Stochastic models for broker inventory in dealership markets with a cash management interpretation

We study the problem of a broker in a dealership market whose buffer content (cash flow) is governed by stochastic price-dependent demand and supply. Three model variants are considered. In the first model, buyers and sellers (borrowers and depositors) arrive independently in accordance with price-dependent compound Poisson streams. The second and the third models are two variants of diffusion approximations. For a certain natural revenue function, taking into account the trade-off between holding and shortage costs (opportunity loss and interest rates), we define relevant cost functionals that lay the groundwork for optimization purposes. The approach in analyzing and computing the cost functionals is based on the optional sampling theorem applied to a certain martingale.