# stochastic integral

## stochastic integral

[stō′kas·tik ′int·ə·grəl]
(mathematics)
An integral used to construct the sample functions of a general diffusion process from those of a Wiener process; it has the form where {Wt , t ≥ 0} is a Wiener process, dWt represents the random disturbances occurring in an infinitesimal time interval dt, and at is independent of future disturbances. Also known as Itô's integral.
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Nayak, Pattanayak and Mishra [1] 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.
2]), in paper [1] we could avoid the use of the stochastic integral to obtain some results from stochastic calculus, e.
X [OMEGA] [right arrow] X, x [member of] X, satisfying the stochastic integral equation:
He will present, "FastSies: A Fast Stochastic Integral Equation Solver for Modeling the Rough Surface Effects," during the Silicon Valley Chapter of the IEEE Solid State Circuits Society Monday, May 15, at 7 p.
The forward looking stochastic integral in discrete time of a process H with respect to a process S is defined by
The proposition below tells us that the sum of an accumulated discounted consumption process and its corresponding discount wealth process can be expressed as a stochastic integral with respect to the Brownian motion.
1]), we do not make use of the stochastic integral.
This calculus exhibits a number of novel features, and McKean finds many in his coverage of Brownian motion, including the issue of Martingale inequality and the law of the iterated logarithm, stochastic integrals and differentials, including Wiener's and Ito's definitions of the stochastic integral, time substitution, and stochastic integrals and differentials for several-dimensional Brownian motion, stochastic integral equations in which d=1, including Lampertoi's methods and Feller's test for explosions, and when d is greater than or equal to 2, including Weyl's Lemma and Brownian motions in a Lie group.
This endeavor entails developing new notions of nonlinear stochastic integrals, and requires a theory that looks beyond the established setting of semimartingales.
Compared to the methods based on Taylor expansion, the proposed symplectic weak second-order methods are implicit, but they are comparable in terms of the number and the complexity of the multiple Ito stochastic integrals or the derivatives of the Hamiltonian functions required.
After a description of the Poisson process and related processes with independent increments, as well as a brief look at Markov processes with a finite number of jumps, the book introduces Brownian motion and develops stochastic integrals and Ito's theory in the context of one-dimensional diffusion processes.

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