The theory of dual control was introduced more than seven decades ago. Although it has provided rich insights to the fields of control, estimation, and system identification, dual control is generally computationally prohibitive. In recent years, however, the use of Koopman operator theory for control applications has been emerging. This paper presents a new reformulation of the stochastic optimal control problem that, employing the Koopman operator, yields a standard LQR problem with the dual control as its solution. We provide a numerical example that demonstrates the effectiveness of the proposed approach compared with certainty equivalence control, when applied to systems with varying observability.