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Linear Algebra and Operator Theory
Published in Harish Parthasarathy, Advanced Probability and Statistics: Applications to Physics and Engineering, 2023
Solving stochastic differential equations driven by discontinuous semimartingales (as a part of linearization of non-linear stochastic systems). Construction of the stochastic integral w.r.t a square integrable Martingale is treated as a Hilbert space isomorphism problem while solving stochastic differential equations driven by semimartinagles using perturbation theory comes under the heading ”linearization of nonlinear systems”. Construction of the stochastic integral w.r.t semi-martingales based onthe Doob-Meyer decomposition.The Doleans-Dade-Meyer-Ito formula for discontinuous semi-martinagleswith application to superpositions of compound Poisson processes and Brownian motion.The generalized Lipshitz conditions for proving existence and uniquenessof solutions to sde’s driven by discontinuous semi-martingales.Applications of stochastic calculus to mathematical finance.Stochastic optimal control for Markov processes.Stochastic nonlinear filtering for Markov processes in the presence of Levymeasurement noise.
Chapter 11: Miscellaneous Topics Used for Engineering Problems
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes and is used to model systems that behave randomly. The main types of stochastic calculus are the Itô calculus and its variational relative known as the Malliavin calculus. For technical reasons the integral is the most useful for general classes of processes. We present here the Itoô lemma. Interested readers may pursue references [19], [30], [50] and [54] for detailed study.
Stochastic Processes
Published in Athanasios Christou Micheas, Theory of Stochastic Objects, 2018
In this section we define two important random objects, stochastic integrals and stochastic differential (or diffusion) equations (SDEs), with applications ranging from biological sciences, physics and engineering to economics. Stochastic calculus is required when ordinary differential and integral equations are extended to involve continuous stochastic processes.
Stabilisation of SDEs and applications to synchronisation of stochastic neural network driven by G-Brownian motion with state-feedback control
Published in International Journal of Systems Science, 2019
Yong Ren, Wensheng Yin, Dongjin Zhu
Due to incomplete information, vague data, imprecise probability, it leads to model Knight uncertainties. Affected by model uncertainty, many researchers investigate the characteristics of model uncertainty in order to provide a framework for theory and applications. To describe model uncertainty, Peng (2007) proposed the notions of G-expectation and G-Brownian motion on sublinear expectation space which provide the new perspective for the stochastic calculus under Knight uncertainty. Moreover, G-Brownian motion, with an interestingly new structure, nontrivially generalises the classical Brownian motion. Stochastic calculus based on G-Brownian motion has concrete applications in uncertainly problems, risk measures, the superhedging in finance and so on. Especially, Black–Scholes–Merton's model was described by G-Brownian motion. Under the G-framework, stochastic differential equations driven by G-Brownian motion (G-SDEs, in short) were firstly introduced by Peng (2008). Since then, Denis, Hu, and Peng (2011) presented function spaces and capacity related to G-expectation. Gao (2009) studied pathwise properties and homeomorphic flows of G-SDEs. Li et al. (2016) used Lyapunov method to study the properties of G-SDEs. Yin and Ren (2017) applied k-vertex G-Lyapunov function to study stochastic coupled systems on networks. For the updated developments on G-stochastic analysis and G-SDEs, one can see the survey paper by Peng (2017).