Linear backward stochastic differential systems of descriptor type with structure and applications to engineering

Linear stochastic degenerate Sobolev equations and applications†

“…The analysis is based on the results of the deterministic problem (15). First, we present mild and strong solvability results and then, by Remark 1 and the assumption A(3)(ii), we prove the well posedness of problem (5).…”

“…For the treatment of the solvability problem one can see [1], [13], [19]. Additionally, the backward problem and the exact controllability of the stochastic descriptor equations (2), (3), have been studied only recently in [13,15]. In this paper, more general equations of the form (4), In infinite dimensions, interesting applications of degenerate equations (4) (or (1)) are obtained if the operator M is a differential and L is a multiplicative one with a function that vanishes in a certain region of the configuration space.…”

An exact null controllability of stochastic singular systems

“…Suppose that x 2 (t) is the inverse stochastic Laplace transform of X 2 (s) obtained from (7). Then, x 2 (t) is the impulse solution to (6) in the sense of the stochastic Laplace transform, or simply, the impulse solution to (6). In this case, if x 1 (t) denotes the solution to (5), then…”

“…Using white noise and fractional white noise, two illustrative applications are presented in a previously conducted study [2]. The basic question of solvability has been formulated and considered [5,6]. Moreover, they propose a normalization procedure, and they completely solve the problem of exact controllability for a class of linear stochastic singular systems.…”

Backward stochastic differential equations with unbounded generators

“…for all y 1 , y 2 ∈ R d , z 1 , z 2 ∈ R d×k , (t, ω) a.e.. Since then, the BSDEs have been studied extensively, and have found wide applicability in areas such as mathematical finance, stochastic control, and stochastic controllability; see, for example, [10], [17], [29], [31], [42], [37], [20], [21], [22] [41], and the references therein. One direction of research has been to weaken the assumption of global Lipschitz condition (2) by assuming only local Lipschitz condition (see [1]), or non-Lipschitz condition of a particular form (see [30], [39]).…”