Consumption–investment problem with pathwise ambiguity under logarithmic utility
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For an investor with intertemporal information about risky assets, we propose a set of càdlàg confidence paths to describe his ambiguity about drift, volatility, and jump of the risky assets. For each possible model, the differential characteristic of log-return processes is a stochastic process and almost surely takes value in the set of confidence paths. Under the framework of the robust consumption–investment problem for logarithmic utility, we prove that a worst-case confidence path exists and can generate a worst-case model and deduce an optimal strategy. A deterministic-to-stochastic paradigm is established and extends the classical martingale method to robust optimization for jump-diffusion model. In numerical analyses, we take a joint ambiguity with two-point jumps as an example to reveal the rule of choosing the worst-case model and the impact of the ambiguity on the optimal strategy.
KeywordsConsumption–investment problem Deterministic-to-stochastic paradigm Càdlàg confidence path Worst-case model Saddle point
Mathematics Subject Classification90C47 93E20 91B28
JEL ClassificationsG11 C61
The authors acknowledge the support from the National Natural Science Foundation of China (Grant Nos. 11471183 and 11871036). The authors also thank the members of the group of Insurance Economics and Mathematical Finance at the Department of Mathematical Sciences, Tsinghua University, for their feedbacks and useful conversations.
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