b in Eq.(l) are bounded, and the following conditions are satisfied:(1) there exists a continuous non-decreasing function p on R+ such that p(0) = 0,I p~2(u) du = +oo, Jo+and Vx,y G E,\\a(x)-a(y)\\^p(\x-y\),(2) there exists a concave non-decreasing function 7 on E+ such that 7(0) = 0,rI 7~1(m) du = +00, Jo+and \/x,y e R\b(x)-b(y)\^(\x-y\), then Eq.(l) has a unique strong solution.The following work is taken from [73]. Firstly we discuss Eq.(5.6) with measurable drift by Zvonkin's method (see [26]) and give the following results:Theorem 13 Suppose that a and b in Eq.(l) are bounded, and the following conditions are satisfied:(1) there exists a continuous non-decreasing function p on R+ such that p(0) = 0,/ 2/ \ 1—jd? = +oo, Jo+ pz{u) + u2and \/x,y G E,\\a(x) -a(y)\\^p(\x-y\).(2) the function / = -—^ is bounded and / G C1\\cr\\then Eq.(l) has a unique strong solution.
Theorem 14 Keep the same assumption as Theorem 13, where p satisfies that0 (u = 0)p(u) = (2)Denote by X(x, ?) the unique strong solution to Eq.(l) with initial value x, then X admits a coutinuous modifications X(x,t) such that W > 0 and19xh X(x, ?) G C([0, t]) is /3-Holder continuous, and for a.a.w, W G R+, X(-, J):1h1 is a homomorphism.Secondly, let G = {p : p is a continuous non-decreasing function on M+,p(0) = 0, Jo+ p~2(x) dx = +oo}. DefineCp = {/ : / is a Borel measurable function on R, 3p e G, such that Va;,?/ e R, \f(x) - f(y)\
Theorem 16 Keep the same assumption as Theorem 15, where p is given by (2). Denote by X(x,-) the pathwise unique solution to Eq.(3) with initial value x, then X admits a coutinuous modifications X(x,t) such that Vt > 0 andl + \/l-e-T* , ?) e C([0, t]) is /3-Holder continuous, and for a.a.w, V￡ e R+, X(-, {):Ih1 is a homomorphism.
引文
[1] Shigekawa I. On a quasi everywhere existence of the local time of the 1-dimensional Brownian motion. Osaka J.Math., 1984, 21: 621-627
[2] Nualart D, Vives J. Smoothness of local time and related Wiener functionals. Potential Analysis, 1992, 1: 257-263
[3] Watanabe S. Fractional order Sobolev spaces on Wiener space. Prob.Theory Re-lat.Fields, 1993, 95: 175-198
[4] Airault H, Ren J, Zhang X. Smoothness of local times of semimartingales. C.R.Acad.Sci.Paris, Serie I, 2000, 330: 719-724
[5] Hu J, Ren J. Infinite dimensinoal quasi continuity, path continuity and ray continuity of functions with fractional regularity. J.Math.Pures Appl, 2001, 80(1): 131—152
[6] Ren J. Quasi sure quadratic variation of smooth martingales. J.Math.Kyoto Univ., 1994, 34(1): 191-206
[7] Liang Z. Quasi sure quadratic variations of two-parameter smooth martingales on the wiener space. J. Math.Kyoto Univ., 1996, 36(3): 619-640
[8] Liu J, Ren J. Quasi sure product variations of two parameter smooth martingales on the Wiener space, Preprint
[9] Bouleau N, Yor M. Sur la variation quadratique des temps locaux de certaines semimartingales. C.R.Acad.Sci. Paris, Serie I, 1981, 292: 491-494
[10] Perkins E. Local time is a semimartingale. Z.Wahrscheinlichkeitstheorie Verw.Gebiete, 1982, 60: 79-117
[11] Revuz D, Yor M. Continuous martingales and Brownian motion. Grund.Math.Wiss., 293, Springer-Verlag, 1991
[12] Alos E, Mazet O, Nualart D. Stochastic calculus with respect to fractional Brown ian motion with Hurst parameter less that 1/2. Stochastic Process.Appl., 2000, 86: 121-139
