Qingshuo Song

Stability of random-switching systems of differential equations

This work is devoted to the stability of random-switching systems of differential equations. After presenting the formulation of random-switching systems, the notion of stability is recalled, and sufficient conditions in terms of the Liapunov function are presented. Then easily verifiable conditions for stability and instability of systems arising in approximation are established. Using a logarithm transformation, necessary and sufficient conditions are derived for systems that are linear in the continuous state component. Several examples are provided as demonstrations. Among other things, a somewhat different behavior from the well-known Hartman-Grobman theorem is observed.

On Optimal Harvesting Problems in Random Environments

This paper investigates the optimal harvesting strategy for a single species living in random environments whose population growth is given by a regime-switching diffusion. Harvesting acts as a (stochastic) control on the size of the population. The objective is to find a harvesting strategy which maximizes the expected total discounted income from harvesting up to the time of extinction of the species; the income rate is allowed to be state- and environment-dependent. This is a singular stochastic control problem, with both the extinction time and the optimal harvesting policy depending on the initial condition. One aspect of receiving payments up to the random time of extinction is that small changes in the initial population size may significantly alter the extinction time when using the same harvesting policy. Consequently, one no longer obtains continuity of the value function using standard arguments for either regular or singular control problems having a fixed time horizon. This paper introduces a new sufficient condition under which the continuity of the value function for the regime-switching model is established. Further, it is shown that the value function is a viscosity solution of a coupled system of quasi-variational inequalities. The paper also establishes a verification theorem and, based on this theorem, an

Mean Exit Times and the Multilevel Monte Carlo Method

\varepsilon

Stochastic Optimization Methods for Buying-Low-and-Selling-High Strategies

-optimal harvesting strategy is constructed under certain conditions on the model. Two examples are analyzed in detail.

Numerical Solutions for Stochastic Differential Games With Regime Switching

Numerical methods for stochastic differential equations are relatively inefficient when used to ap- proximate mean exit times. In particular, although the basic Euler-Maruyama method has weak order equal to one for approximating the expected value of the solution, the order reduces to one half when it is used in a straightforward manner to approximate the mean value of a (stopped) exit time. Consequently, the widely used standard approach of combining an Euler-Maruyama discretization with a Monte Carlo simulation leads to a computationally expensive procedure. In this work, we show that the multilevel approach developed by Giles (Oper. Res., 56 (2008), pp. 607-617) can be adapted to the mean exit time context. In order to justify the algorithm, we analyze the strong error of the discretization method in terms of its ability to approximate the exit time. We then show that the resulting multilevel algorithm improves the expected computational complexity by an order of magnitude, in terms of the required accuracy. Numerical results are provided to illustrate the analysis.

Outperforming the Market Portfolio with a Given Probability

Abstract This article is concerned with a numerical method using stochastic approximation approach for an optimal trading (buy and sell) strategy. The underlying asset price is governed by a mean-reverting stochastic process. The objective is to buy and sell the asset so as to maximize an overall expected return. One of the advantages of our approach is that the underlying asset is model free. Only mean reversion is required. Slippage cost is imposed on each transaction. Convergence of the algorithms is provided. Numerical examples are reported to demonstrate the results.

On the Continuity of Stochastic Exit Time Control Problems

This paper is concerned with numerical methods for stochastic differential games of regime-switching diffusions. Numerical methods using Markov chain approximation techniques are developed. A new proof of the existence of a saddle point for the stochastic differential game is provided. This new proof enables us to treat certain systems with nonseparable (in controls) structure. Convergence of the algorithms is derived by means of weak convergence methods. In addition, examples are also provided for demonstration purposes.

Rates of Convergence of Numerical Methods for Controlled Regime-Switching Diffusions with Stopping Times in the Costs

Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show that this value function is the smallest nonnegative viscosity supersolution of a nonlinear PDE. As in Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.], we do not assume the existence of an equivalent local martingale measure, but merely the existence of a local martingale deflator.

Weak convergence methods for approximation of the evaluation of path-dependent functionals

We determine a weaker sufficient condition than that of Theorem 5.2.1 in Fleming and Soner (2006) for the continuity of the value functions of stochastic exit time control problems.

Outperformance Portfolio Optimization Via the Equivalence of Pure and Randomized Hypothesis Testing

This work is concerned with rates of convergence of Markov chain approximation methods for controlled switching diffusions. The cost function is defined on an infinite horizon with stopping times and without discount. Displaying both continuous dynamics and discrete events, the discrete events are modeled by continuous-time Markov chains to delineate a random environment and other random factors that cannot be represented by diffusion processes. This paper presents a first attempt using a probabilistic approach to treat such rates of convergence problems. In addition, in contrast to the significant developments in the literature using partial differential equation (PDE) methods for the approximation of controlled diffusions, there do not yet appear to be any PDE results to date for rates of convergence of numerical solutions for controlled switching diffusions, to the best of our knowledge. Although some of the working conditions in this paper such as the one-dimensional continuous state variable, nondegenerate diffusions, and control only on the drift may be seemingly strong, they are adequate as the starting point for using this new approach to treat the rates of convergence problems. Moreover, in the literature, to prove the convergence using Markov chain approximation methods for control problems involving cost functions with stopping (even for uncontrolled diffusion without switching), an added assumption was used to avoid the so-called tangency problem. As a by-product of our approach, by modifying the value function, it is demonstrated that the anticipated tangency problem will not arise in the sense of convergence in probability and in the sense of