scholarly journals Tangential existence and comparison, with applications to single and multiple integration

2018 ◽  
Vol 37 (1) ◽  
pp. 21-52
Author(s):  
Olav Kallenberg
Keyword(s):  
Continuous Time ◽  
Comparison Theorems ◽  
Stochastic Integrals ◽  
Multiple Integration ◽  
Multiple Stochastic Integrals ◽  
Local Characteristics ◽  
Independent Increments ◽  
General Existence

TANGENTIAL EXISTENCE AND COMPARISON, WITH APPLICATIONS TO SINGLE AND MULTIPLE INTEGRATIONTwo semi-martingales with respect to a common filtration are said to be tangential if they have the same local characteristics. When the latter are non-random, the underlying semi-martingale is known to have independent increments. We show that every semi-martingale has a tangential process with conditionally independent increments. We also extend the Zinn–Hitchenko and related tangential comparison theorems to continuous time. Combining those results, we obtain some surprisingly general existence,convergence, and tightness criteria for broad classes of single and 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

Characterization of stable processes by identically distributed stochastic integrals

1980 ◽  
Vol 12 (3) ◽  
pp. 689-709 ◽  
Author(s):  
M. Riedel
Keyword(s):  
Stochastic Process ◽  
Stable Process ◽  
Convergence In Probability ◽  
Stochastic Integrals ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Subject

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

scholarly journals Homogenization for deterministic maps and multiplicative noise

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130201 ◽  
Author(s):  
Georg A. Gottwald ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Multiplicative Noise ◽  
Stochastic Integrals ◽  
Deterministic System ◽  
One Dimensional ◽  
Stable Lévy Process ◽  
Fast Flow

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.

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.

Asymptotic likelihood theory for diffusion processes

1975 ◽  
Vol 12 (2) ◽  
pp. 228-238 ◽  
Author(s):  
B. M. Brown ◽  
J. I. Hewitt
Keyword(s):  
Central Limit Theorem ◽  
Maximum Likelihood ◽  
Limit Theorem ◽  
Continuous Time ◽  
Diffusion Processes ◽  
Maximum Likelihood Estimates ◽  
Stochastic Integrals ◽  
Ratio Test ◽  
Chi Squared ◽  
Asymptotic Likelihood

We investigate the large-sample behaviour of maximum likelihood estimates (MLE's) of the parameters of a diffusion process, which is observed throughout continuous time. The results (limit normal distribution for the MLE and an asymptotic chi-squared likelihood ratio test) correspond exactly to classical asymptotic likelihood results, and follow easily from a central limit theorem for stochastic integrals.

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