[13] Alos E, Mazet O, Nualart D. Stochastic calculus with respect to Gaussian processes. Ann.Probab., 2001, 29: 766-801
[14] Alos E, Nualart D. Stochastic integration with respect to the fractional Brownian motion. Stochastics and Stochastics Rep., 2003, 75: 129—152
[15] Dai W, Heycle C. Ito's formula with respect to fractional Brownian motion and its application. J.Appl.Math.Stochastic Anal., 1996, 9: 439-448
[16] Decreusefond L, Ustunel A. Stochastic analysis of the fractional Brownian motion. Potential Anal, 1998, 10: 177-214
[17] Duncan T, Hu Y, Pasik-Duncan B. Stochastic calculus for fractional Brownian motion I.Theory. SIAM J. Control Optim., 2000, 38: 582-612
[18] Hu Y, φksendal B. Fractional white noise calculus and applications to finance. Department of Mathematics, University of Oslo, 1999, 10, Preprint
[19] Lin S. Stochastic analysis of fractional Brownian motions. Stochastic and Stochastic Rep., 1995, 55: 121-140
[20] Zahle M. Integration with respect to fractal functions and stochastic calculus I.. Prob.Theory Relat. Fields, 1998, 111(3): 333-374
[21] Zahle M. Integration with respect to fractal functions and stochastic calculus II.. 1999, Preprint
[22] Leger S, Pontier M. Drap brownien fractionnaire. C.R.Acad.Sci.Paris, Serie I, 1999, 329: 893-898
[23] Ayache A, Leger S, Pontier M. Drap brownien fractionnaire. Potential Anal, 2002, 17: 31-43
[24] Rogers L. Arbitrage with fractional Brownian motion. Math.Finance, 1997, 7: 95-105
[25] Yamada T, Watanabe S. On the uniqueness of solutions of stochastic differential equations. J.Math.Kyoto Univ., 1971, 11: 155-167
[26]Zvonkin A. A transformation of the phase space of a process that removes the drift. Math.USSR Sbornik 2, 1974, 129-149
Dim. Anal. Quantum Probab., 2004, 7:155～159
[90] Osterwalder K, Schrader R. Axios for Euclidean Green's function Ⅰ. Comm. Math. Phys., 1973,31:83～112
[91] Osterwalder K, Schrader R. Axios for Euclidean Green's function Ⅱ. Comm. Math. Phys., 1975, 42:281～305
[92] 吴莺.重正化核的Poisson随机积分表示.应用数学,2005,18:484～488
[93] Accard L, Lu Y G, Volovich I. Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations. Publications of ⅡAS (Kyoto), 1997.
[94] Accard L, Lu Y G. The Wigner semi-circle law in quantum electro dynamics. Commun.Math.Phys., 1996, 180:605～632
[95] Lytvynov E, Rebenko A L, Shchepan'uk G V. Wick calculus on spaces of generalized functions of compound Poisson white noise. Rep. Math. Phys., 1997, 39:219～248
[96] Bogolubov N N, Logunov A A et al. General Principles of Quantum Field Theory. Kulwer, 1990.
[97] Sicong J. The q-deformed binomial distribution and its asymptotic behavior. J. Phys. A: Math. Gen., 1994, 27: 493～499
[98] Gasper G, Rahman M. Hypergeometric Series. Cambridge: Cambridge Univ. Press, 1990.
[99] Sicong J. The q-deformaed binomial distribution and its asymptotic behaviour. J. Phys. A: Math. Gen., 1994, 27:493～499
[100] Allaway W. Some properties of the q-Hermite polynomials. Can. J. Math., 1980, 32: 686～694
[101] Jackson F. On q-definite integrals. Q. J. Pure Appl. Math., 1910, 41:193～203
[102] Huang Z Y, Li P Y, Wu Y. Explicit forms of q-deformed Lévy-Meixner polynomials and their generating functions, preprint 2005.
