Abstract
Under the hypotheses that the risk reserve is governed by a Wiener process with not continuous income function, the Author proves an existence and uniqueness result in suitable Sobolev spaces for the weak solution of a differential equation involving the expectation of the discounted dividend payments, with boundary conditions related to the presence of a constant dividend barrier. Finally the Author considers the special case of a continuous income function, in which there exists an unique strong solution of the above problem.
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