Consumption–investment problem with pathwise ambiguity under logarithmic utility

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.