Robust Utility Maximization With Lévy Processes

Abstract

We study a robust portfolio optimization problem under model uncertainty for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets, that is, possible instantaneous drift, volatility, and jump characteristics of the price process. We show that an optimal investment strategy exists and compute it in semi-closed form. Moreover, we provide a saddle point analysis describing a worst-case model.

Constrained portfolio-consumption strategies with uncertain parameters and borrowing costs

“…In the following, we give some intuitive explanations of different consumption patterns. From the expression of f P and c * (t), we know that the optimal consumption c * t = c * (t) achieves the 2 In the case of c = 0, the results are similar to those in Table 2 except that cI 123 0 + c(t)I 12 123 +cI 4 12 and cI 23 0 + c(t)I 4 23 are replaced by c(t)I 12 0 + cI 4 12 and c(t), respectively. Note that when c = 0, since λ > 0, the last two rows about the optimal consumption are then irrelevant.…”

“…We obtain explicit solutions and properties of the optimal strategies. Although explicit solutions are derived under one risky asset setting, our method can be applied to study the multiple risky assets setting as in [23], and similar results will still hold, albeit with more complicated situations.…”

“…Since A and B are convex and compact, we may apply the minmax theorem (see Theorem B on pp. 131 in [25] or section 3 in [23]), and deduce that there exists a saddle point ( x * π , x * c ; x * µ , x * Σ ) such that (9) holds. Moreover, the compactness of A and B implies that…”

“…4 Explicit solutions of the optimal strategies and worst-case In this section, we further compute the saddle point (x * π ; x * µ , x * σ ) of the function g(·; ·, ·) given in (23). It then follows from Theorems 3.2 and 3.3 that the saddle point provides an explicit solution for the worst-case parameters and the optimal portfolio of the maxmin problem (5) Under Assumption 4.1, the worst-case parameters (µ * , σ * ) and the optimal portfolio π * are given as follows:…”

“…By the expression (27) and c * s = x * c,P (q P (s)), the parameters (R, π, π, µ, µ, σ, σ) will effect the optimal consumption through the channel of the opportunity process e qP (s) , which is the investor's extra utilities obtained by optimizing over all the admissible portfolio-consumption strategies (least affected by model uncertainty) in the remaining horizon [s, T ]. A closer look at the ODE (15) for q P (s) tells us that those parameters will only enter into the future investment contributing factor g(x * π ; x * µ , x * σ ) in (23). Increasing the borrowing cost R will make the future investment contributing factor g(x * π ; x * µ , x * σ ) smaller, so the opportunity process will also become smaller, i.e.…”