Continuous-time stochastic analysis of optimal experimentation and of derivative asset pricing
This thesis applies continuous-time stochastic techniques to problems in economics of information and financial economics. The first part of the thesis uses non-linear filtering and stochastic control theory to study a continuous-time model of optimal experimentation by a monopolist who faces an unknown demand curve subject to random changes. It is shown that different probabilities of a demand curve switch can lead to qualitatively very different optimal behaviour. Moreover, the dependence of the optimal policy on these switching probabilities is discontinuous. This suggests that a market or an economy embedded in a changing environment may alter its behaviour dramatically if the volatility of the environment passes a critical threshold. The second part of the thesis studies continuous-time models of derivative asset pricing. First, a review of the so-called direct approach to debt option pricing emphasises the principal modeling problems of this approach and highlights the shortcomings of certain models proposed in the literature. Next, the connection between martingale measures and numeraire portfolios is exploited in problems of option pricing with strict upper and lower bounds on the underlying financial variable. This leads to a new decomposition of option prices in terms of exercise probabilities calculated under particular martingale measures and allows a simple proof of certain generalisations of the Black-Scholes option price formula. Finally, martingale methods are used to examine pricing formulae for general contingent claims, yielding a new method for inferring state prices from a given pricing formula. It is shown that if price processes are continuous semimartingales and the pricing formula is sufficiently regular, then the latter uniquely determines the risk-neutral law of the underlying asset price.