Open quantum systems are of wide interest owing to their ubiquity and rich physical phenomena; however, the features that make the systems interesting also render their simulation challenging on near-term quantum hardware. Various algorithms for simulating open quantum systems on quantum hardware have been proposed. Except for variational algorithms, all previous proposals either require ancilla qubits to implement the non-unitary evolution or exploit hardware-specific noise channels. Here we propose an ancilla-free, non-variational dynamic quantum algorithm for simulating the jump stochastic Schrödinger equation (SSE) on near-term quantum hardware. In the most common unravelling, time evolution is generated from the SSE through a series of non-unitary continuous-time evolutions as well as probabilistic "quantum jumps," or discontinuous changes in the quantum state associated with Lindblad operators. The continuous non-unitary evolution can be implemented by the quantum imaginary time evolution algorithm. We implement the quantum jumps using mid-circuit measurements and sub-microsecond readout capability recently introduced on transmon-based IBM Quantum devices. Our approach enables the simulation of open quantum systems without variational optimization or any ancilla qubits.