Martingale

The Martingale Sinkhorn Algorithm ArXiv ID: 2310.13797 “View on arXiv” Authors: Unknown Abstract We develop a numerical method for the martingale analogue of the Benamou-Brenier optimal transport problem, which seeks a martingale interpolating two prescribed marginals which is closest to the Brownian motion. Recent contributions have established existence and uniqueness for the optimal martingale under finite second moment assumptions on the marginals, but numerical methods exist only in the one-dimensional setting. We introduce an iterative scheme, a martingale analogue of the celebrated Sinkhorn algorithm, and prove its convergence in arbitrary dimension under minimal assumptions. In particular, we show that convergence holds when the marginals have finite moments of order $p > 1$, thereby extending the known theory beyond the finite-second-moment regime. The proof relies on a strict descent property for the dual value of the martingale Benamou–Brenier problem. While the descent property admits a direct verification in the case of compactly supported marginals, obtaining uniform control on the iterates without assuming compact support is substantially more delicate and constitutes the main technical challenge. ...

A greedy algorithm for habit formation under multiplicative utility ArXiv ID: 2305.04748 “View on arXiv” Authors: Unknown Abstract We consider the problem of optimizing lifetime consumption under a habit formation model, both with and without an exogenous pension. Unlike much of the existing literature, we apply a power utility to the ratio of consumption to habit, rather than to their difference. The martingale/duality method becomes intractable in this setting, so we develop a greedy version of this method that is solvable using Monte Carlo simulation. We investigate the behaviour of the greedy solution, and explore what parameter values make the greedy solution a good approximation to the optimal one. ...