Optimal cash management under uncertainty

We solve an agent’s optimization problem of meeting demands for cash over time with cash deposited in bank or invested in stock. The stock pays dividends and uncertain capital gains, and a commission is incurred in buying and selling of stock. We use a stochastic maximum principle to obtain explicitly the optimal transaction policy.     

Optimal payout policy in presence of downside risk

We analyze the determination of a value maximizing dividend payout policy for a broad class of cash reserve processes modeled as spectrally negative jump diffusions. We extend previous results based on continuous diffusion models and characterize the value of the optimal dividend distribution strategy explicitly. We also characterize explicitly the values as well as the optimal dividend thresholds for a class of associated optimal liquidation and sequential lump sum dividend control problems. Our results indicate that both the value as well as the marginal value of the optimal policies are increasing functions of policy flexibility in the discontinuous setting as well.      

Pricing and risk management of interest rate swaps

This paper reformulates the valuation of interest rate swaps, swap leg payments and swap risk measures, all under stochastic interest rates, as a problem of solving a system of linear equations with random perturbations. A sequence of uniform approximations which solves this system is developed and allows for fast and accurate computation. The proposed method provides a computationally efficient alternative to Monte Carlo based valuations and risk measurement of swaps. This is demonstrated by conducting numerical experiments and so our method provides a potentially important real-time application for analysis and calculation in markets.     

一类脉冲型平稳最佳随机控制之研究

本文研究了一类平稳的脉冲控制模型，不仅证明了最佳脉冲控制的存在性而且构造出一个最佳控制．     

Cost/risk balanced management of scarce resources using stochastic programming

We consider the situation when a scarce renewable resource should be periodically distributed between different users by a Resource Management Authority (RMA). The replenishment of this resource as well as users demand is subject to considerable uncertainty. We develop cost optimization and risk management models that can assist the RMA in its decision about striking the balance between the level of target delivery to the users and the level of risk that this delivery will not be met. These models are based on utilization and further development of the general methodology of stochastic programming for scenario optimization, taking into account appropriate risk management approaches. By a scenario optimization model we obtain a target barycentric value with respect to selected decision variables. A successive reoptimization of deterministic model for the worst case scenarios allows the reduction of the risk of negative consequences derived from unmet resources demand. Our reference case study is the distribution of scarce water resources. We show results of some numerical experiments in real physical systems.     

A note on convergence of an approximate hedging portfolio with liquidity risk

When the underlying asset price depends on activities of traders, hedging errors include costs due to the illiquidity of the underlying asset and the size of this cost can be substantial. Cetin et al. (2004), Liquidity risk and arbitrage pricing theory, Finance and Stochastics, 8(3), 311-341, proposed a hedging strategy that approximates the classical Black–Scholes hedging strategy and produces zero liquidity costs. Here, we compute the rate of convergence of the final value of this hedging portfolio to the option payoff in case of a European call option; i.e. we see how fast its hedging error converges to zero. The hedging strategy studied here is meaningful due to its simple liquidity cost structure and its smoothness relative to the classical Black–Scholes delta.     

Optimal life cycle portfolio choice with variable annuities offering liquidity and investment downside protection

This paper assesses optimal life cycle consumption and portfolio allocations when households have access to Guaranteed Minimum Withdrawal Benefit (GMWB) variable annuities over their adult lifetimes. Our contribution is to evaluate demand for these products which provide access to equity investments with money-back guarantees, longevity risk hedging, and partially-refundable premiums, in a realistic world with uncertain labor and capital market income as well as mortality risk. Others have predicted that consumers will only purchase such annuities late in life, but we show that they will optimally purchase GMWBs prior to retirement, consistent with their recent rapid uptick in sales. Additionally, many individuals optimally adjust their portfolios and consumption streams along the way by taking cash withdrawals from the products. These products can substantially enhance consumption, by up to 10% for those who experience highly unfavorable experiences in the stock market.     

Optimal reinsurance with a rescuing procedure

We consider a large insurance company whose reserve is modeled by a diffusion process. The management of the insurance company makes a decision on reinsurance in order to reduce the insurance risk. An optimal decision is the one which minimizes the expected time to reach a goal before the reserve reaches a ruin level. We introduce a rescuing procedure to deal with the case that the company is “too big to fail”. We disclose that the optimal decision of the management heavily depends on how much time the company needs to wait for rescuing when it gets in trouble.     

