Geometric BSDEs

Roger Laeven, Emanuela Rosazza Gianin and Marco Zullino

Papers from arXiv.org

Abstract: We introduce Geometric Backward Stochastic Differential Equations (GBSDEs) and two-driver BSDEs, which arise naturally in the geometric dynamics of dynamic return risk measures and of recursive portfolio choice. Through a reduction to auxiliary ordinary BSDEs with logarithmic and singular quadratic (LN-Q) growth rate $y|\ln (y)|+|z|^2/y$, we establish existence, regularity, uniqueness and stability of solutions, under both bounded and unbounded driver coefficients and terminal conditions, and we transfer these results to the original two-driver equations. We then deploy the theory in two applications. We solve a portfolio optimization problem under stochastic differential utility, in which the opportunity process satisfies an endogenously derived two-driver BSDE and optimality is established via our two-driver comparison theorem. We further apply GBSDEs to dynamic return and star-shaped risk measures, including (robust) $L^p$-norms, and characterize their positive homogeneity, star-shapedness and multiplicative convexity.

Date: 2024-05, Revised 2026-08
New Economics Papers: this item is included in nep-rmg
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