Linear Backward Stochastic Differential Equations of Descriptor Type: Regular Systems

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

“…However, in the present version of the paper, this case would not be considered further as the interest reader can look the [16] and the relevant references therein for further details. Here, the case of a general matrix C can also be dealt with in a similar fashion to that in [16], but as shown there such conditions are less explicit. Similar derivation can be done even for our present case of symmetric/skew-symmetric coefficients, but at significant notational complexity, and hence it is omitted.…”

“…The case when the matrix coefficient B 0 ≠ , (and inevitably u(t) is also appeared) can be discussed separately and presented similarly as in paper [16], where the normalizability process can be also derived assuming a special type for the controller u(t), see for instance [22,21]. However, in the present version of the paper, this case would not be considered further as the interest reader can look the [16] and the relevant references therein for further details. Here, the case of a general matrix C can also be dealt with in a similar fashion to that in [16], but as shown there such conditions are less explicit.…”

“…Linear backward stochastic differential equations (BSDEs) of descriptor type without structural coefficients were introduced by Gashi and Pantelous in their recent paper [16] extending further the ideas and the results presented in [1,20] which were based on generalized stochastic processes of Dirac type. In that paper, the main equation is given by …”

“…Finally, it should be mentioned that two illustrative applications are presented in [21] using white noise and fractional white noise. More recently, in [16], the basic question of solvability has formulated and considered. There, under the assumption that the pencil is regular, but without assuming any particular structure for the matrices, and using fundamental tools of matrix pencil theory, such as the Weierstrass complex canonical form, the necessary and sufficient conditions for the existence of a unique solution pair are derived.…”

“…From the considerations in [16], it is apparent that in order to get explicit and simple criteria for solvability, the special structure of matrix coefficients of (2.1) must be exploited. Therefore, in this paper we consider Eq.…”