Bilinear Stochastic Differential Equation Research Articles

Stratonovich framework into Markovian satellite dynamics of Carleman-based filtering using energy function observables approach

In this paper the satellite system in which fluctuating aerodynamic torque and the radial perturbation about the pitch motion of orbit is analyzed. The objectives of the paper are three-fold. The first objective is the ‘Carleman linearization representation into the nonlinear stochastic evolution of the Markov process’. The second is to filter process and measurement noises in the satellite dynamics parameters in the Carleman setting. The third is to find the stability and convergence condition of satellite dynamics. The first is achieved by unifying the generating function, Carleman embedding, Itô stochastic differential rules and the finite closure with Kronecker algebra. Concerning the second objective, we recast the finite-dimensional Stochastic Differential Equations (SDE) of the satellite dynamics into a finite system of bilinear SDE via the Carleman embedding. The third objective is to achieve the Lyapunov function and asymptotic stability condition for the satellite stochastic dynamics involving the ‘Stratonovich differential’. In this paper, we demonstrate the utility of the satellite dynamics filtering in the Carleman-based filtering via ‘its convergence analysis as well as its superiority with available methods’, i.e., the benchmark extended Kalman filter, Gaussian second-order filter and Kushner–Stratonovich higher-order filter. From simulation performed it can be said that the Carleman-based filtering is superior than other benchmark filters in terms of their Absolute Filtering Error (AFE) of conditional means and conditional variances of the satellite dynamics states.

On the Theory of a Nonlinear Dynamic Circuit Filtering

Stochastic Differential Equations (SDEs) describe physical systems to account for random forcing terms in the evolution of the state trajectory. The noisy sampling mixer, a component of digital wireless communications, can be regarded as a potential case from the dynamical systems’ viewpoint. The universality of the noisy sampling mixer is attributed to the fact that it adopts the structure of a nonlinear SDE and its linearized version becomes a time-varying bilinear SDE. This paper develops a mathematical theory for the nonlinear noisy sampling mixer from the filtering viewpoint. Since the filtering of stochastic systems hinges on the structure of dynamical systems and observation equation set up, we consider three ‘filtering models’. The first model, accounts for a nonlinear SDE coupled with a nonlinear observation equation. In the second model, we consider a bilinear SDE with a linear observation equation to achieve the nonlinear sampling filtering. Note that the bilinear SDE coupled with the linear observation is a consequence of the Carleman linearization to the nonlinear SDE and the nonlinear observation equation. In the third model, we consider a Stratonovich SDE coupled with a nonlinear observation equation. The filtering equation of this paper can be further utilized to guide the design process of the noisy sampling mixer.

Non-linear filtering for bilinear stochastic differential systems: A Stratonovich perspective

Bilinear stochastic differential equations have found applications to model turbulence in autonomous systems as well as switching uncertainty in non-linear dynamic circuits. In signal processing and control literature, bilinear stochastic differential equations are ubiquitous, since they capture non-linear qualitative characteristics of dynamic systems as well as offer closed-form solutions. The novelties of the paper are two: we weave bilinear filtering for the Stratonovich stochasticity. Then this paper unfolds the usefulness of bilinear filtering for switched dynamic systems. First, the Stratonovich stochasticity is embedded into a vector ‘bilinear’ time-varying stochastic differential equations. Then, coupled non-linear filtering equations are achieved. Finally, the non-linear filtering results are applied to an appealing bilinear stochastic Ćuk converter circuit. This paper also encompasses a system of coupled bilinear filtering equations for the vector input Brownian motion case. This paper brings the notions of systems theory, that is, bilinearity, Stratonovich stochasticity, non-linear filtering techniques and switched electrical networks together. Numerical simulation results are presented to demonstrate that the proposed bilinear filter can achieve much better and accurate filtering performance than the conventional Extended Kalman Filter (EKF).

