Independence of a Class of Multiple Stochastic Integrals

1999 ◽  
pp. 249-259 ◽  
Author(s):  
Nicolas Privault
Keyword(s):  
Stochastic Integrals ◽  
Multiple Stochastic Integrals

The approximation of multiple stochastic integrals

1992 ◽  
Vol 10 (4) ◽  
pp. 431-441 ◽  
Author(s):  
P.E. Kloeden ◽  
E. Platen ◽  
I.W. Wright
Keyword(s):  
Stochastic Integrals ◽  
Multiple Stochastic Integrals

A Nearly Exact Reformulation of the Girsanov Linearization for Stochastically Driven Nonlinear Oscillators

2013 ◽  
Vol 80 (2) ◽  
Author(s):  
Tara Raveendran ◽  
D. Roy ◽  
R. M. Vasu
Keyword(s):  
Sampling Strategy ◽  
Nonlinear Oscillators ◽  
Linearization Method ◽  
Stochastic Integrals ◽  
Rejection Sampling ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Multiple Stochastic Integrals ◽  
Nikodym Derivative

The Girsanov linearization method (GLM), proposed earlier in Saha, N., and Roy, D., 2007, “The Girsanov Linearisation Method for Stochastically Driven Nonlinear Oscillators,” J. Appl. Mech.,74, pp. 885–897, is reformulated to arrive at a nearly exact, semianalytical, weak and explicit scheme for nonlinear mechanical oscillators under additive stochastic excitations. At the heart of the reformulated linearization is a temporally localized rejection sampling strategy that, combined with a resampling scheme, enables selecting from and appropriately modifying an ensemble of locally linearized trajectories while weakly applying the Girsanov correction (the Radon–Nikodym derivative) for the linearization errors. The semianalyticity is due to an explicit linearization of the nonlinear drift terms and it plays a crucial role in keeping the Radon–Nikodym derivative “nearly bounded” above by the inverse of the linearization time step (which means that only a subset of linearized trajectories with low, yet finite, probability exceeds this bound). Drift linearization is conveniently accomplished via the first few (lower order) terms in the associated stochastic (Ito) Taylor expansion to exclude (multiple) stochastic integrals from the numerical treatment. Similarly, the Radon–Nikodym derivative, which is a strictly positive, exponential (super-) martingale, is converted to a canonical form and evaluated over each time step without directly computing the stochastic integrals appearing in its argument. Through their numeric implementations for a few low-dimensional nonlinear oscillators, the proposed variants of the scheme, presently referred to as the Girsanov corrected linearization method (GCLM), are shown to exhibit remarkably higher numerical accuracy over a much larger range of the time step size than is possible with the local drift-linearization schemes on their own.

On some inequalities of multiple stochastic integrals for normal martingales

1998 ◽  
Vol 64 (3-4) ◽  
pp. 161-176
Author(s):  
Tsung-Ming Chad ◽  
Ching-sung Chou
Keyword(s):  
Stochastic Integrals ◽  
Multiple Stochastic Integrals

scholarly journals An estimate about multiple stochastic integrals with respect to a normalized empirical measure

2005 ◽  
Vol 42 (3) ◽  
pp. 295-341 ◽  
Author(s):  
P. Major
Keyword(s):  
Product Space ◽  
Random Variables ◽  
Measurable Space ◽  
Analogous Problem ◽  
Empirical Measure ◽  
Stochastic Integrals ◽  
Good Bound ◽  
Multiple Stochastic Integrals ◽  
Ito Integral ◽  
Random Integral

Let a sequence of iid. random variables ξ 1 , …, ξ n be given on a measurable space ( X,X ) with distribution µ together with a function f ( x1 , …, xk ) on the product space ( Xk , Xk ). Let µ n denote the empirical measure defined by these random variables and consider the random integral \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$J_{n,k} (f) = \frac{{n^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{k!}}\smallint 'f(u_1 ,...,u_k )(\mu _n (du_1 ) - \mu (du_1 ))...(\mu _n (du_k ) - \mu (du_k )),$$ \end{document} where prime means that the diagonals are omitted from the domain of integration. A good bound is given on the probability P (| Jn,k ( f )| > x ) for all x > 0 which is similar to the estimate in the analogous problem we obtain by considering the Gaussian (multiple) Wiener-Itô integral of the function f . The proof is based on an adaptation of some methods of the theory of Wiener-Itô integrals. In particular, a sort of diagram formula is proved for the random integrals Jn,k ( f ) together with some of its important properties, a result which may be interesting in itself. The relation of this paper to some results about U -statistics is also discussed.

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