The opportunity process for optimal consumption and investment with power utility

We study the utility maximization problem for power utility random fields in a semimartingale financial market, with and without intermediate consumption. The notion of an opportunity process is introduced as a reduced form of the value process of the resulting stochastic control problem. We show how the opportunity process describes the key objects: optimal strategy, value function, and dual problem. The results are applied to obtain monotonicity properties of the optimal consumption.     

Backward stochastic partial differential equations related to utility maximization and hedging

We study the utility maximization problem, the problem of minimization of the hedging error and the corresponding dual problems using dynamic programming approach. We consider an incomplete financial market model, where the dynamics of asset prices are described by an ℝ^d-valued continuous semimartingale. Under some regularity assumptions, we derive the backward stochastic PDEs for the value functions related to these problems, and for the primal problem, we show that the strategy is optimal if and only if the corresponding wealth process satisfies a certain forward SDE. As examples we consider the mean-variance hedging problem and the cases of power, exponential, logarithmic utilities, and corresponding dual problems. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 45, Martingale Theory and Its Application, 2007.     

Causal optimal transport and its links to enlargement of filtrations and continuous-time stochastic optimization

The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are also used in order to give robust transport-based estimates for the value of having additional information, as well as model sensitivity with respect to the reference measure, for the classical stochastic optimization problems of utility maximization and optimal stopping.     

部分信息下资产收益率发生紊乱的最优投资策略

本文研究了在部分信息且市场利率非零的情形下,资产预期收益率发生紊乱(disorder)时,终端净财富的期望指数效用最大化问题.利用半鞅和倒向随机微分方程(BSDE)刻画价值过程的方法,获得了最优交易策略和价值过程的明确表达式,推广了一般框架下最优投资组合的研究结果.     

BSDEs in utility maximization with BMO market price of risk

This article studies quadratic semimartingale BSDEs arising in power utility maximization when the market price of risk is of BMO type. In a Brownian setting we provide a necessary and sufficient condition for the existence of a solution but show that uniqueness fails to hold in the sense that there exists a continuum of distinct square-integrable solutions. This feature occurs since, contrary to the classical Itô representation theorem, a representation of random variables in terms of stochastic exponentials is not unique. We study in detail when the BSDE has a bounded solution and derive a new dynamic exponential moments condition which is shown to be the minimal sufficient condition in a general filtration. The main results are complemented by several interesting examples which illustrate their sharpness as well as important properties of the utility maximization BSDE.     

Portfolio optimization for a large investor under partial information and price impact

This paper studies portfolio optimization problems in a market with partial information and price impact. We consider a large investor with an objective of expected utility maximization from terminal wealth. The drift of the underlying price process is modeled as a diffusion affected by a continuous-time Markov chain and the actions of the large investor. Using the stochastic filtering theory, we reduce the optimal control problem under partial information to the one with complete observation. For logarithmic and power utility cases we solve the utility maximization problem explicitly and we obtain optimal investment strategies in the feedback form. We compare the value functions to those for the case without price impact in Bäuerle and Rieder (IEEE Trans Autom Control 49(3):442–447, 2004) and Bäuerle and Rieder (J Appl Prob 362–378, 2005). It turns out that the investor would be better off due to the presence of a price impact both in complete-information and partial-information settings. Moreover, the presence of the price impact results in a shift, which depends on the distance to final time and on the state of the filter, on the optimal control strategy.     

Risk aversion asymptotics for power utility maximization

We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to infinity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models. The limits are related to exponential and logarithmic utility. To derive these results, we combine approaches from optimal control, convex analysis and backward stochastic differential equations (BSDEs).     

