AdaBoost is a meta ensemble learning algorithm. The most important theoretical property behind it is "Boosting", which also plays an important role in cost sensitive learning. However, available cost sensitive Boosting algorithms, such as AdaCost, AdaC1, AdaC2, AdaC3, CSB0, CSB1 and CSB2, are just heuristic. They add cost parameters into voting weight calculation formula or sample weights updating strategy of AdaBoost, so that the algorithms are forced to focus on samples with higher misclassification costs. However, these heuristic modifications have no theoretical foundations. The worst thing is that they break the most important theoretical property of AdaBoost, namely “Boosting”. Compared to AdaBoost which converges to optimal Bayes decision rule, those cost sensitive algorithms do not converge to cost sensitive decision rule. This paper studies the problem of designing cost sensitive Boosting algorithms strictly under Boosting theory. First, two new loss functions are constructed by making exponential loss and logit loss cost sensitive. It can be proved that the new loss functions are Fisher consistent in cost sensitive setting, therefore optimizing them finally leads to cost sensitive Bayes decision rule. Performing gradient decent in functional space to optimize these two loss functions then results in new cost sensitive Boosting algorithms: AsyB and AsyBL. Experimental results on synthetic Gaussian data prove that in comparison with other cost sensitive Boosting algorithms, AsyB and AsyBL always better approximate cost sensitive Bayes decision rule. Experimental results on UCI datasets further prove that AsyB and AsyBL generate better cost sensitive classifiers with lower misclassification costs and the misclassification costs decrease exponentially with iterations.