Utilizing forward infinite horizon stochastic integral
equations, we propose the finite-time random periodic Lipschitz shadowing theorem of SDEs.
One important property for the stochastic integral
By the definition of a stochastic integral
, formula (34) defines a Gaussian process with zero mathematical expectation and a covariance matrix
The two most commonly used stochastic integral
types are the Ito integral (Ito 1951) and the Stratonovich integral (Stratonovich 1966).
They cover the stochastic integral
and Itf formula,Ornstein-Uhlenbeck processes and stochastic differential equations, and random attractors.
For any h [member of] [L.sup.2]([0, T] x R), we define the stochastic integral
of h with respect to L:
Kolarova, "An application of stochastic integral
equations to electrical networks," Acta Electrotechnica et Informatica, vol.
The k-th boundary p-adic free stochastic integral
[mathematical expression not reproducible] of T for the j-th p-adic w-s motion [[??].sub.P,J], is defined to be
Then, for [F.sub.0] every measurable E valued random variable [x.sub.0] [member of] [L.sub.2]([OMEGA], E), and control u [member of] [U.sub.ad], the stochastic evolution equation has a unique mild solution x [member of] [B.sup.a.sub.[infinity]] (I, E) in the sense that it satisfies the following stochastic integral
in which X is a real martingale and Y is the stochastic integral
, with respect to X, of a certain predictable process H = [([H.sub.t]).sub.t[greater than or equal to]0] taking values in [-1,1].
Nayak, Pattanayak and Mishra  proved that the random Fourier-Stieltjes series(RFS) (1 ) converges in probability to the stochastic integral
Since the stochastic integral
on the right-hand side of Eq.(15) is locally integrable Martingale under a real world probability measure P, then