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.
From: Roger Laeven [view email]
[v1]
Wed, 15 May 2024 11:22:06 UTC (42 KB)
[v2]
Thu, 18 Jul 2024 09:07:22 UTC (42 KB)
[v3]
Tue, 9 Sep 2025 11:01:04 UTC (47 KB)
[v4]
Tue, 18 Aug 2026 07:36:00 UTC (49 KB)
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