Continuous-time limits of multi-period cost-of-capital valuation

We consider multi-period cost-of-capital valuation of a liability cashflow subject to repeated capital requirements that are partly financed by capital injections from capital providers with limited liability. Limited liability means that, in any given period, the capital provider is not liable for further payment in the event that the capital provided at the beginning of the period turns out to be insufficient to cover both the current-period payments and the updated value of the remaining cash flow. The liability cash flow is modeled as a continuous-time stochastic process on [0, T]. The multi-period structure is given by apartition of [0, T] into subintervals, and on the corresponding finite set of times a discrete-time value process is defined. Our main objectiveis the analysis of existence and properties of continuous-time limits of discrete-time value processes corresponding to a sequence of partitions whose meshes tend to zero. Moreover, we provide explicit and interpretable valuation formulas for a wide class of cash flow models