[103] Anshelevich M. Free martingale polynomials. J. Funct. Anal., 2003, 201:228～261
[104] Anshelevich M. Appell polynomials and their relatives. Int. Math. Res. Not., 2004, 65: 3469～3531
[105] Nica A. A one-parameter family of transforms, linearizing convolution laws for probability distributions. Commum. Math. Phys., 1995, 168:187～207
[41] Coutin L, Nualart D, Tudor C. Tanaka formula for the fractional Brownian motion. Stochastic Process.Appl., 2001, 94:301～315
[42] Erraoui M, Nualart D, Ouknine Y. Hyperbolic stochastic partial diferential equations with additive fractional Brownian sheet. Stochastic Dynamics, 2003, 3:121～139
[43] Da Prato, Zabczyk J. Stochastic equations in infinite dimensions. Cambridge University Press, 1992
[44] Ikeda N, Watanabe S. Stochastic differential equations and diffusion processes. North-Holland/Kodanska, 1989
[45] 黄志远.随机分析学基础(第二版).北京:科学出版社,2001
[46] Zvonkin A, Krylov N. On strong solutions of stochastic differential equations (in Russian). Proc.School-Seminal on Theor.Random Processes, Vilnius, Part Ⅱ, 1975, 9～88
[47] Adams R. Sobolev Spaces. New-York:Academic Press, 1978
[48] Airault H, Ren J. Modulus of continuity of the canonic Brownian motion "on the group of diffeomorphisms of the circle". J. Funet.Anal., 2002, 196(2): 395～426
[49] Bass R. Probabilistic techniques in analysis. Springer-Verlag, 1995
[50] Bihari I, A generalization of a lemma of Belmman and its application to uniqueness problem of differential equations. Acta Math.Acad.Sci.Hungar, 1956, 71～94
[51] Brzezniak Z, Carroll A. Approximations of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces. Lecture notes in mathematics, Springer-Verlag Heidelberg, 2003, 1832:251～289
[52] Cairoli R. Sur une équation différentielle stochastique. C.R.Acad.Sci.Paris Série A, 1972, 274:1173～1742
[53] Cairoli R, Walsh J. Stochastic integrals in the plane. Acta Mathematica, 1975, 134: 111～183
[54] Cao G. Existence and pathwise uniqueness of solutions to stochastic differential equations in Banach space. To appear in Mathematica Applicata(China)
[55] Cao G, He K. On a type of stochastic differential equations driven by countably many Brownian motions. J. Funct.Anal., 2003, 203(1):262～285
[56] 曹桂兰,何凯.由可数个Brown单驱动的随机微分方程解的存在唯一性.数学物理学报(中文版)(已录用)
[57] Cao G, He K. Strongly Markovian property of solutions on a type of SDEs driven by countably many Brownian motions, Preprint
[58] Cao G, He K. Quasi sure p-variation of fractional Brownian motion, Preprint
[59] Cao C, He K. Quasi sure p-variation of fractional Brownian sheet, Preprint
[60] Cao G, He K, Zhang X. Successive approximations of infinite dimensional SDEs with jump, Preprint
[61] Carmona L, Coutin G. Stochastic integration with respect to fractional Brownian motion, 1998, Preprint
[62] Carmona L, Coutin G, Montseny. Stochastic integration with respect to fractional Brownian motion. Annales de l'Institut H. Poincare, 2003, 39(1): 27～68
[63] Chen X. The path uniqueness of the solution of the two-parameter Poisson type stochastic differential equation. Chinese Sci.Bull., 1988, 33(9): 649～652
[64] Chen X. Comparison theorem of the strong solution of the two-parameter Poisson type stochastic differential equation. Chinese Sci.Bull., 1989, 34(12): 881～884
[65] Chevalier L. Martingales continues a deux parametres. Bull.Sci.Math., 1982, 106: 19～62
[66] Feyel D, de la Pradelle A. Fractional integrals and Brownian processes. Potential Anal., 1996, 10: 273～288
[67] 龚光鲁.随机微分方程引论(第二版).北京:北京大学出版社,2000
[68] He K, Cao G. Successive approximations of solutions to stochastic differential equations with jump in the plane. To appear in Mathematica Applicata(China)
[69] 何凯,曹桂兰.连续局部鞅的无穷维随机积分表示.华中科技大学学报(自然科学版)(已录用)
[70] 何凯,曹桂兰.由可数个Brown运动驱动的随机微分方程的唯一强解
[71] 何凯,曹桂兰.无穷维半线性倒向随机微分方程的逐次逼近
[72] He K, Ren J. Quasi sure quadratic variation of local times of smooth semimartingales. Bull.Sci.Math., 2004, 128(2): 91-104
[73] He K, Zhang X. One dimensional SDEs with distributional drifts. Preprint
[74] 何声武,汪嘉冈,严加安. 半鞅与随机分析. 北京:科学出版社,1995
[75] Hitsuda M, Watanabe H. On stochastic integrals with respect to an infinite number of Brownian motions and its applications. Proc.of Intern.Symp.SDE., 1976, 57—74
[76] Hu Y, Peng S. Adapeted solution of a backward semilinear stochastic evolution equation. Stochastic Anal. Appl, 1991, 9: 445-459