A half-wave rectifier circuit as a bilinear stochastic differential system

Abstract Bilinear stochastic differential systems have found striking applications in diverse fields, e.g. autonomous systems under noise influence, population models, theoretical biology and switched electrical networks. Because of their surprising applications, this paper develops the closed-form solution for a Stratonovich time-varying bilinear Stochastic Differential Equation (SDE). Subsequently, the analytic result of this paper is applied to a simple, but an appealing half-wave rectifier electronic circuit. Furthermore, we derive the mean and variance expressions of the state satisfying the rectifier SDE. The stochastic differential rule in combination with the concept of characteristic function of the Gaussian stochastic process plays the key role to develop the theory of this paper. The results of this paper will be useful to accomplish the noise analysis of bilinear systems arising from systems and control literature.

On a Non-linear Electronic Circuit Filtering

The stochastic versions of non-linear dynamic circuits are formalized using non-linear stochastic differential equations. Stochastic differential equations (SDEs) are exploited to analyse dynamical systems in noisy environments. A potential application of the SDEs can be regarded as `stochastic processes in electronic circuits'. The noisy sampling mixer, a component of digital wireless communications, is an appealing and standard case from the dynamical systems' viewpoint. It assumes the structure of a non-linear SDE, and its linearized version becomes time-varying bilinear SDE. This paper derives the filtering equations for the noisy non-linear sampling mixer circuit utilizing the filtering density evolution equation. The filtering model for the stochastic problem of concern here comprises the following: (1) a non-linear SDE describing the noisy sampling mixer and (2) a non-linear noisy observation equation. It is interesting to note that the filtered estimate accounts for observations. On the other hand, the predicted estimate does not account for the observation terms in evolution equations. As a result of this, the filtered estimate confirms the greater accuracy of estimated state trajectory in contrast to the predicted trajectory. The filtering equation of this paper can be further utilized for control of the noisy sampling mixer, where the observations are available.

Some new results on time-varying Stratonovich and Itô bilinear stochastic systems

The bilinear stochasticity of dynamical systems is attributed to the input–output coupling term, where the input is a random input and the state is the output of dynamical systems. Stochastically influenced bilinear systems are described via bilinear stochastic differential equations. In this paper, first we construct a mathematical method for the closed-form solution to a scalar Stratonovich time-varying bilinear stochastic differential equation driven by a vector random input as well as the Itô counterpart. Second, the analytic results of the paper are applied to an electrical circuit that assumes the structure of a bilinear stochastic dynamic circuit. The noise analysis of the bilinear dynamic circuit is achieved by deriving the mean and variance equations as well. The theory of this paper hinges on the ‘Stratonovich calculus’, conversion of the Stratonovich integral into the Itô integral and characteristic function of the vector Brownian motion. The results of this paper will be useful for research communities looking for estimation and control of bilinear stochastic differential systems.

Interest rate models on Lie groups

This paper examines an alternative approach to interest rate modeling, in which the nonlinear and random behavior of interest rates is captured by a stochastic differential equation evolving on a curved state space. We consider as candidate state spaces the matrix Lie groups; these offer not only a rich geometric structure, but—unlike general Riemannian manifolds—also allow for diffusion processes to be constructed easily without invoking the machinery of stochastic calculus on manifolds. After formulating bilinear stochastic differential equations on general matrix Lie groups, we then consider interest rate models in which the short rate is defined as linear or quadratic functions of the state. Stochastic volatility is also augmented to these models in a way that respects the Riemannian manifold structure of symmetric positive-definite matrices. Methods for numerical integration, parameter identification, pricing, and other practical issues are addressed through examples.

MULTI-DIMENSIONAL GEOMETRIC BROWNIAN MOTIONS, ONSAGER-MACHLUP FUNCTIONS, AND APPLICATIONS TO MATHEMATICAL FINANCE

The solutions of the following bilinear stochastic differential equation are studied dxt=∑k=1mAtkxtdwk(t)+Btxtdtwhere Atk, are (deterministic) continuous matrix-valued functions of t and wm(t) are m independent standard Brownian motions. Conditions are given such that the solution is positive if the initial condition is positive. The equation the most probable path must satisfy is also derived and applied to a mathematical finance problem.

Rate of convergence for an estimator in a portfolio and consumption model

The rate of convergence in a sample path sense is given for a strongly consistent, recursive estimator. This estimator is for the unknown average return rate of the risky asset that is a parameter in a bilinear stochastic differential equation for the wealth in a portfolio selection and consumption model.

Lyapunov number of bilinear stochastic differential equations

Lyapunov number of bilinear stochastic differential equations