Band 28, Heft 2

In this paper, we propose a new numerical method for 1-D backward stochastic differential equations (BSDEs for short) without using conditional expectations. The approximations of the solutions are obtained as solutions of a backward linear system generated by the terminal conditions. Our idea is inspired from the extended Kalman filter to non-linear system models by using a linear approximation around deterministic nominal reference trajectories.

We wish to study a class of optimal controls for problems governed by forward-backward doubly stochastic differential equations (FBDSDEs). Firstly, we prove existence of optimal relaxed controls, which are measure-valued processes for nonlinear FBDSDEs, by using some tightness properties and weak convergence techniques on the space of Skorokhod 𝔻{\mathbb{D}} equipped with the S -topology of Jakubowski. Moreover, when the Roxin-type convexity condition is fulfilled, we prove that the optimal relaxed control is in fact strict. Secondly, we prove the existence of a strong optimal controls for a linear forward-backward doubly SDEs. Furthermore, we establish necessary as well as sufficient optimality conditions for a control problem of this kind of systems. This is the first theorem of existence of optimal controls that covers the forward-backward doubly systems.

We discuss nonparametric estimation of a trend coefficient in models governed by a stochastic differential equation driven by a sub-fractional Brownian motion with small noise.

We study the problem of deepening the Jessen–Wintner theorem for asymmetric Bernoulli convolutions. In particular, we investigate the Lebesgue structure of a random incomplete sum of series, whose terms are reciprocal to Jacobsthal–Lucas numbers.

The G -Elliptic law under the G -Lindeberg condition for the independent pairs of the entries of a random matrix is proven.