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。

Non-zero-sum Stochastic Differential Portfolio Game under the Jump Diffusion Model

In this paper, we investigate a nonzero-sum portfolio game problem between two investors which aims at maximizing their mean-variance utility under the jump diffusion model. The financial market is assumed to consist of one risk-free asset and one risky asset whose price process is governed by the jump diffusion process. By introducing the relative performance concerns, we formulate the nonzero-sum game as two utility maximization problems. Each investor is assumed to maximize his mean-variance utility of the difference between his terminal wealth and that of his competitor. By applying the stochastic control theory, analytical expressions of the equilibrium investment strategies and equilibrium value functions are obtained. Finally, some numerical examples are performed to illustrate the influence of model parameters on the equilibrium investment strategies. The results indicate that consideration of jump risks improves the investor's utility; meanwhile, the relative performance concerns of the investor increase the amount invested in the risky asset, which implies that the competition would lead the investor to be much more risk-seeking.