STOCHASTIC MODEL OF LAKE SYSTEM INVASION AND ITS OPTIMAL CONTROL: NEURODYNAMIC PROGRAMMING AS A SOLUTION METHOD

Abstract We develop a stochastic model for the process of spread of an aquatic invader in a lake system. The invader is transported along with recreation boats, and treatment of boats allows one to control the invader spread. Optimization of invasion and control costs leads to a dynamic programming problem. However, standard stochastic dynamic programming (SDP) algorithms allow us to solve the problem for at most 13–14 lakes, which is significantly less than the size of systems arising in applications. We have developed a new technique, which is based upon the ideas of reinforcement learning and neurodynamic programming. This approach allows us to obtain approximate but reasonable control policy for essentially greater lake systems. The resulting numerical technique is comparatively easy to implement, and it can be applied to other spatially extended optimal control problems as well.     

基于价格极值构建有效价差的广义矩估计

基于Corwin和Schultz(2012)提出的有效价差的High-Low估计,结合价格极值信息得到新的一阶矩条件,构造了有效价差的广义矩估计。随后通过随机数值模拟比较了基于价格极值的广义矩估计(GMM)与Roll的协方差估计、Bayes估计以及Corwin和Schultz的High-Low估计在多种不同状态下的估计精度。数值模拟结果显示,无论在交易连续的理想状态下还是交易不连续且波动率相对不高的非理想状态下,GMM估计的精度均高于其余三种估计;基于我国股票市场的实例分析,也表明GMM估计的估计精度优于其余三种估计。因此,GMM估计为度量金融资产的交易成本提供了一种有效方法。     

Optimal control of information epidemics modeled as Maki Thompson rumors

We model the spread of information in a homogeneously mixed population using the Maki Thompson rumor model. We formulate an optimal control problem, from the perspective of single campaigner, to maximize the spread of information when the campaign budget is fixed. Control signals, such as advertising in the mass media, attempt to convert ignorants and stiflers into spreaders. We show the existence of a solution to the optimal control problem when the campaigning incurs non-linear costs under the isoperimetric budget constraint. The solution employs Pontryagin’s Minimum Principle and a modified version of forward backward sweep technique for numerical computation to accommodate the isoperimetric budget constraint. The techniques developed in this paper are general and can be applied to similar optimal control problems in other areas.We have allowed the spreading rate of the information epidemic to vary over the campaign duration to model practical situations when the interest level of the population in the subject of the campaign changes with time. The shape of the optimal control signal is studied for different model parameters and spreading rate profiles. We have also studied the variation of the optimal campaigning costs with respect to various model parameters. Results indicate that, for some model parameters, significant improvements can be achieved by the optimal strategy compared to the static control strategy. The static strategy respects the same budget constraint as the optimal strategy and has a constant value throughout the campaign horizon. This work finds application in election and social awareness campaigns, product advertising, movie promotion and crowdfunding campaigns.     

面对成比例交易成本的情况下外汇期权的定价

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor's expected utility. Option prices are determined from the investor's basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility.Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

FOREIGN CURRENCY OPTION PRICING WITH PROPORTIONAL TRANSACTION COSTS

The purpose of present work is to examine the financial problem of finding the universal reservation prices of a European call option written on exchange rate when there is proportional transaction costs of trading foreign currency in the market. An approach is suggested to compute the reservation bid-ask price of foreign currency call option based on maximizing the investor＇s expected utility. Option prices are determined from the investor＇s basic portfolio selection problem, without the need to solve a more complex optimization problem involving the insertion of the option payoffs into the terminal value function. Option prices are computed numerically in a Markov chain approximation for the case of exponential utility. Numerical results show that the option price bounds are almost independent of the alternative risk aversion parameter, but the bounds of NT region becomes narrower and the range of values of the initial holding for which the fair price lies within the bid-ask spread is shifted to a lower value when the risk aversion parameter increases.     

A numerical algorithm for pricing electricity derivatives for jump-diffusion processes based on continuous time lattices

We present a numerical algorithm for pricing derivatives on electricity prices. The algorithm is based on approximating the generator of the underlying price process on a lattice of prices, resulting in an approximation of the stochastic process by a continuous time Markov chain. We numerically study the rate of convergence of the algorithm for the case of the Merton jump-diffusion model and apply the algorithm to calculate prices and sensitivities of both European and Bermudan electricity derivatives when the underlying price follows a stochastic process which exhibits both fast mean-reversion and jumps of large magnitude.     

Stochastic uncapacitated hub location

We study stochastic uncapacitated hub location problems in which uncertainty is associated to demands and transportation costs. We show that the stochastic problems with uncertain demands or dependent transportation costs are equivalent to their associated deterministic expected value problem (EVP), in which random variables are replaced by their expectations. In the case of uncertain independent transportation costs, the corresponding stochastic problem is not equivalent to its EVP and specific solution methods need to be developed. We describe a Monte-Carlo simulation-based algorithm that integrates a sample average approximation scheme with a Benders decomposition algorithm to solve problems having stochastic independent transportation costs. Numerical results on a set of instances with up to 50 nodes are reported.     