[77] Karatzas I, Shreve S. Brownian motion and stochastic calculus. New York: Springer- Verlag, 1988
[78] Kaspi H, Rosen J. p-variation for families of local times on lines. Sem.Prob.XXXIV. Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg New York, 2000, 1729, 171-184
[79] Knoche C, Frieler K. Solutions of stochastic differential equations in infinite dimensional Hilbert spaces and their dependence on initial data. Bielefeld university, 2001
[80] Le Gall. Applications du temps local aux equations differentielles stochastiques uni- dimensionnelles. Sem.Prob.XVII. Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York, 1983, 986, 15-31
[81] Liang Z. Existence and pathwise uniqueness of solutions for stochastic differential equations with respect to martingales in the plane. Stochastic Process.Appl., 1999, 83: 303-317
[82] Liang Z, Zheng M. Estimates on moments of the solution to S.D.E. with respect to martingales in the plane. Stochastic Process.Appl., 1996, 62: 263—276
[83] Liu J. On the existence and uniqueness of solutions to stochastic differential equationsof mixed Brownian and Poissonian sheet type. Stochastic Process.Appl., 2001, 94: 339-354
[84] Ma J, Yong J. Forward-backward stochastic differential equations and their applications. New York: Springer-Verlag, 1999
[85] Malliavin P. The canonic diffusion above the diffeomorphism group of the circle. C.R.Acad.Sci.Paris, Serie I, 1999, 329: 325-329
[86]Mandelbrot B, Van Ness J. Fractional Brownian motions, fractional noises and applications. SIAM Rev, 1968, 10: 422-437
[87] Mao X. Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stochastic Process.Appl, 1995, 58: 281-292
[88]Marcus M, Rosen J. p-variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments. Ann.Probab., 1992, 20: 1685—1713
[89]Mazziotto G. Equation du filtrage pour un processes de Possion melange a deux indices. Stochastic, 1980, 4: 89-119
[90]Neidhardt A. Stochastic integrals in 2-uniformly smooth Banach spaces. University of Wisconsin, 1978
[91] Nie Z. Uniqueness on two-parameter Ito's stochastic differential equations. Acta Math.Sinica., 1987, 30: 179-186
[92] Pardoux E, Peng S. Adapeted solution of a backward stochastic differential equation. Systems Control Lett., 1990, 14: 55-61
[93] Perkins E. Local time and pathwise uniqueness for stochastic differential equations. Sem.Prob.XVI. Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York, 1982, 920, 201-208
[94] Protter P. Stochastic intergration and differential equation: a new approach. New York: Springer, 1990
[95] Reid J. Estimate on moments of the solution to stochastic differential equations in the plane. Ann.Probab., 1983, 11: 656-668
[96] Ren J, Zhang X. Quasi sure analysis of two-parameter stochastic differential equations. Stochastic and Stochastic Rep., 2002, 72(3-4): 251-276
[97] Rogers L, Williams D. Diffusions, Markov processes and martingales. 2. 2nd Edition. Cambridge: Cambridge University press, 2000
[98]Rosen J. p-variation of the local times of stable processes and intersection local time. Seminar on Stochastic processes, 1991(Boston), Progress in Probability, Birkhauser, Boston, 1993,33, 157-168
[99]Situ R. On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process.Appl., 1997, 66: 209—236
[100] Situ R. Backward stochastic differential equations with jumps and applications. Guangzhou: Guangdong Science and Technology Press, 2000
[101] Stroock D, Varadhan S, Multidimensional Diffusion Processes. Springer-Verlag, Berlin-Heidelberg-New York, 1979
[102] Taniguchi T. Successive approximations to solutions of stochastic differential equations. J.Differ. Equations, 1992, 96: 152～169
[103] Wentzell A. A course in the theory of stochastic processes. McGraw-Hill Inc, 1981
[104] Wong E, Zakai M. Martingale and stochastic integrals for processes with multidimensional parameter. Z.Wahrscheinlichkeitstheorie Verw.Gebiete., 1974, 29: 109～122
[105] Wong E, Zakai M. Weak martingales and stochastic integrals in the plane. Ann.Probab., 1976, 4: 570～586
[106] Wong E, Zakai M. Differentiation formula for stochastic integral in the plane. Stochastic Process.Appl., 1978, 6: 339～349
[107] Yamada T. On the successive approximation of solutions of stochastic differential equations. J.Math.Kyoto Univ., 1981, 21: 501～515
[108] Yamada T, Ogura Y. On the strong comparison theorems for solutions of stochastic differential equations. Z.Wahrscheinlichkeitstheorie Verw.Gebiete, 1981, 56: 3～19
