FBSDE

AcronymDefinition
FBSDEForward-Backward Stochastic Differential Equation
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By that theorem, the decoupled FBSDE (1), (5), and (C1) has a unique solution [(r, [lambda]), q, [[??].sub.1](t)] such that |q(t)| is bounded and [[??].sub.1](t) [member of] [L.sub.H.sup.2]([0, T], [R.sup.n]).
By Theorem 1 of Zhang (2006) again, the decoupled FBSDE (1), (5), and (C1) has a unique solution [(r, [lambda]), Q, [[??].sub.2](t)], in which |Q(t)| is bounded and [[eta].sub.2](t) [member of] [L.sub.H.sup.2]([0, T], [R.sup.n]
Accordingly, we employ a framework of forward-backward stochastic differential equations (FBSDEs) to avoid difficulties with the VMM framework.
In a general model setting, we obtain the technical conditions for the existence and uniqueness of these solutions using the theory of FBSDEs. Explicit solutions are obtained in a subpopulation mortality setting.
Thus we fix [epsilon] and proceed to study the unconstrained problem (54) in the context of Section 4: consider the controlled FBSDE system consisting of
According to Lemma 1, [mathematical expression not reproducible] can be found as the solution of the fully coupled system of FBSDE (72) and (75).