  We study the model theory of finitely ramified henselian valued fields of
fixed initial ramification, obtaining versions of the Ax-Kochen-Ershov
principle as follows. We identify the induced structure on the residue field
and show that once the residue field is endowed with this structure, the theory
of the valued field is determined by the theories of the enriched residue field
and the value group. Similarly, we show that the existential theory of the
valued field is determined by the positive existential theory of the enriched
residue field. We also prove that an embedding of finitely ramified henselian
valued fields is existentially closed as soon as the induced embeddings of
value group and residue field are existentially closed. This last result
requires no enrichment of the residue field, in analogy to the corresponding
result for model completeness, which holds by results of Ershov and Ziegler.
