Stochastic Differential Equations In Infinite Dimensional Space

In this paper, we mainly study stochastic differential equations in infinite dimensional space. Our paper consists mainly of the following contents:1 SDEs in Hilbert spaceLet K and H be two real separate Hilbert spaces, and (U, ε, n) be a σ-finite measurable space. (Ω, F, P; F_t) is a complete filtered probability space, W is a K-valued cylindrical Q-Brownian motion, and p is a [/-valued stationary Poisson point process with quasi left continuous path, N_p, N_p and n is the counting measure, the compensating measure and the characteristic measure of p respectively. U₀ ∈ε with n(U — U₀) < +∞.Consider the following non-Markovian stochastic differential equations with jumps in H:In paper [54], we prove the following result: Theorem 1 Suppose that(1) there exists a function H(t,u) : R₊ × R₊ → R₊ such that:(1a) H(t,u) is locally integrable in t for each fixed u ∈ R₊ and is continuous nonde-creasing in u for each fixed t ∈ R₊,(1b) Vt > 0, X ∈ L_loc^p(D(H)) such that(1c) VK > 0 and any initial value u₀≥4^p-1E(||X₀||^p), the differential equationhas a global solution.(2) there exists a function G(t,u) : K+ x E+ i—> R+ such that:(2a) G(t,u) is locally integrable in ï¿¡ for each fixed u G M.+ and is continuous nonde-creasing in u for each fixed t G R+, and G(t, 0) = 0,(2b) W > 0, X,y G ï¿¡foc(D(#)) such that-E / \\f(t,X,u)-f(t,Y,u)rn(du) \Ju0||/(t,X,?)-/(t,yjU)fn(d?)G(t,E(sup||Xr-yr|(2c) \/K > 0, if a nonnegative function Z satisfies that Vï¿¡ G R_|,Zt^K f G(s,Zs)ds, Jothen Z = 0.then Eq.(l) has a pathwise unique solution.We should bring the reader's attention to the following fact, the conditions in Theorem 1 is so weak that most of the well known results about non-lipschitz SDE can be as special cases of it. In addition, this kind of equation is an extension of classical stochastic differential equations, and we will give a simple example which includes the Brownian motion on the group of diffeomorphism of the circle (see [2, 1]).Noticing that the existence of pure jump's integration term in Eq.(l), we could meet the moment estimate problem of this term when we consider the existence of the solution by successive approximations. So we prove the following Burkholder-Davis-Gundy's inequality:Theorem 2 For p>2, / G T?OC(H), if Mt = Jo* /^ f(s,u)Np(ds, du), then there exists a constant Kp>t > 0 such that Vi e R+,2 BSDEs in Hilbert spaceConsider the following semilinear backward stochastic differential equations in H (Q = !)â– â– rp rpXT=Xt+ f (a(s, X8) + Ya) dWs + f b(s, Xs, Ys) ds. (2)Jt JiIn paper [67], We prove the following result: Theorem 3 Suppose that(1) there exists a function H(t,u) : [0,T] x E+ h-> R+ such that:rp(la) H(t,u) satisfies JQ H(s,u)ds < +oo for each fixed u G ]R+ and is continuous nondecreasing in u for each fixed t € [0, T],(lb) there exists a constant C > 0, W e [0,T], X e C2(n,H), Y e ï¿¡2(fi,ï¿¡2), such thatE\\a(t,X)\\l2 < E\\b(t,X,Y)\\2 <(lc) \/K > 0 and any final value ut^O, the differential equationhas a solution in [0,T].(2) there exists a function G(t, u) : [0, T] x R+ i—> R+ such that:rp(2a) G(t,u) satisfies JQ G(s,u)ds < +oo for each fixed u G R+ and is continuous nondecreasing in u for each fixed t G [0, T], and G(t, 0) = 0,(2b) Vt G [0,T], X,X; G ï¿¡2(n,H), Y,Y' G ï¿¡2(fi,ï¿¡2) such that E||6(ï¿¡,X,y)-6(i,X',y')H2 < G(t,(2c) VAT > 0, if a nonnegative function Z on [0,T] satisfies that Vt G [0,T],iTlthen Z = 0.then Eq.