Integral Representation of Martingales

We prove results on exact asymptotics as for the means and probabilities , where , , is an -dimensional Wiener process, is a positive continuous function (potential) satisfying certain conditions, , and are prescribed vectors. The results are obtained by a new method developed in this paper, the Laplace method for the occupation time of a multi-dimensional Wiener process. We consider examples of monomial and radial potentials and prove results on exact asymptotics of small deviations for the probabilities as with a fixed .

In this technical note, we restage the positive effects of Gaussian white noises for continuous-time driftless input-affine nonholonomic systems. Compared with previous studies, we found that the positive effects are hidden in the shadows of nonuniqueness of stochastic integrals over multi-dimensional Wiener processes. Furthermore, we found that the positive effects are artificially restaged by numerical calculations with use of Euler scheme for ordinary differential equations. Our results demonstrate that the positive effects act as “hidden independent control inputs” for nonholonomic systems. This implies that our results provide a constructive procedure for designing control inputs such that the origins become $p$ -th moment exponentially or finite-time stable.

We investigate the rate of convergence of the Wong-Zakai approximations for second-order stochastic PDEs of parabolic type driven by a multi-dimensional Wiener process W.

T–S fuzzy affine model based non-synchronized state estimation for nonlinear Itô stochastic systems

On the number of crossings of a partli reflecting hyperplane by a multidimensional wiener process

We consider interest rate models where the forward rates are allowed to be driven by a multidimensional Wiener process as well as by a marked point process. Assuming a deterministic volatility structure, and using ideas from systems and control theory, we investigate when the input-output map generated by such a model can be realized by a finite dimensional stochastic differential equation. We give necessary and sufficient conditions, in terms of the given volatility structure, for the existence of a finite dimensional realization and we provide a formula for the determination of the dimension of a minimal realization. The abstract state space for a minimal realization is shown to have an immediate economic interpretation in terms of a minimal set of benchmark forward rates, and we give explicit formulas for bond prices in terms of the benchmark rates as well as for the computation of derivative prices.

Stabilization for T–S model based uncertain stochastic systems

Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge–Kutta family

We consider cadlag local martingales M with initial value zero and jumps larger than a for some a larger than or equal to -1, and prove Novikov-type criteria for an exponential local martingale to be a uniformly integrable martingale. We obtain criteria using both the quadratic variation and the predictable quadratic variation. We prove optimality of the coefficients in the criteria. As a corollary, we obtain a verbatim extension of the classical Novikov criterion for continuous local martingales to the case of local martingales with initial value zero and nonnegative jumps.

A Visual Classification of Local Martingales

In this chapter, we consider studies involving cadlag processes. We pay particular attention to semi-martingales with respect to a given filtration; these processes will always be taken with cadlag paths. We present the definition of stochastic integrals with respect to a square integrable martingale, and we extend the definition to stochastic integrals with respect to a local martingale. Then, we introduce semi-martingales, quadratic covariation processes for semi-martingales and some general versions of Ito’s formula and Girsanov’s theorem. We give necessary and sufficient conditions for the existence of an equivalent martingale measure. We end the chapter with a brief survey of valuation in an incomplete market.

Square Integrable Martingales and Structure of the Functionals on a Wiener Process

After introducing cylindrical Gaussian random variables and Hilbert space valued Gaussian random variables, we define cylindrical Wiener process and Hilbert space valued Wiener process in a natural way. We develop the Itô stochastic integral with respect to cylindrical and Hilbert space valued Wiener processes simultaneously as they share many common features. Our construction of the Itô integral is consistent with the construction of a stochastic integral with respect to square integrable martingales, however, regarding measurability, we assume only that the integrand is adapted, following the classical ideas as presented by Liptser and Shiryayev (Statistics of Stochastic Processes. Nauka, ) and by Øksendal (Stochastic Differential Equations. Springer, New York, ). We provide detailed proofs of properties of the Itô integral and follow with the Martingale Representation Theorem. Our unified approach to stochastic integration with respect to cylindrical and Hilbert space valued Wiener processes allowed us to present the Stochastic Fubini Theorem and the Itô Formula in both cases in an almost identical way.

Let be a standard d-dimensional Brownian motion and L a smooth functional on the space C = Cd[0, 1] of continuous functions from [0,1] to Rd, such that Denote Thenis a square integrable martingale which can be represented as a stochastic integral; consequently the random variable has the representationFor Fréchet differentiable functions L, Clark(1) gave the following formula for the integrand e:

Consider A Closed E of the Real Line R and a (Measureble) Function Defined on R and Strictly Concave on Every Component of R/E. The Present Paper Deals with the Following Marticgale Problem: Find a Continuous Local Martingale (X,F), Given on a Family () of Probability Spaces With for every such that x is stopped at the debut of e, and the process is a local martingale up to We show that this martingale problem possesses a unique solution. This solution is a strong Markov continuous local martingale. Furthermore, it is a pure continuous local martingale and therefore satisfies the previsible representation property. Conversely, for every strong Markov continuous local martingale (X,F) we can find a closed subset E of R and a function f strictly concave on every component of R\E such that (X, F) is the unique solution of the above martingale problem. In the particular case that E and f(X) we recover the theorem of P. Levy on the martingale characterization of the Brownian motion.