Valuation of variable annuities with Guaranteed Minimum Withdrawal Benefit under stochastic interest rate

This paper develops an efficient direct integration method for pricing of the variable annuity (VA) with guarantees in the case of stochastic interest rate. In particular, we focus on pricing VA with Guaranteed Minimum Withdrawal Benefit (GMWB) that promises to return the entire initial investment through withdrawals and the remaining account balance at maturity. Under the optimal (dynamic) withdrawal strategy of a policyholder, GMWB pricing becomes an optimal stochastic control problem that can be solved using backward recursion Bellman equation. Optimal decision becomes a function of not only the underlying asset but also interest rate. Presently our method is applied to the Vasicek interest rate model, but it is applicable to any model when transition density of the underlying asset and interest rate is known in closed-form or can be evaluated efficiently. Using bond price as a numéraire the required expectations in the backward recursion are reduced to two-dimensional integrals calculated through a high order Gauss–Hermite quadrature applied on a two-dimensional cubic spline interpolation. The quadrature is applied after a rotational transformation to the variables corresponding to the principal axes of the bivariate transition density, which empirically was observed to be more accurate than the use of Cholesky transformation. Numerical comparison demonstrates that the new algorithm is significantly faster than the partial differential equation or Monte Carlo methods. For pricing of GMWB with dynamic withdrawal strategy, we found that for positive correlation between the underlying asset and interest rate, the GMWB price under the stochastic interest rate is significantly higher compared to the case of deterministic interest rate, while for negative correlation the difference is less but still significant. In the case of GMWB with predefined (static) withdrawal strategy, for negative correlation, the difference in prices between stochastic and deterministic interest rate cases is not material while for positive correlation the difference is still significant. The algorithm can be easily adapted to solve similar stochastic control problems with two stochastic variables possibly affected by control. Application to numerical pricing of Asian, barrier and other financial derivatives with a single risky asset under stochastic interest rate is also straightforward.     

Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework

This paper investigates an optimal investment strategy of DC pension plan in a stochastic interest rate and stochastic volatility framework. We apply an affine model including the Cox–Ingersoll–Ross (CIR) model and the Vasicek mode to characterize the interest rate while the stock price is given by the Heston’s stochastic volatility (SV) model. The pension manager can invest in cash, bond and stock in the financial market. Thus, the wealth of the pension fund is influenced by the financial risks in the market and the stochastic contribution from the fund participant. The goal of the fund manager is, coping with the contribution rate, to maximize the expectation of the constant relative risk aversion (CRRA) utility of the terminal value of the pension fund over a guarantee which serves as an annuity after retirement. We first transform the problem into a single investment problem, then derive an explicit solution via the stochastic programming method. Finally, the numerical analysis is given to show the impact of financial parameters on the optimal strategies.     

A numerical method to price European derivatives based on the one factor LIBOR Market Model of interest rates

We consider the problem of pricing European interest rate derivatives based on the LIBOR Market Model (LMM) with one driving factor. We derive a closed-form approximation of the transition probability density functions associated to the stochastic dynamical systems that describe the behaviour of the forward LIBOR interest rates in the LMM. These approximate formulae are based on a truncated power series expansion of the solutions of the Fokker–Planck equations associated to the LMM. The approximate probability density functions obtained are used to price European interest rate derivatives using the method of discounted expectations. The resulting integrals are low dimensional when the most commonly traded European interest rate derivatives are considered, and they can be computed efficiently using elementary numerical quadrature schemes (i.e. Simpson’s rule). The algorithm obtained is very well suited for parallel computing and is tested on the problem of pricing several derivatives including an European swaption and an interest rate spread option. In both cases, the method proposed in this paper appears to be accurate (i.e. relative error of order 10⁻², 10⁻³, or even 10⁻⁴) and approximately between 278 and 63 000 times faster than previous methods based on the Monte Carlo simulation of the LMM stochastic dynamical systems.

The website http://www.econ.univpm.it/pacelli/ballestra/finance/w2 contains material that helps the understanding of this paper and makes available to the interested users the computer programs that implement the numerical method proposed.     

One approach to the problem of nonparametric estimation in statistics of random processes based on the method of ill-posed problem

We consider the problem of estimation of integrated volatility, i.e., of the integral of the diffusion coefficient squared, in a stochastic differential equation for a random process that corresponds to geometric Brownian motion. In additon to purely theoretical interest, this problem is of interest for applications since the problem of evaluation of integrated volatility for financial assets is an important part of financial engineering topics. In the present paper, we suggest a new approach to the above-mentioned problem. We derive an integral equation whose solution determines the value of integrated volatility. This integral equation is a typical ill-posed problem of mathematical physics. The main idea of the proposed reduction of the original problem to an ill-posed problem consists of making its solution robust with respect to anomalous values of statistical data which are generated, for example, by market microstructure effects, such as the bid-ask spread. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 117–128.     

On stochastic lot-sizing problems with random lead times

We give multi-stage stochastic programming formulations for lot-sizing problems where costs, demands and order lead times follow a general discrete-time stochastic process with finite support. We characterize the properties of an optimal solution and give a dynamic programming algorithm, polynomial in the scenario tree size, when orders do not cross in time.     

