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Abstract. The issue of giving an explicit description of the flow of information concerning the time of bankruptcy of a company (or a state) arriving on the market is tackled by defining a bridge process starting from zero and conditioned to be equal to zero when the default occurs. This enables to catch some empirical facts on the behavior of financial markets: when the bridge process is away from zero, investors can be relatively sure that the default will not happen immediately. However, when the information process is close to zero, market agents should be aware of the risk of an imminent default. In this sense the bridge process leaks information concerning the default before it occurs. The objective of this first paper on Brownian bridges on stochastic intervals is to provide the basic properties of these processes.

We consider one-dimensional stochastic differential equations with generalized drift which involve the local time L X of the solution process:where b is a measurable real function, B is a Wiener process and ν denotes a set function which is defined on the bounded Borel sets of the real line R such that it is a finite signed measure on B([−N, N ]) for every N ∈ N. This kind of equation is, in dependence of using the right, the left or the symmetric local time, usually studied under the atom condition ν({x}) < 1/2, ν({x}) > −1/2 and |ν({x})| < 1, respectively. This condition allows to reduce an equation with generalized drift to an equation without drift and to derive conditions on existence and uniqueness of solutions from results for equations without drift. The main aim of the present note is to treat the cases ν({x}) ≥ 1/2, ν({x}) ≤ −1/2 and |ν({x})| ≥ 1, respectively, for some x ∈ R, and we give a complete description of the features of equations with generalized drift and their solutions in these cases.

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