  We consider the phase retrieval problem in which one tries to reconstruct a
function from the modulus of its wavelet transform. We study the unicity and
stability of the reconstruction. In the case where the wavelets are Cauchy
wavelets, we prove that the modulus of the wavelet transform uniquely
determines the function up to a global phase. We show that the reconstruction
operator is continuous but not uniformly continuous. We describe how to
construct pairs of functions which are far away in $L^2$-norm but whose wavelet
transforms are very close, in modulus. The principle is to modulate the wavelet
transform of a fixed initial function by a phase which varies slowly in both
time and frequency. This construction seems to cover all the instabilities that
we observe in practice; we give a partial formal justification to this fact.
Finally, we describe an exact reconstruction algorithm and use it to
numerically confirm our analysis of the stability question.
