Abstract
In this paper, we study general time-inconsistent stochastic control models which are driven by a stochastic differential equation with random jumps. Specifically, the time-inconsistency arises from the presence of a non-exponential discount function in the objective functional. We consider equilibrium, instead of optimal, solution within the class of open-loop controls. We prove an equivalence relationship between our time-inconsistent problem and a time-consistent problem such that the equilibrium controls for the time-consistent problem coincide with the equilibrium controls for the time-inconsistent problem. We establish two general results which characterize the open-loop equilibrium controls. As special cases, a generalized Merton's portfolio problem and a linear-quadratic problem are discussed.
Highlights
We consider in this paper stochastic control problems when the system under consideration is governed by a SDE of the following type dX (s) =b (s, X (s), u (s)) ds + σ (s, X (s), u (s)) dW (s)+ c (s, X (s−), u (s−), z) N, (1) ZX (t) = x, and for any fixed initial pair (t, x), the objective is to maximize the expected utility functional TJ (t, x; u (·)) = Et,x ν (t, s) f (s, u (s)) ds + ν (t, T ) h (X (T )), (2)t over the set of the admissible controls
The common assumption in most of the existing literature is that the discount rate of time preference is constant over time, leading to the exponential form of the discount function: ν (t, s) = e−δ(s−t) and ν (t, T ) = e−δ(T −t), 2010 Mathematics Subject Classification
We present a stochastic verification theorem which provides a sufficient condition for equilibrium controls of Problem (N)
Summary
We consider in this paper stochastic control problems when the system under consideration is governed by a SDE of the following type dX (s) =. X (t) = x, and for any fixed initial pair (t, x), the objective is to maximize the expected utility functional T. T over the set of the admissible controls. In the above model b, σ, c, f and h are deterministic functions. Ν (t, s) f (s, u (s)) is the discounted local utility and ν (t, T ) h (X (T )) is the terminal utility, where ν (·, ·) represents the discount function. The common assumption in most of the existing literature is that the discount rate of time preference is constant over time, leading to the exponential form of the discount function:. Primary: 93E20, 91B08, 91B70; Secondary: 62P20, 60H30, 60H10, 49N90, 93E25
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