This paper addresses implementation of IIR digital filters using stochastic computing. Stochastic computing requires fewer logic gates and is inherently fault-tolerant. Thus, these structures are well suited for deep sub-micron technologies. While it is easy to realize FIR digital filters using stochastic computing, implementation of IIR digital filters is non-trivial. Stochastic logic assumes independence of input signals; however, the feedback in IIR digital filters leads to correlation of input signals and the independence assumption is violated. The novelty of this paper lies in demonstrating that, despite the feedback in IIR filters, these filters can be implemented using stochastic logic. The key to stochastic implementation is selection of an IIR filter structure where the states are orthogonal and are, therefore, uncorrelated. Two architectures are presented for stochastic IIR digital filter. Both architectures are based on the lattice filter representation where the states are orthogonal. The first is based on a state-space description of the IIR filter derived from the lattice filter structure. The second is based on transforming the lattice IIR digital filter into an equivalent form that can exploit the novel scaling approach developed in our prior work for inner product computations. Our experimental results show that the two proposed architectures for stochastic IIR digital filters can lead to one to two orders of magnitude reduction in the output error-to-signal power ratio, compared to stochastic implementations using direct-form IIR filters. Furthermore, for higher-order filters, while stochastic direct-form structures fail to function correctly, the state-space and lattice based stochastic IIR digital filters always filter the input signals in a functionally correct manner.