Full-Observation Case

Abstract

Our problem consists in deciding when to intervene on the fully-observable value of the guarantee or loan along its lifetime T (in the partial observation case, where the guarantor can only monitor the value of the guarantee through costly audits, the problem consists also in deciding when it is best to audit). First of all, we have the value of the guarantee at time t, y(t), modelled as a diffusionwith drift and diffusion parameters, α(y(t)t) and σ(y(t) t), respectively, and which depend on the underlying process. This means that when no control is exerted one has dy(t) = α(y(t), t)dt+σ(y(t), t)dz(t), where z(t) denotes the standard Wiener process. When control can be exerted through the parameters, we obtain for the controlled diffusionthe model

$$ d_y \left( t \right) = \alpha \left( {y\left( t \right),e\left( t \right),t} \right)dt + \sigma \left( {y\left( t \right),e\left( t \right),t} \right)dz\left( t \right) $$

where e(t) is an admissible control law, that is one which is non-anticipative with respect to the process z(t). Notice that unless σ(y,t) ≡ 0, realizations of the process y(t) cannot force to be non-negative with this model. To make sure that y(t) is always positive, we can choose a Geometric Brownian Motion with reflection at the barrier y = 0, as a model for the assets’ value process. However, it is also possible to have negative values for y(t) with arbitrarily small probability, by an appropriate selection of the parameters μand σ.