Minimization Algorithms Based on Supervisor and Searcher Cooperation

In the present work, we explore a general framework for the design of new minimization algorithms with desirable characteristics, namely, supervisor-searcher cooperation. We propose a class of algorithms within this framework and examine a gradient algorithm in the class. Global convergence is established for the deterministic case in the absence of noise and the convergence rate is studied. Both theoretical analysis and numerical tests show that the algorithm is efficient for the deterministic case. Furthermore, the fact that there is no line search procedure incorporated in the algorithm seems to strengthen its robustness so that it tackles effectively test problems with stronger stochastic noises. The numerical results for both deterministic and stochastic test problems illustrate the appealing attributes of the algorithm.     

Dynamic uncapacitated lot sizing with random demand under a fillrate constraint

This paper deals with the single-item dynamic uncapacitated lot sizing problem with random demand. We propose a model based on the “static uncertainty” strategy of Bookbinder and Tan (1988). In contrast to these authors, we use exact expressions for the inventory costs and we apply a fillrate constraint. We present an exact solution method and modify several well-known dynamic lot sizing heuristics such that they can be applied for the case of dynamic stochastic demands. A numerical experiment shows that there are significant differences in the performance of the heuristics whereat the ranking of the heuristics is different from that reported for the case of deterministic demand.     

Decision making in a dynamic environment

In this paper we consider the decision problem affected by an unwanted dynamic parameter. We show that this problem can be solved by an adaptive joint maximum-likelihood (ML) strategy, which makes use of a stochastic gradient algorithm. We point out that the bandwidth of the stochastic gradient algorithm is an important design parameter, which greatly influences the decision error probability.     

手数约束和凹交易费下的离散投资组合模型及算法

本文建立带手数约束和凹交易费的离散投资组合模型,给出求解该模型的一种精确算法。该算法是一个基于拉格朗日松弛和次梯度对偶搜索的分枝定界算法。为测试算法的有效性,用随机产生的数据对模型进行数值实验。作为其应用,用沪深300指数的真实数据实证检验该模型,并与不含交易费用的离散投资组合模型进行数值比较分析。数值分析表明算法能在合理的时间内给出模型的投资组合策略, 对解决中小规模的离散投资组合问题是有效的。     

Simulation of suspended substance transport on the continental shelf: Horizontal dispersion

Suspended substance dispersion in a water body is simulated in the case when the spread area is considerably larger than the depth of the water body. A model of horizontal dispersion of pollutants is formulated and analyzed. Numerical approaches to the computation of suspended substance dispersion in a water body are discussed. A meshless stochastic numerical algorithm is proposed that combines the advantages of two well-known techniques, namely, the discrete cloud method and the stochastic discrete particle method. The performance the method and its features are demonstrated by comparing numerical results with the exact solution to the model problem of turbulent dispersion of a pollution plume produced by a continuous source of suspension.     

Random-Direction Optimization Algorithms with Applications to Threshold Controls

This work develops a class of stochastic optimization algorithms. It aims to provide numerical procedures for solving threshold-type optimal control problems. The main motivation stems from applications involving optimal or suboptimal hedging policies, for example, production planning of manufacturing systems including random demand and stochastic machine capacity. The proposed algorithm is a constrained stochastic approximation procedure that uses random-direction finite-difference gradient estimates. Under fairly general conditions, the convergence of the algorithm is established and the rate of convergence is also derived. A numerical example is reported to demonstrate the performance of the algorithm.     

Enhancing trading strategies with order book signals

We use high-frequency data from the Nasdaq exchange to build a measure of volume imbalance in the limit order (LO) book. We show that our measure is a good predictor of the sign of the next market order (MO), i.e., buy or sell, and also helps to predict price changes immediately after the arrival of an MO. Based on these empirical findings, we introduce and calibrate a Markov chain-modulated pure jump model of price, spread, LO and MO arrivals and volume imbalance. As an application of the model, we pose and solve a stochastic control problem for an agent who maximizes terminal wealth, subject to inventory penalties, by executing trades using LOs. We use in-sample-data (January to June 2014) to calibrate the model to 11 equities traded in the Nasdaq exchange and use out-of-sample data (July to December 2014) to test the performance of the strategy. We show that introducing our volume imbalance measure into the optimization problem considerably boosts the profits of the strategy. Profits increase because employing our imbalance measure reduces adverse selection costs and positions LOs in the book to take advantage of favourable price movements.     

The influence of demand variability on the performance of a make-to-stock queue

Variability, in general, has a deteriorating effect on the performance of stochastic inventory systems. In particular, previous results indicate that demand variability causes a performance degradation in terms of inventory related costs when production capacity is unlimited. In order to investigate the effects of demand variability in capacitated production settings, we analyze a make-to-stock queue with general demand arrival times operated according to a base-stock policy. We show that when demand inter-arrival distributions are ordered in a stochastic sense, increased arrival time variability indeed leads to an augmentation of optimal base-stock levels and to a corresponding increase in optimal inventory related costs. We quantify these effects through several numerical examples.