On Infinite Order Systems of Stochastic Differential Equations Arising in the Theory of Optimal Non-Linear Filtering

1973 ◽  
Vol 17 (2) ◽  
pp. 218 ◽  
Author(s):  
B. L. Rozovskii ◽  
A. N. Shiryaev
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Infinite Order ◽  
Linear Filtering ◽  
Non Linear

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

2020 ◽  
Vol 42 (10) ◽  
pp. 1755-1768
Author(s):  
Sandhya Rathore ◽  
Shambhu N Sharma ◽  
Dhruvi Bhatt ◽  
Shaival Nagarsheth
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Dynamic Systems ◽  
Autonomous Systems ◽  
Linear Filtering ◽  
Electrical Networks ◽  
Closed Form Solutions ◽  
Dynamic Circuits ◽  
Non Linear ◽  
And Control

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).

scholarly journals Milstein-type semi-implicit split-step numerical methods for nonlinear stochastic differential equations with locally Lipschitz drift terms

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 1-12 ◽  
Author(s):  
Burhaneddin Izgi ◽  
Coskun Cetin
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Rates Of Convergence ◽  
Ginzburg Landau ◽  
Linear Growth Condition ◽  
Locally Lipschitz ◽  
Non Linear ◽  
Ginzburg Landau Equations

We develop Milstein-type versions of semi-implicit split-step methods for numerical solutions of non-linear stochastic differential equations with locally Lipschitz coefficients. Under a one-sided linear growth condition on the drift term, we obtain some moment estimates and discuss convergence properties of these numerical methods. We compare the performance of multiple methods, including the backward Milstein, tamed Milstein, and truncated Milstein procedures on non-linear stochastic differential equations including generalized stochastic Ginzburg-Landau equations. In particular, we discuss their empirical rates of convergence.

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