A Connection between Singular Stochastic Control and Optimal Stopping

We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.     

A Connection between Singular Stochastic Control and Optimal Stopping

We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.     

具有随机利率、随机变差的最优投资和联合比例-超额损失再保险

对跳-扩散风险模型,研究了最优投资和再保险问题.保险公司可以购买再保险减少理赔,保险公司还可以把盈余投资在一个无风险资产和一个风险资产上.假设再保险的方式为联合比例-超额损失再保险.还假设无风险资产和风险资产的利率是随机的,风险资产的方差也是随机的.通过解决相应的Hamilton-Jacobi-Bellman(HJB)方程,获得了最优值函数和最优投资、再保险策略的显示解.特别的,通过一个例子具体的解释了得到的结论.     

跳扩散模型下基金平衡管理的最优脉冲控制

在基金市值波动服从跳扩散过程, 基金持有的罚金成本为当前基金水平的二次函数及存在交易费的假设下研究了无穷时域的基金平衡管理的最小成本模型. 利用随机最优脉冲控制的拟变分不等式理论建立了判定定理,得到了最优脉冲控制策略的存在性, 同时通过构造方法给出了解的数学结构形式.     

A Stochastic Linear Quadratic Optimal Control Problem with Generalized Expectation

Abstract

In this article, we initiate a study on optimal control problem for linear stochastic differential equations with quadratic cost functionals under generalized expectation via backward stochastic differential equations.     

Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain

Abstract

We consider stochastic optimal control problems in Banach spaces, related to nonlinear controlled equations with dissipative non linearities: on the nonlinear term we do not impose any growth condition. The problems are treated via the backward stochastic differential equations approach, that allows also to solve in mild sense Hamilton Jacobi Bellman equations in Banach spaces. We apply the results to controlled stochastic heat equation, in space dimension 1, with control and noise acting on a subdomain.     

Robust risk management

Estimating the probabilities by which different events might occur is usually a delicate task, subject to many sources of inaccuracies. Moreover, these probabilities can change over time, leading to a very difficult evaluation of the risk induced by any particular decision. Given a set of probability measures and a set of nominal risk measures, we define in this paper the concept of robust risk measure as the worst possible of our risks when each of our probability measures is likely to occur. We study how some properties of this new object can be related with those of our nominal risk measures, such as convexity or coherence. We introduce a robust version of the Conditional Value-at-Risk (CVaR) and of entropy-based risk measures. We show how to compute and optimize the Robust CVaR using convex duality methods and illustrate its behavior using data from the New York Stock Exchange and from the NASDAQ between 2005 and 2010.     

Analysis of a chance-constrained new product risk model with multiple customer classes

We consider the non-convex problem of minimizing a linear deterministic cost objective subject to a probabilistic requirement on a nonlinear multivariate stochastic expression attaining, or exceeding a given threshold. The stochastic expression represents the output of a noisy system featuring the product of mutually-independent, uniform random parameters each raised to a linear function of one of the decision vector’s constituent variables. We prove a connection to (i) the probability measure on the superposition of a finite collection of uncorrelated exponential random variables, and (ii) an entropy-like affine function. Then, we determine special cases for which the optimal solution exists in closed-form, or is accessible via sequential linear programming. These special cases inspire the design of a gradient-based heuristic procedure that guarantees a feasible solution for instances failing to meet any of the special case conditions. The application motivating our study is a consumer goods firm seeking to cost-effectively manage a certain aspect of its new product risk. We test our heuristic on a real problem and compare its overall performance to that of an asymptotically optimal Monte-Carlo-based method called sample average approximation. Numerical experimentation on synthetic problem instances sheds light on the interplay between the optimal cost and various parameters including the probabilistic requirement and the required threshold.     

Optimal risk transfer for agents with germs

We introduce a new class of risk measures called generalized entropic risk measures (GERMS) that allow economic agents to have different attitudes towards different sources of risk. We formulate the problem of optimal risk transfer in terms of these risk measures and characterize the optimal transfer contract. The optimal contract involves what we call intertemporal source-dependent quotient sharing, where agents linearly share changes in the aggregate risk reserve that occur in response to shocks to the system over time, with scaling coefficients that depend on the attitudes of each agent towards the source of risk causing the shock. Generalized entropic risk measures are not dilations of a common base risk measure, so our results extend the class of risk measures for which explicit characterizations of the optimal transfer contract can be found.