Stability conditions of continuous-time switched stochastic dynamical systems driven by a Brownian motion and a Markov modulated compound Poisson process are provided. The mode signal, which manages the transition between subsystems, is modeled as a Markov chain. The state variables of the switched stochastic system are subject to jumps of random size occurring at random instances. The intensity of the occurrences, as well as the size of these jumps are modulated by the mode signal. A comparison approach is employed to show the almost sure asymptotic stability of the zero solution. Finally, an illustrative numerical example is presented to demonstrate the efficacy of our results.