随机通胀风险下基于半鞅理论的最优投资策略

本文在半鞅理论框架下,构建包括可交易风险资产、不可交易风险资产和未定权益的金融投资模型。在考虑随机通胀风险和获取部分市场信息的情形下,研究投资经理人终端真实净财富指数效用最大化问题。运用滤波理论、半鞅和倒向随机微分方程(BSDE)理论,求解带有随机通胀风险的最优投资策略和价值过程精确解。数值分析结果发现,可交易风险资产最优投资额随着预期通胀率的增加而减少,投资价值呈先增后减态势。当通胀波动率无限接近可交易风险资产名义价格波动率时,通胀风险可完全对冲,投资人会不断追加在可交易风险资产的投资额,以期实现终端真实净财富期望指数效用最大化。研究结果为金融市场的投资决策提供更加科学的理论参考。     

A maximum principle for Markov regime-switching forward–backward stochastic differential games and applications

In this paper, we present an optimal control problem for stochastic differential games under Markov regime-switching forward–backward stochastic differential equations with jumps. First, we prove a sufficient maximum principle for nonzero-sum stochastic differential games problems and obtain equilibrium point for such games. Second, we prove an equivalent maximum principle for nonzero-sum stochastic differential games. The zero-sum stochastic differential games equivalent maximum principle is then obtained as a corollary. We apply the obtained results to study a problem of robust utility maximization under a relative entropy penalty and to find optimal investment of an insurance firm under model uncertainty.     

跳扩散模型下的非零和随机微分投资组合博弈

现实经济中,当股票价格受到一些重大信息影响而发生突发性的跳跃时,用跳扩散过程来描述股票价格的趋势更符合实际情况。基于这一观察,本文研究跳扩散模型下包含两个投资者的非零和投资组合博弈问题。假设金融市场中包含一种无风险资产和一种风险资产,其中风险资产的价格动态用跳扩散模型来描述。将该非零和博弈问题构造成两个效用最大化问题,每个投资者的目标是最大化终端时刻自身财富与其竞争对手财富差的均值-方差效用。运用随机控制理论,得到了均衡投资策略以及相应值函数的解析表达。最后通过数值仿真算例分析了模型相关参数变动对均衡投资策略的影响。仿真结果显示:当股价发生不连续跳跃,投资者在构造投资策略时考虑跳跃风险可以显著增加其效用水平;同时,随着博弈竞争的加剧,投资者为了在竞争中取得更好的表现,往往会采取更加激进的投资策略,增加对风险资产的投资。     

Two-agent Pareto optimal cooperative investment in general semimartingale model

We consider the problem of expected utility maximization for the two-agent case in general semimartingale model. For this case a cooperative investment game is posed as follows: firstly collect both agents' capital as a whole at the initial time, and invest it in a trading strategy. Then at some time T ₀ one agent quits cooperation and terminates investment, so they divide the wealth and each of them gets a part. During the time interval [T ₀, T], the other agent invests her capital in a new trading strategy herself. By stochastic optimization methods with the help of the theory of backward stochastic differential equations, we give a characterization of Pareto optimal cooperative strategies and a characterization of situations where cooperation strictly Pareto dominates non-cooperation in our model.     

Asymptotic analysis of utility-based hedging strategies for small number of contingent claims

We study the linear approximation of utility-based hedging strategies for small number of contingent claims. We show that this approximation is actually a mean-variance hedging strategy under an appropriate choice of a numéraire and a risk-neutral probability. In contrast to previous studies, we work in the general framework of a semimartingale financial model and a utility function defined on the positive real line.     

Optimal reinsurance and investment with unobservable claim size and intensity

We consider the optimal reinsurance and investment problem in an unobservable Markov-modulated compound Poisson risk model, where the intensity and jump size distribution are not known but have to be inferred from the observations of claim arrivals. Using a recently developed result from filtering theory, we reduce the partially observable control problem to an equivalent problem with complete observations. Then using stochastic control theory, we get the closed form expressions of the optimal strategies which maximize the expected exponential utility of terminal wealth. In particular, we investigate the effect of the safety loading and the unobservable factors on the optimal reinsurance strategies. With the help of a generalized Hamilton–Jacobi–Bellman equation where the derivative is replaced by Clarke’s generalized gradient as in Bäuerle and Rieder (2007), we characterize the value function, which helps us verify that the strategies we constructed are optimal.     

A control approach to robust utility maximization with logarithmic utility and time-consistent penalties

We propose a stochastic control approach to the dynamic maximization of robust utility functionals that are defined in terms of logarithmic utility and a dynamically consistent convex risk measure. The underlying market is modeled by a diffusion process whose coefficients are driven by an external stochastic factor process. In particular, the market model is incomplete. Our main results give conditions on the minimal penalty function of the robust utility functional under which the value function of our problem can be identified with the unique classical solution of a quasilinear PDE within a class of functions satisfying certain growth conditions. The fact that we obtain classical solutions rather than viscosity solutions facilitates the use of numerical algorithms, whose applicability is demonstrated in examples.     

