Grant: Discovery Grants Program - Individual

In financial modelling it is assumed that the only information which can be used in decision making is present and past information.  Otherwise, investors would be able to make riskless profits with positive probability.  Mathematical models of financial markets are usually considered ill-posed if they allow for riskless profits with positive probability.   Nevertheless, we recognize that some information about the future behaviour of certain derivative security prices is available to all market participants and must be regarded as a piece of current information.  The simplest example is that the redemption value of a bond is known at issue.Forward-backward stochastic differential equations are an effective mathematical tool for incorporating present knowledge of, or constraints on, the future behaviour of an asset price in a rigorous fashion while still preserving the restriction that economic agents cannot use future information to make trades which result in riskless profits.  One set of equations models the evolution of the price of an asset (or factors of the economy effecting prices) forward in time based on present and past information. The second set of equations can be viewed as evolving backward in time from a random terminal condition, however, it is constrained to do so in a way that is predictable based on current and past information.  The research in this proposal applies forward-backward stochastic differential equations to various bonds, futures contracts and forward contracts.  The goal is to extend the existence and uniqueness results for these equations to a wider class of models that better model observed financial data.  We also plan to study the application of these equations to a wider class of derivative securities such as options on bonds, futures, and forwards.  We plan to extend numerical methods for forward-backward equations to a wider class of modelsrelated to the financial applications.