[109] 严加安,彭实戈,方诗赞,吴黎明.随机分析选讲.北京:科学出版社,2000
[110] Yeh J. Existence of strong solutions for stochstic differential equations in the plane. Pac.J.Math., 1981, 97: 217～247
[111] Yeh J. Existence of weak solutions to stochstic differential equations in the plane with continuous coefficients. Trans.Amer.Math.Soc., 1985, 290: 345～361
[112] Yeh J. Uniqueness of strong solutions to S.D.E. in the plane with deterministic boundary process. Pac.J. Math., 1987, 128: 391～400
[113] 周少甫.(非Lipschitz系数)倒向随机微分方程和随机微分效用:[博士学位论文].华中科技大学图书馆,2000
[114] Zhou S, Cao X. Adapeted solution of a backward semilinear stochastic evolution equation with non-Lipschitz coefficients. Acta Math.Sci., 2003, 23B(4): 485～493
[2] Nualart D, Vives J. Smoothness of local time and related Wiener functionals. Potential Analysis, 1992, 1: 257-263
[3] Watanabe S. Fractional order Sobolev spaces on Wiener space. Prob.Theory Re-lat.Fields, 1993, 95: 175-198
[4] Airault H, Ren J, Zhang X. Smoothness of local times of semimartingales. C.R.Acad.Sci.Paris, Serie I, 2000, 330: 719-724
[5] Hu J, Ren J. Infinite dimensinoal quasi continuity, path continuity and ray continuity of functions with fractional regularity. J.Math.Pures Appl, 2001, 80(1): 131—152
[6] Ren J. Quasi sure quadratic variation of smooth martingales. J.Math.Kyoto Univ., 1994, 34(1): 191-206
[7] Liang Z. Quasi sure quadratic variations of two-parameter smooth martingales on the wiener space. J. Math.Kyoto Univ., 1996, 36(3): 619-640
[8] Liu J, Ren J. Quasi sure product variations of two parameter smooth martingales on the Wiener space, Preprint
[9] Bouleau N, Yor M. Sur la variation quadratique des temps locaux de certaines semimartingales. C.R.Acad.Sci. Paris, Serie I, 1981, 292: 491-494
[10] Perkins E. Local time is a semimartingale. Z.Wahrscheinlichkeitstheorie Verw.Gebiete, 1982, 60: 79-117
[11] Revuz D, Yor M. Continuous martingales and Brownian motion. Grund.Math.Wiss., 293, Springer-Verlag, 1991
[12] Alos E, Mazet O, Nualart D. Stochastic calculus with respect to fractional Brown ian motion with Hurst parameter less that 1/2. Stochastic Process.Appl., 2000, 86: 121-139
[13] Alos E, Mazet O, Nualart D. Stochastic calculus with respect to Gaussian processes. Ann.Probab., 2001, 29: 766-801
[14] Alos E, Nualart D. Stochastic integration with respect to the fractional Brownian motion. Stochastics and Stochastics Rep., 2003, 75: 129—152
[15] Dai W, Heycle C. Ito's formula with respect to fractional Brownian motion and its application. J.Appl.Math.Stochastic Anal., 1996, 9: 439-448
[16] Decreusefond L, Ustunel A. Stochastic analysis of the fractional Brownian motion. Potential Anal, 1998, 10: 177-214
[17] Duncan T, Hu Y, Pasik-Duncan B. Stochastic calculus for fractional Brownian motion I.Theory. SIAM J. Control Optim., 2000, 38: 582-612
[18] Hu Y, φksendal B. Fractional white noise calculus and applications to finance. Department of Mathematics, University of Oslo, 1999, 10, Preprint
[19] Lin S. Stochastic analysis of fractional Brownian motions. Stochastic and Stochastic Rep., 1995, 55: 121-140
[20] Zahle M. Integration with respect to fractal functions and stochastic calculus I.. Prob.Theory Relat. Fields, 1998, 111(3): 333-374
[21] Zahle M. Integration with respect to fractal functions and stochastic calculus II.. 1999, Preprint
[22] Leger S, Pontier M. Drap brownien fractionnaire. C.R.Acad.Sci.Paris, Serie I, 1999, 329: 893-898
[23] Ayache A, Leger S, Pontier M. Drap brownien fractionnaire. Potential Anal, 2002, 17: 31-43
[24] Rogers L. Arbitrage with fractional Brownian motion. Math.Finance, 1997, 7: 95-105
[25] Yamada T, Watanabe S. On the uniqueness of solutions of stochastic differential equations. J.Math.Kyoto Univ., 1971, 11: 155-167
[26]Zvonkin A. A transformation of the phase space of a process that removes the drift. Math.USSR Sbornik 2, 1974, 129-149
Dim. Anal. Quantum Probab., 2004, 7:155～159
[90] Osterwalder K, Schrader R. Axios for Euclidean Green's function Ⅰ. Comm. Math. Phys., 1973,31:83～112
[91] Osterwalder K, Schrader R. Axios for Euclidean Green's function Ⅱ. Comm. Math. Phys., 1975, 42:281～305
[92] 吴莺.重正化核的Poisson随机积分表示.应用数学,2005,18:484～488
[93] Accard L, Lu Y G, Volovich I. Interacting Fock spaces and Hilbert module extensions of the Heisenberg commutation relations. Publications of ⅡAS (Kyoto), 1997.
