Robust utility maximization of terminal wealth with drift and volatility uncertainty

ABSTRACT

We give explicit solutions for utility maximization of terminal wealth problem $u\left(X_T\right)$ in the presence of Knightian uncertainty $\underset{\pi \in {\mathrm{\Pi }}_{[0,T]}^{\mathrm{a}\mathrm{d}}}{sup}\underset{\theta \in {\mathrm{\Theta }}_{[0,T]}}{inf}\mathbb{E}\left[u\right(X_T^{\pi ,\theta }\left)\right]$ in continuous time $[0,T]$. We assume there is uncertainty on both drift and volatility of the underlying stocks, which induce nonequivalent measures on canonical space of continuous paths Ω. We take that the uncertainty set resides in compact sets that are time dependent. In this framework, we solve the robust optimization problem with logarithmic, power and exponential utility functions, explicitly. Numerical simulations revealing the effects of uncertainty on the dynamics are also presented.

Keywords:

• Knightian uncertainty
• mathematical finance
• optimal control

2010 Mathematics Subject Classifications:

• 91B16
• 93E20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Nazarbayev University.