Quasi Sure Analysis Of Local Time And Fractional Brownian Motion

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=0holds 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.