Jumps in a stochastic optimization: Self-financing portfolio for risk-loving investors

We optimized consumer/ investor behaviour, subject to self-financing constraint by using  stochastic dynamics system with jumps. Our aim in this paper is to compare a stochastic optimization model with and without jumps in a self-financing Portfolio Model, for a risk lover investors. In this paper, our contributed to the literature is to introduce an analytical solution of the utility maximizing model and investigate the consequences of jumps and bequest to the economic system. A previous model by Gazioglu & Bastiyali-Hafavi (2010) used optimization, only with a Brownian motion during optimization. In this paper, we introduce jumps into a Poisson process with various intensities.  We compare the model with and without jumps for the risk lover investors.  Furthermore, as a form of wealth, we compare the results with inheritance (with bequest ) and without inheritance (bequest).