U Jin Choi

An algorithm for optimal portfolio selection problem with transaction costs and random lifetimes

Abstract We study a consumption and portfolio selection problem in the presence of proportional transaction costs. In order to explore the effects of the expectation about the length of an investor’s lifetime on her optimal consumption and investment, we generalize Constantinides’ [2] optimal investment model with transaction costs by randomizing the investor’s lifetime. We convert the problem into a free boundary problem with two free boundaries and obtain an optimal consumption and investment policy. We provide a numerical algorithm for this free boundary problem and prove convergence of a numerical solution obtained by the algorithm to a true solution. By using numerical results, we investigate the effect of investor’s expected lifetime on liquidity premia due to transaction costs.

Optimal consumption and portfolio selection problem with downside consumption constraints

Abstract We study a general optimal consumption and portfolio selection problem of an infinitely-lived investor whose consumption rate process is subjected to downside constraint. That is, her consumption rate is greater than or equals to some positive constant. We obtain the general optimal policies in an explicit form using martingale method and Feynman–Kac formula. We derive some numerical results of optimal consumption and portfolio in the special case of a constant relative risk aversion (CRRA) utility function.

Almost sure convergence of galerkin approximations for a heat equation with a random initial condition

We analyze probabilistic convergences of random Galerkin approximations for a heat equation with a random initial condition. Almost sure L2-convergence results for both continuous time and discrete time Galerkin approximations are obtained by the Borel-Cantellis lemma. A criterion for determining the sample size is suggested.

Prime number grid scheme to approximate function spaces with their low dimensional subspaces

We introduce a method of approximating C^j[0,1], j = 0,1,..., with its low dimensional subspaces of uniform grids generated by prime numbers. This is very efficient in terms of computational complexity and suitable for parallel computation.

Almost sure convergence of a finite element approximations for the random Sturm-Liouville boundary value problem

Abstract This is a continuation of our previous work [3]. Convergent properties of the finite element approximations and a sample mean for the random Sturm-Liouville boundary value problem is discussed.

A Lattice Method for Lookback Options with Regime-Switching Volatility

We develop a lattice method for pricing lookback options in a regime-switching market environment. We assume the market is governed by a two-state Markov chain and stock volatility can change whenever the market environment changes. We develop a method which resolves the bias in the binomial method of Babbs (2000), and combine it with the pentanomial method of Bollen (1998). Our method can be used for American-style lookback options as well as European-style lookback options and is a simple but efficient way for pricing them compared with the methods currently available. We also analyze the convergence of the proposed method.

An algebraic geometric method to calculate branches near a bifurcation point

Abstract Systems of nonlinear algebraic equations with a parameter arises in many branches of mathematics and engineering. Computation of the solution curve is usually hindered by the existence of a bifurcation point, where the rank of the corresponding matrix is not full and usually there is another branch curve passing it. We introduce a method from algebraic geometry to find the branch curve. A computational example is provided.