Summary: This paper illustrates the natural role that Hellinger processes can play in solving problems from finance. We propose an extension of the concept of Hellinger process applicable to entropy distance and \(f\)-divergence distances, where \(f\) is a convex logarithmic function or a convex power function with general order \(q,0\neq q<1\). These concepts lead to a new approach to Merton’s optimal portfolio problem and its dual in general Lévy markets.