Heston随机波动率模型下带负债的投资组合博弈

本文研究了Heston随机波动模型下两个投资人之间的随机微分投资组合博弈问题。假设金融市场上存在价格过程服从常微分方程的无风险资产和价格过程服从Heston随机波动率模型的风险资产。该博弈问题被构造成两个效用最大化问题,每个投资者的目标是最大化终止时刻个人财富与竞争对手财富差的效用。首先,我们应用动态规划原理,得出了相应值函数所满足的HJB方程。然后,得到了在幂期望效用框架下非零和博弈的均衡投资策略和值函数的显式表达。最后,借助数值模拟,分析了模型中的参数对均衡投资策略和值函数的影响,从而为资产负债管理提供一定的理论指导。     

Optimal control of Markovian jump processes with partial information and applications to a parallel queueing model

We consider a stochastic control problem over an infinite horizon where the state process is influenced by an unobservable environment process. In particular, the Hidden-Markov-model and the Bayesian model are included. This model under partial information is transformed into an equivalent one with complete information by using the well-known filter technique. In particular, the optimal controls and the value functions of the original and the transformed problem are the same. An explicit representation of the filter process which is a piecewise-deterministic process, is also given. Then we propose two solution techniques for the transformed model. First, a generalized verification technique (with a generalized Hamilton–Jacobi–Bellman equation) is formulated where the strict differentiability of the value function is weaken to local Lipschitz continuity. Second, we present a discrete-time Markovian decision model by which we are able to compute an optimal control of our given problem. In this context we are also able to state a general existence result for optimal controls. The power of both solution techniques is finally demonstrated for a parallel queueing model with unknown service rates. In particular, the filter process is discussed in detail, the value function is explicitly computed and the optimal control is completely characterized in the symmetric case.     

Optimal Consumption and Portfolio with Ambiguity to Markovian Switching

This paper considers the problem of maximizing expected utility from consumption and terminal wealth under model uncertainty for a general semimartingale market, where the agent with an initial capital and a random endowment can invest. To find a solution to the investment problem we use the martingale method. We first prove that under appropriate assumptions a unique solution to the investment problem exists. Then we deduce that the value functions of primal problem and dual problem are convex conjugate functions. Furthermore we consider a diffusion-jump-model where the coefficients depend on the state of a Markov chain and the investor is ambiguity to the intensity of the underlying Poisson process. Finally, for an agent with the logarithmic utility function, we use the stochastic control method to derive the Hamilton-Jacobi-Bellmann (HJB) equation. And the solution to this HJB equation can be determined numerically. We also show how thereby the optimal investment strategy can be computed.     

Stochastic utilities with subsistence and satiation: Optimal life insurance purchase,consumption and investment

We introduce stochastic utilities such that utility of any fixed amount of interest is a stochastic process or random variable. Also, there exist stochastic (or random) subsistence and satiation levels associated with stochastic utilities. Then, we consider optimal consumption, life insurance purchase and investment strategies to maximize the expected utility of consumption, bequest and pension with respect to stochastic utilities. We use the martingale approach to solve the optimization problem in two steps. First, we solve the optimization problem with an equality constraint which requires that the present value of consumption, bequest and pension is equal to the present value of initial wealth and income stream. Second, if the optimization problem is feasible, we obtain the explicit representations of the replicating life insurance purchase and portfolio strategies. As an application of our general results, we consider a family of stochastic utilities which have hyperbolic absolute risk aversion (HARA).     

Super-replication and utility maximization in large financial markets

We study the problems of super-replication and utility maximization from terminal wealth in a semimartingale model with countably many assets. After introducing a suitable definition of admissible strategy, we characterize superreplicable contingent claims in terms of martingale measures. Utility maximization problems are then studied with the convex duality method, and we extend finite-dimensional results to this setting. The existence of an optimizer is proved in a suitable class of generalized strategies: this class has also the property that maximal expected utility is the limit of maximal expected utilities in finite-dimensional submarkets. Finally, we illustrate our results with some examples in infinite dimensional factor models.