You need a probability distribution in order to define/use a risk measure. When dealing with a ODE, I can imagine three cases that allow you to use a risk measure:

1) Your ODE solution is the expected value of an SDE solution (like in Feynman-Kac framework). So you come back from ODE to SDE and you obtain a distribution that makes sense from a risk perspective.

2) Your dynamics is deterministic but not uniquely determined - maybe there is an interval of possible initial conditions or your observation time is stochastic. In this case you could describe your system with a statistical mechanics ensemble and evaluate risk on the resulting distribution.

3) Your PDE is a Schroedinger Equation, so the solution is a wave function psi and you can perform a risk measure over the resulting probability distribution |psi|^2.

Otherwise, if your system is deterministic, the initial conditions are fully given and you just want to know the system state in a specific time in the future, there is no possible risk evaluation. Hope this helps.

A common scenario is that either the initial conditions are not known exactly, or the model description is imperfect. In order to estimate the probability distribution underlying the risk analysis you need to iterate/integrate you dynamical system which includes these errors and therefore producing an uncertain estimate (either an estimate of a future state characterised by a pdf or an ensemble of trajectories). This would certainly come up, for example, in weather or climate predictions. See (of example): Physica D, (2001) 151: 125-141; Physica D (2004) 196: 224-242; QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY (2007) 133: 1309-1325 .