(2) has a unique solution.3 SDEs in Banach spaceLet E be a M-type 2 real separable Banach space, F be a real separable Banach space, and W be a F-valued Wiener process.Consider the following non-Markovian stochastic differential equations in E:Xt = X0+ [ 0.We have the following theorem:rpTheorem 8 Let A be a non-negative function such that Jo A(s) ds < +oo, p\ and pi be concave nondecreasing functions on R+ such that pi(0) = p2(0) = 0 such thatu-du = +00.lfa(t,co,0) E C2([O,T],Rm0l2),b(t,uj,O,O) E C2([0,T},Rm), and Vt E [O,T],X,X' e Y,Y' elm?l2, such that\\a(t,X)-a(t,X')\\ < y/Mt)Pi(\X X'\) a-s-\b(t,X,Y)-b(t,X',r)\ < ^X({jp2(\X-X>\) + C\\Y-Y'\\ a.s.,then Eq.(6) has a unique solution.5 Two parameter SDEs with jumpsThis part of work will be published in Mathematica Applicata(China) (see [64]), where we only consider two parameter time homogeneous Markovian stochastic differential equations driven by a Brown sheet, here we extend to study non-Markovian type equations driven by an infinite sequence of independent Brownian sheets.Let W = (W1, W2, ? ? ?), {W\j = 1,2,---} is an infinite sequence of independent Brownian sheets on (fi, jF, IP; Tt), which is a complete filtered probability space. T is a Poisson sheet, A is the intensity measure of Us, and N = T — A.Consider the following two parameter non-Markovian stochastic differential equations with jumps:Xz = Xo+(7)By the method similar to Theorem 1 we can prove the following result: Theorem 9 Suppose that(1) there exists a function H(z,u) : R^ x M.+ hh> R+ such that:(la) H(z,u) is locally integrable in z with respect to dBz for each fixed u e M+ and is continuous nondecreasing in u for each fixed z € R+, where Bz = z + Az,(lb) VzeR2+,X E APE(\\a(z,X)\n +M(\b(z,X)n +E(\a(z,X)n +E(\f3(z,X)n(lc) Vi^ > 0 and any initial value (i.e. on axes) ?o^5p¹E (||Xo||p), the differential equationdu = KH(z,u)dBzhas a global solution.(2) there exists a function G(z, u) : W?+ x R+ h^ R+ such that:(2a) G(z, u) is locally integrable in z with respect to dBz for each fixed u G M+ and is continuous nondecreasing in u for each fixed z G R^_, and G(z, 0) = 0,(2b) Vz G R2+, X,Y G ï¿¡foc(?) such that+E(\a(z,X) - a(z,Y)\p) +E(\(3(z,X) - (3(z,Y)\p)(2c) Vi^ > 0, if a nonnegative function Z satisfies that Vz ethen Z = 0.then Eq.(7) has a pathwise unique solution.6 SDEs driven by countably many Brownian sheetsConsider the following two parameter non-Markovian stochastic differential equations: X)<%, (8)where W = (W1, W2, ? ? ?), {W^,j = 1,2,---} is an infinite sequence of independent Brownian sheets on (Q,!F,W; jTt), which is a complete filtered probability space.We give a condition on existence of the unique strong solution to the equation with non-Lipschitz coefficients. This part of work will be published in Acta Math.Sci.(A) (see [50]).Theorem 10 Let A be a two parameter non-negative, locally integrable function, pi and p2 be concave nondecreasing functions on R+ such that pi(0) = p2(0) = 0 and 3p^2, such that-du = +oo.If ||a(z,0)|| and \b(z,0)\ € ï¿¡foc, and V^ e R2+, X,Y G 2U, such that\b(z,X)-b(z,Y)\ <...