Stochastic Processes: Applications in Finance and Insurance

Martingales in Finance

Let us consider a continuous time arbitrage free financial market with one risk-free investment (bond) and one risky asset (stock). All processes are assumed to be defined on the complete probability space $(\Omega ,{\mathcal{F}}_{T},({\mathcal{F}}_{t}),P)$ and adapted to the filtration $({\mathcal{F}}_{t}),\ \ t \leq T.$ The bond yields a constant rate of return r ≥ 0 over each time period. The risk-free bond represents an accumulation factor and its price process B equals

$$d{B}_{t} = r{B}_{t}dt,\ t \in [0,T],\ {B}_{0} = 1,$$

(1)

or B _t = e ^rt. The evolution of the stock price S _t is described by the linear stochastic differential equation

$$d{S}_{t} = {S}_{t}(\mu dt + \sigma d{W}_{t}),\ t \in [0,T],\ {S}_{0} = S,$$

(2)

where the expected rate of return μ and the volatility coefficient σare constants....