Application Of Stochastic Analysis In Financial Engineering

This paper is focused on two problems in financial economics.The first one is the optimal consumption/investment problem formulated by Merton (1971),while the second one is option pricing. These problems are unified by their reliance on the theory of stochastic differential equations to model the trading of risky securities in continuous time. We solve these two problems by applying the theory of stochastic calculus and differential equation to them. In the first problem, this theory allows us to characterize the value function and optimal consumption process in a context more general than considered hereofore. We subsequently specialize the model to the case of constant coefficients, so as to illustrate the use of the Hamilton-Jacobi-Bellman equation in Stochastic control.Then we use the conclusion gotten in the first problem to solve the option pricing problem and finally end up with the Black&Scholes option valuation formula.