In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is...In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is used that yields convergence of the Newton iteration to a solution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Some appropriate assumptions on the closedness and smoothness of the starting value and the solution are shown to lead to optimal convergence rates.展开更多
Given a continuous semimartingale M = （Mt）t≥〉0 and a d-dimensional continuous process of locally bounded variation V = （V^1,……, V^d）, the multidimensional Ito Formula states that f（Mt, Vt） - f（M0, V0） = ...Given a continuous semimartingale M = （Mt）t≥〉0 and a d-dimensional continuous process of locally bounded variation V = （V^1,……, V^d）, the multidimensional Ito Formula states that f（Mt, Vt） - f（M0, V0） = ∫[0, t] Dx0f（Ms, Vs）dMs＋∑i=1^d∫[0, t] Dxi F（Ms, Vs）dVs^i＋1/2∫[0, t] Dx0^2 f（Ms, Vs）d 〈M〉s if f（x0,……,xd） is of C^2-type with respect to x0 and of C^1-type with respect to the other arguments This formula is very useful when solving various optimal stopping problems based on Brownian motion. However, in such application the function f typically fails to satisfy the stated conditions in that its first partial derivative with respect to x0 is only absolutely continuous. We prove that the formula remains true for such functions and demonstrate its use with two examples from Mathematical Finance.展开更多
文摘In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is used that yields convergence of the Newton iteration to a solution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Some appropriate assumptions on the closedness and smoothness of the starting value and the solution are shown to lead to optimal convergence rates.
基金Partially supported by the Deutsche Forschungsgemeinschaft(DFG) under Grant SCHM 677/7-1
文摘Given a continuous semimartingale M = （Mt）t≥〉0 and a d-dimensional continuous process of locally bounded variation V = （V^1,……, V^d）, the multidimensional Ito Formula states that f（Mt, Vt） - f（M0, V0） = ∫[0, t] Dx0f（Ms, Vs）dMs＋∑i=1^d∫[0, t] Dxi F（Ms, Vs）dVs^i＋1/2∫[0, t] Dx0^2 f（Ms, Vs）d 〈M〉s if f（x0,……,xd） is of C^2-type with respect to x0 and of C^1-type with respect to the other arguments This formula is very useful when solving various optimal stopping problems based on Brownian motion. However, in such application the function f typically fails to satisfy the stated conditions in that its first partial derivative with respect to x0 is only absolutely continuous. We prove that the formula remains true for such functions and demonstrate its use with two examples from Mathematical Finance.
