  We establish existence of Predictable Forward Performance Processes (PFPPs)
in complete markets, which has been previously shown only in the binomial
setting. Our market model can be a discrete-time or a continuous-time model,
and the investment horizon can be finite or infinite. We show that the main
step in construction of PFPPs is solving a one-period problem involving an
integral equation, which is the counterpart of the functional equation found in
the binomial case. Although this integral equation has been partially studied
in the existing literature, we provide a new solution method using the Fourier
transform for tempered distributions. We also provide closed-form solutions for
PFPPs with inverse marginal functions that are completely monotonic and
establish uniqueness of PFPPs within this class. We apply our results to two
special cases. The first one is the binomial market and is included to relate
our work to the existing literature. The second example considers a generalized
Black-Scholes model which, to the best of our knowledge, is a new result.
