The ergodic rate performance and limits of orthogonal frequency-division multiple access (OFDMA) cognitive radios (CRs) is studied under imperfect cross-link knowledge. We propose a novel stochastic interference management and exploitation technique to mitigate and control the imposed interference by CRs on licensed users in underlay spectrum sharing. The optimum downlink channel-adaptive resource allocation (RA) algorithm is designed to maximize the CRs functionality subject to the satisfying average transmit and probabilistic peak interference power constraints. An expression of the cumulative density function (cdf) of CRs' received signal-to-interference-plus-noise ratio (SINR) is developed to evaluate the resultant ergodic rate. Simulation studies are conducted to examine the proposed RA algorithm and investigate the impact of parameters uncertainty on the overall system performance.