[94] Accard L, Lu Y G. The Wigner semi-circle law in quantum electro dynamics. Commun.Math.Phys., 1996, 180:605～632
[95] Lytvynov E, Rebenko A L, Shchepan'uk G V. Wick calculus on spaces of generalized functions of compound Poisson white noise. Rep. Math. Phys., 1997, 39:219～248
[96] Bogolubov N N, Logunov A A et al. General Principles of Quantum Field Theory. Kulwer, 1990.
[97] Sicong J. The q-deformed binomial distribution and its asymptotic behavior. J. Phys. A: Math. Gen., 1994, 27: 493～499
[98] Gasper G, Rahman M. Hypergeometric Series. Cambridge: Cambridge Univ. Press, 1990.
[99] Sicong J. The q-deformaed binomial distribution and its asymptotic behaviour. J. Phys. A: Math. Gen., 1994, 27:493～499
[100] Allaway W. Some properties of the q-Hermite polynomials. Can. J. Math., 1980, 32: 686～694
[101] Jackson F. On q-definite integrals. Q. J. Pure Appl. Math., 1910, 41:193～203
[102] Huang Z Y, Li P Y, Wu Y. Explicit forms of q-deformed Lévy-Meixner polynomials and their generating functions, preprint 2005.
[103] Anshelevich M. Free martingale polynomials. J. Funct. Anal., 2003, 201:228～261
[104] Anshelevich M. Appell polynomials and their relatives. Int. Math. Res. Not., 2004, 65: 3469～3531
[105] Nica A. A one-parameter family of transforms, linearizing convolution laws for probability distributions. Commum. Math. Phys., 1995, 168:187～207
[41] Coutin L, Nualart D, Tudor C. Tanaka formula for the fractional Brownian motion. Stochastic Process.Appl., 2001, 94:301～315
[42] Erraoui M, Nualart D, Ouknine Y. Hyperbolic stochastic partial diferential equations with additive fractional Brownian sheet. Stochastic Dynamics, 2003, 3:121～139
[43] Da Prato, Zabczyk J. Stochastic equations in infinite dimensions. Cambridge University Press, 1992
[44] Ikeda N, Watanabe S. Stochastic differential equations and diffusion processes. North-Holland/Kodanska, 1989
[45] 黄志远.随机分析学基础(第二版).北京:科学出版社,2001
[46] Zvonkin A, Krylov N. On strong solutions of stochastic differential equations (in Russian). Proc.School-Seminal on Theor.Random Processes, Vilnius, Part Ⅱ, 1975, 9～88
[47] Adams R. Sobolev Spaces. New-York:Academic Press, 1978
[48] Airault H, Ren J. Modulus of continuity of the canonic Brownian motion "on the group of diffeomorphisms of the circle". J. Funet.Anal., 2002, 196(2): 395～426
[49] Bass R. Probabilistic techniques in analysis. Springer-Verlag, 1995
[50] Bihari I, A generalization of a lemma of Belmman and its application to uniqueness problem of differential equations. Acta Math.Acad.Sci.Hungar, 1956, 71～94
[51] Brzezniak Z, Carroll A. Approximations of the Wong-Zakai type for stochastic differential equations in M-type 2 Banach spaces with applications to loop spaces. Lecture notes in mathematics, Springer-Verlag Heidelberg, 2003, 1832:251～289
[52] Cairoli R. Sur une équation différentielle stochastique. C.R.Acad.Sci.Paris Série A, 1972, 274:1173～1742
[53] Cairoli R, Walsh J. Stochastic integrals in the plane. Acta Mathematica, 1975, 134: 111～183
[54] Cao G. Existence and pathwise uniqueness of solutions to stochastic differential equations in Banach space. To appear in Mathematica Applicata(China)
[55] Cao G, He K. On a type of stochastic differential equations driven by countably many Brownian motions. J. Funct.Anal., 2003, 203(1):262～285
[56] 曹桂兰,何凯.由可数个Brown单驱动的随机微分方程解的存在唯一性.数学物理学报(中文版)(已录用)
[57] Cao G, He K. Strongly Markovian property of solutions on a type of SDEs driven by countably many Brownian motions, Preprint
[58] Cao G, He K. Quasi sure p-variation of fractional Brownian motion, Preprint
