Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities

Robust Utility Maximization With Lévy Processes

“…Recently, Biagini and Pinar () also construct a saddle point in a setting where the uncertainty in the drift may depend on the realization of the volatility in a specific way. Finally, Fouque, Pun, and Wong () consider a stochastic volatility model with uncertain correlation (but known drift) and describe an asymptotic closed‐form solution. In this paper, our main contribution is to exhibit and solve a problem that includes uncertainty about fairly general models while remaining very tractable.…”

Non-zero-sum reinsurance games subject to ambiguous correlations

“…Each insurer has her own confidence interval for ρ, where the bounds could be different (different constraints sets), and she maximizes her expected utility of her relative terminal surplus with respect to that of her counterparty by choosing her proportional reinsurance protection under the worst-case scenario of ρ. We show that the associated reinsurance game with ambiguous correlation fits naturally into the two-dimensional G-Brownian motion framework that is first introduced in [11] and has been subsequently applied to other stochastic control problems, as shown in [5,7].…”

“…Analogous to [7,8], we have the following verification theorem. (6) is the unique deterministic continuous viscosity solution of the following Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation:…”

“…Although the Nash equilibrium in Section 3 also includes the case when insurer k is competitive (κ k > 0) and insurer m is cooperative (κ m < 0), for k = m ∈ {1, 2}, we choose not to study this case for there is no clear economic rationale on the establishment of a game between one competitive and one cooperative insurers. See [2,6,14] for the optimization under the relative performance concerns when κ k ∈ [0, 1]; and [7] for the maximin formulation of the robust portfolio optimization with ambiguous correlation.…”

“…However, it is important to note that W (k) (t) and B (k) (t) are not equivalent.Replacing the original Brownian motionW (k) (t) with G-Brownian motion B (k) (t) in (4), the dynamics ofX q k ,qm k (t) is given by dX q k ,qm k (t) = [δ k + µ k q k (t) − κ k µ m q m (t)]dt + (σ k q k (t), −κ k σ m q m (t))d B (k) (t)(5)withX q k ,qm k (0) =x k . As in[5,7], the worst-case utility function is defined asU t,x k ,q k k := − E k [−U k (X q k ,qm k (T ))|F (k) t ] = inf Pρ∈P Θ k E Pρ [U k (X q k ,qm k (T ))|F (k) t ], where F (k) t = σ({ B (k) (s)} t s=0 ). The admissible set of the insurer k's reinsurance strategies isQ k = {q k (t) ∈ F (k)t |q k (t) ∈ [0, 1]}.…”