This note is devoted to the study of the absolute convergence of Bernstein polynomials. It is proved that for each x∈ , the sequence of the Bernstein polynomials of a function of bounded variation is absolutely su...This note is devoted to the study of the absolute convergence of Bernstein polynomials. It is proved that for each x∈ , the sequence of the Bernstein polynomials of a function of bounded variation is absolutely summable by |C,1| method. Moreover, the estimate of the remainders of the |C,1| sum of the sequence of the Bernstein polynomials is obtained.展开更多
In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （...In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （e.g., the multiple Malliavin integration-by-parts formula）, and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.展开更多
文摘This note is devoted to the study of the absolute convergence of Bernstein polynomials. It is proved that for each x∈ , the sequence of the Bernstein polynomials of a function of bounded variation is absolutely summable by |C,1| method. Moreover, the estimate of the remainders of the |C,1| sum of the sequence of the Bernstein polynomials is obtained.
基金supported by Shanghai University Young Teacher Training Program(Grant No.slg14032)National Natural Science Foundations of China(Grant Nos.11501366 and 11571206)
文摘In this work, we theoretically analyze the convergence error estimates of the Crank-Nicolson （C-N） scheme for solving decoupled FBSDEs. Based on the Taylor and ItS-Taylor expansions, the Malliavin calculus theory （e.g., the multiple Malliavin integration-by-parts formula）, and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.
