This paper presents contributions to the mathematical core of risk and uncertainty management in compliance with the principles of New Budgeting laid out in 2008 by the Danish Ministry of Transport to be used in large infrastructure projects. Basically, the new principles are proposed in order to prevent future budget overruns. One of the central ideas is to introduce improved risk management processes and the present paper addresses this particular issue. A relevant cost function in terms of unit prices and quantities is developed and an event impact matrix with uncertain impacts from independent uncertain risk events is used to calculate the total uncertain risk budget. Cost impacts from the individual risk events on the individual project activities are kept precisely track of in order to comply with the requirements of New Budgeting. Additionally, uncertain likelihoods for the occurrence of risk events are allowed for in order to adequately reflect the complexity of realistic budgeting in the presence of uncertainty. Uncertainties are modelled in accordance with the basic principles of possibility and represented by fuzzy intervals, corresponding to lack of knowledge. Application is demonstrated by means of a numerical test case and the cumulated distribution function of the uncertain total risk budget is calculated.