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 is that

By the definition of a

stochastic integral [18], 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 equation:

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 [1] proved that the random Fourier-Stieltjes series(RFS) (1 ) converges in probability to the

stochastic integralSince the

stochastic integral on the right-hand side of Eq.(15) is locally integrable Martingale under a real world probability measure P, then