[59] Cao C, He K. Quasi sure p-variation of fractional Brownian sheet, Preprint
[60] Cao G, He K, Zhang X. Successive approximations of infinite dimensional SDEs with jump, Preprint
[61] Carmona L, Coutin G. Stochastic integration with respect to fractional Brownian motion, 1998, Preprint
[62] Carmona L, Coutin G, Montseny. Stochastic integration with respect to fractional Brownian motion. Annales de l'Institut H. Poincare, 2003, 39(1): 27～68
[63] Chen X. The path uniqueness of the solution of the two-parameter Poisson type stochastic differential equation. Chinese Sci.Bull., 1988, 33(9): 649～652
[64] Chen X. Comparison theorem of the strong solution of the two-parameter Poisson type stochastic differential equation. Chinese Sci.Bull., 1989, 34(12): 881～884
[65] Chevalier L. Martingales continues a deux parametres. Bull.Sci.Math., 1982, 106: 19～62
[66] Feyel D, de la Pradelle A. Fractional integrals and Brownian processes. Potential Anal., 1996, 10: 273～288
[67] 龚光鲁.随机微分方程引论(第二版).北京:北京大学出版社,2000
[68] He K, Cao G. Successive approximations of solutions to stochastic differential equations with jump in the plane. To appear in Mathematica Applicata(China)
[69] 何凯,曹桂兰.连续局部鞅的无穷维随机积分表示.华中科技大学学报(自然科学版)(已录用)
[70] 何凯,曹桂兰.由可数个Brown运动驱动的随机微分方程的唯一强解
[71] 何凯,曹桂兰.无穷维半线性倒向随机微分方程的逐次逼近
[72] He K, Ren J. Quasi sure quadratic variation of local times of smooth semimartingales. Bull.Sci.Math., 2004, 128(2): 91-104
[73] He K, Zhang X. One dimensional SDEs with distributional drifts. Preprint
[74] 何声武,汪嘉冈,严加安. 半鞅与随机分析. 北京:科学出版社,1995
[75] Hitsuda M, Watanabe H. On stochastic integrals with respect to an infinite number of Brownian motions and its applications. Proc.of Intern.Symp.SDE., 1976, 57—74
[76] Hu Y, Peng S. Adapeted solution of a backward semilinear stochastic evolution equation. Stochastic Anal. Appl, 1991, 9: 445-459
[77] Karatzas I, Shreve S. Brownian motion and stochastic calculus. New York: Springer- Verlag, 1988
[78] Kaspi H, Rosen J. p-variation for families of local times on lines. Sem.Prob.XXXIV. Lecture Notes in Mathematics, Springer-Verlag, Berlin Heidelberg New York, 2000, 1729, 171-184
[79] Knoche C, Frieler K. Solutions of stochastic differential equations in infinite dimensional Hilbert spaces and their dependence on initial data. Bielefeld university, 2001
[80] Le Gall. Applications du temps local aux equations differentielles stochastiques uni- dimensionnelles. Sem.Prob.XVII. Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York, 1983, 986, 15-31
[81] Liang Z. Existence and pathwise uniqueness of solutions for stochastic differential equations with respect to martingales in the plane. Stochastic Process.Appl., 1999, 83: 303-317
[82] Liang Z, Zheng M. Estimates on moments of the solution to S.D.E. with respect to martingales in the plane. Stochastic Process.Appl., 1996, 62: 263—276
[83] Liu J. On the existence and uniqueness of solutions to stochastic differential equationsof mixed Brownian and Poissonian sheet type. Stochastic Process.Appl., 2001, 94: 339-354
[84] Ma J, Yong J. Forward-backward stochastic differential equations and their applications. New York: Springer-Verlag, 1999
[85] Malliavin P. The canonic diffusion above the diffeomorphism group of the circle. C.R.Acad.Sci.Paris, Serie I, 1999, 329: 325-329
[86]Mandelbrot B, Van Ness J. Fractional Brownian motions, fractional noises and applications. SIAM Rev, 1968, 10: 422-437
[87] Mao X. Adapted solutions of backward stochastic differential equations with non-Lipschitz coefficients. Stochastic Process.Appl, 1995, 58: 281-292
