Continuous-time portfolio selection with liability: Mean–variance model and stochastic LQ approach

In this paper we formulate a continuous-time mean–variance portfolio selection model with multiple risky assets and one liability in an incomplete market. The risky assets’ prices are governed by geometric Brownian motions while the liability evolves according to a Brownian motion with drift. The correlations between the risky assets and the liability are considered. The objective is to maximize the expected terminal wealth while minimizing the variance of the terminal wealth. We derive explicitly the optimal dynamic strategy and the mean–variance efficient frontier in closed forms by using the general stochastic linear-quadratic (LQ) control technique. Several special cases are discussed and a numerical example is also given.     

Reversible coagulation–fragmentation processes and random combinatorial structures: Asymptotics for the number of groups

The equilibrium distribution of a reversible coagulation‐fragmentation process (CFP) and the joint distribution of components of a random combinatorial structure (RCS) are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (= components) in the case a(k) = qk^p?1, k ≥ 1, q, p > 0, where a(k), k ≥ 1, is the parameter function that induces the invariant measure. The result obtained is compared with the ones for logarithmic RCS's and for RCS's, corresponding to the case p < 0. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

Invariant approach to optimal investment–consumption problem: the constant elasticity of variance (CEV) model

The optimal investment–consumption problem under the constant elasticity of variance (CEV) model is solved using the invariant approach. Firstly, the invariance criteria for scalar linear second‐order parabolic partial differential equations in two independent variables are reviewed. The criteria is then employed to reduce the CEV model to one of the four Lie canonical forms. It is found that the invariance criteria help in transforming the original equation to the second Lie canonical form and with a proper parameter selection; the required transformation converts the original equation to the first Lie canonical form that is the heat equation. As a consequence, we find some new classes of closed‐form solutions of the CEV model for the case of reduction into heat equation and also into second Lie canonical form. The closed‐form analytical solution of the Cauchy initial value problems for the CEV model under investigation is also obtained. Copyright © 2016 John Wiley & Sons, Ltd.