Stability Estimating in Optimal Sequential Hypotheses Testing We study the stability of the classical optimal sequential probability ratio test based on independent identically distributed observations ₁,₂,\dots when testing two simple hypotheses about their common density f: f=₀ versus f=₁. As a functional to be minimized, it is used a weighted sum of the average (under ₀) sample number and the two types error probabilities. We prove that the problem is reduced to stopping time optimization for a ratio process generated by ₁,₂,\dots with the density ₀. For \ta_* being the corresponding optimal stopping time we consider a situation when this rule is applied for testing between ₀ and an alternative \tilde ₁, where \tilde ₁ is some approximation to ₁. An inequality is obtained which gives an upper bound for the expected cost excess, when \ta_* is used instead of the rule \tilde\ta_* optimal for the pair (₀,\tilde ₁). The inequality found also estimates the difference between the minimal expected costs for optimal tests corresponding to the pairs (₀,₁) and (₀,\tilde ₁).