McKean-Vlasov-Type And Two-Time-Scale Stochastic Differential Equations And Related Applications

In this dissertation,we consider the numerical approximation of the solution to McKean-Vlasov stochastic differential equations(SDEs),the averaging principle for twotime-scale system with time delay based on the chemical Langevin equations,the averaging principle for two-time-scale McKean-Vlasov stochastic systems,and a class of mean-field linear-quadratic control problems.Compared to classical SDEs,the coefficients of McKeanVlasov SDEs depend on the distribution.The dissertation consists of the following five chapters:Chapter 1 briefly introduces some backgrounds and the research,the basic knowledge of stochastic space,the L-derivative with respect to the measure and the Ito formula for measure-dependent functions,and some general inequalities,and then gives the arrangement of this dissertation.Chapter 2 examines the strong convergence of the Euler-Maruyama scheme for the McKean-Vlasov SDEs under the local Lipschitz conditions with respect to the state variable.For classical SDEs,it is well known that local Lipschitz type conditions and some growth conditions guarantee the existence and uniqueness of the global solution and the strong convergence of the Euler-Maruyama scheme by using the standard truncation technique and the stopping time technique;see[41,71,117].Nevertheless,for McKean-Vlasov SDEs,under the local Lipschitz conditions with respect to the state variable,there exist fundamental difficulties to prove the existence and uniqueness of the solution and the strong convergence of the Euler-Maruyama scheme.To this end,we use Euler-like sequence of interpolations and partition of the sample space.Since the coefficients depend on the distribution,we need to approximate the distribution by the empirical measure.To proceed,the stochastic interacting particle system is used as a bridge.Chapter 3 investigates the averaging principle for the two-time-scale stochastic system with time delay based on the chemical Langevin equation.We prove the asymptotic behavior of the sequence of slow-varying process as ??0 by applying the perturbed test function.At first,we make a time-scale transformation for the fast-varying equation and obtain a new equation.We prove that any order moment of the solution to this equation is uniformly bounded with respect to time and this equation has a unique invariant measure.With this invariant measure,we then get the averaged equation and prove that the sequence of slowvarying process converges weakly to the weak solution of the averaged equation,the key step is the estimate of the partial derivatives of the transition probability density and stationary density.Chapter 4 establishes the averaging principle for the McKean-Vlasov SDEs.We prove the asymptotic behavior of the sequence of slow-varying process as ??0 by using the martingale method.At first,we make a time-scale transformation for the fast-varying equation and obtain a new McKean-Vlasov equation.We prove that the existence and uniqueness of the solution to this equation,the moment of the solution is uniformly bounded with respect to time,and this equation has a unique invariant measure.Since the corresponding semigroup is non-linear,the methods of proving the existence of invariant measure for classical SDEs is invalid.Inspired by[106],we obtain the invariant measure by showing that a measure sequence is Cauchy in Polish space Pp(Rd).Then,with this invariant measure,we get the averaged equation and prove that the sequence of slow-varying process converges weakly to the weak solution of the averaged equation,the key point is to use discrete approximation and the uniform boundedness of the moment of the slow-varying process.Chapter 5 analyzes the partially observable mean-field linear-quadratic control problem.Applying separation principle and fully observable mean-field linear-quadratic control problem,we obtain the semi-explicit solution in terms of Riccati equations.In particular,we can get the explicit solution in the special case.