Let \((\xi_n)\) be a sequence of i.i.d. random variables such that \(\xi_n\geq-1\) and \(r\leq E[\xi_n] <\infty\), where \(\xi_n\) is the rate of return of a certain stock at time \(n\) \((r\) being the deterministic rate of return of a risk-free bond). Let \(x_n\) \((c_n)\) be the total capital (consumption) at time \(n\) where \(0\leq c_n\leq x_n\). At time \(n\), \(b_n(x_n-c_n)\) is invested into bonds, and \((1-b_n) (x_n-c_n)\) is invested into stocks where \(0\leq b_n\leq 1\). Then the recursion \[ x_{n+1}= \bigl[b_n(1+r) +(1-b_n) (1+\xi_n) \bigr](x_n- c_n),\;n\geq 0 \] holds where \(x_0=x\) is the initial capital. With consumption \(c_n\) a certain satisfaction \(g(c_n)\) is associated (for a given function \(g)\). A portfolio strategy is given by a sequence \(u=(b_n,c_n)\). Let \(0< \gamma<1\) be a given discount factor. The value function is given by \[ w(x)= \sup_uE^u_x \left[ \sum^\infty_{i=0} \gamma^ig(x_i) \right] \] (the supremum taken over all admissible strategies \(u)\). Let \({\mathcal W}\) denote the family of all continuous functions \(f:[0,\infty [\to R\) such that \(\sup_{x\geq 0}|f(x)|/(x+1) < \infty\). The authors show that if \(g\in {\mathcal W}\), then there exists a unique \(w\in {\mathcal W}\) satisfying the corresponding Bellman equation, and \(w\) coincides with the above value function for a certain optimal strategy. The authors also study the above problem in the case of proportional transaction costs for buying and selling stocks.