[88]Marcus M, Rosen J. p-variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments. Ann.Probab., 1992, 20: 1685—1713
[89]Mazziotto G. Equation du filtrage pour un processes de Possion melange a deux indices. Stochastic, 1980, 4: 89-119
[90]Neidhardt A. Stochastic integrals in 2-uniformly smooth Banach spaces. University of Wisconsin, 1978
[91] Nie Z. Uniqueness on two-parameter Ito's stochastic differential equations. Acta Math.Sinica., 1987, 30: 179-186
[92] Pardoux E, Peng S. Adapeted solution of a backward stochastic differential equation. Systems Control Lett., 1990, 14: 55-61
[93] Perkins E. Local time and pathwise uniqueness for stochastic differential equations. Sem.Prob.XVI. Lecture Notes in Mathematics, Springer, Berlin Heidelberg New York, 1982, 920, 201-208
[94] Protter P. Stochastic intergration and differential equation: a new approach. New York: Springer, 1990
[95] Reid J. Estimate on moments of the solution to stochastic differential equations in the plane. Ann.Probab., 1983, 11: 656-668
[96] Ren J, Zhang X. Quasi sure analysis of two-parameter stochastic differential equations. Stochastic and Stochastic Rep., 2002, 72(3-4): 251-276
[97] Rogers L, Williams D. Diffusions, Markov processes and martingales. 2. 2nd Edition. Cambridge: Cambridge University press, 2000
[98]Rosen J. p-variation of the local times of stable processes and intersection local time. Seminar on Stochastic processes, 1991(Boston), Progress in Probability, Birkhauser, Boston, 1993,33, 157-168
[99]Situ R. On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process.Appl., 1997, 66: 209—236
[100] Situ R. Backward stochastic differential equations with jumps and applications. Guangzhou: Guangdong Science and Technology Press, 2000
[101] Stroock D, Varadhan S, Multidimensional Diffusion Processes. Springer-Verlag, Berlin-Heidelberg-New York, 1979
[102] Taniguchi T. Successive approximations to solutions of stochastic differential equations. J.Differ. Equations, 1992, 96: 152～169
[103] Wentzell A. A course in the theory of stochastic processes. McGraw-Hill Inc, 1981
[104] Wong E, Zakai M. Martingale and stochastic integrals for processes with multidimensional parameter. Z.Wahrscheinlichkeitstheorie Verw.Gebiete., 1974, 29: 109～122
[105] Wong E, Zakai M. Weak martingales and stochastic integrals in the plane. Ann.Probab., 1976, 4: 570～586
[106] Wong E, Zakai M. Differentiation formula for stochastic integral in the plane. Stochastic Process.Appl., 1978, 6: 339～349
[107] Yamada T. On the successive approximation of solutions of stochastic differential equations. J.Math.Kyoto Univ., 1981, 21: 501～515
[108] Yamada T, Ogura Y. On the strong comparison theorems for solutions of stochastic differential equations. Z.Wahrscheinlichkeitstheorie Verw.Gebiete, 1981, 56: 3～19
[109] 严加安,彭实戈,方诗赞,吴黎明.随机分析选讲.北京:科学出版社,2000
[110] Yeh J. Existence of strong solutions for stochstic differential equations in the plane. Pac.J.Math., 1981, 97: 217～247
[111] Yeh J. Existence of weak solutions to stochstic differential equations in the plane with continuous coefficients. Trans.Amer.Math.Soc., 1985, 290: 345～361
[112] Yeh J. Uniqueness of strong solutions to S.D.E. in the plane with deterministic boundary process. Pac.J. Math., 1987, 128: 391～400
[113] 周少甫.(非Lipschitz系数)倒向随机微分方程和随机微分效用:[博士学位论文].华中科技大学图书馆,2000
[114] Zhou S, Cao X. Adapeted solution of a backward semilinear stochastic evolution equation with non-Lipschitz coefficients. Acta Math.Sci., 2003, 23B(4): 485～493