Filtration consistent nonlinear expectations and evaluations of contingent claims

We will study the following problem. Let Xt, t ∈ [0, T], be an Rd-valued process defined on a time interval t ∈ [0, T]. Let Y be a random value depending on the trajectory of X. Assume that, at each fixed time t ≤ T, the information available to an agent (an individual, a firm, or even a market) is the trajectory of X before t. Thus at time T , the random value of Y (ω) will become known to this agent. The question is: how will this agent evaluate Y at the time t? We will introduce an evaluation operator εt[Y ] to define the value of Y given by this agent at time t. This operator εt[·] assigns an (X s)0≤s≤T -dependent random variable Y to an (X s)0≤s≤t-dependent random variable εt[Y]. We will mainly treat the situation in which the process X is a solution of a SDE (see equation (3.1)) with the drift coefficient b and diffusion coefficient σ containing an unknown parameter θ = θt. We then consider the so called super evaluation when the agent is a seller of the asset Y . We will prove that such super evaluation is a filtration consistent nonlinear expectation. In some typical situations, we will prove that a filtration consistent nonlinear evaluation dominated by this super evaluation is a g-evaluation. We also consider the corresponding nonlinear Markovian situation. © Springer-Verlag 2004.