We optimized consumer/investor behaviour, subject to self-financing constraint by using stochastic dynamics system with jumps. Our aim in this article was to compare a stochastic optimization model with and without jumps in a self-financing portfolio model, for the risk averse investors. In this article, our contribution to the literature was to introduce an analytical solution of the utility maximizing model and to investigate the consequences of jumps and bequest to the economic system. A previous model by Gazioglu and Bastiyali-Hafavi (2010) used optimization, only with a Brownian motion during optimization. In this article, we introduced jump difussion to Brownian motion during optimization. We compared the model with and without jumps for the risk averse investors. Furthermore, as a form of wealth, we compared the results with and without bequest.