In this thesis, we study a class of reaction-diffusion equations of the form $\partial u\partial t=Lu+\varphi u-1kuk+1$ where $L$ is the stochastic generator of a Markov process, $\varphi$ is a function of the space variables and $k\in R\setminus \{0\}$. An important example, in the case when $k>0$, is equations of the FKPP-type. We also give an example from the theory of utility maximisation problems when such equations arise and in this case . We introduce a new representation, for the solution of the equation, as the optimal value of an optimal control problem. We also give a second representation which can be seen as a dual problem to the first optimisation problem. We note that this is a new type of dual problem and we compare it to the standard Lagrangian dual formulation.

By choosing controls in the optimisation problems we obtain upper and lower bounds on the solution to the PDE. We use these bounds to study the speed of the wave front of the PDE in the case when $L$ is the generator of a suitable Lévy process.