  This paper extends the results of the article [C. Kl\"{u}ppelberg and S. M.
Pergamenchtchikov. Optimal consumption and investment with bounded downside
risk for power utility functions. In Optimality and Risk: {\it Modern Trends in
Mathematical Finance. The Kabanov Festschrift}, pages 133-169, 2009] to a
jump-diffusion setting. We show that under the assumption that only positive
jumps in the asset prices are allowed, the explicit optimal strategy can be
found in the subset of admissible strategies satisfying the same risk
constraint as in the pure diffusion setting. When negative jumps probably
happen, the regulator should be more conservative. In that case, we suggest to
impose on the investor's portfolio a stricter constraint which depends on the
probability of having negative jumps in the assets during the whole considered
horizon.
