Nonexistence and nonuniqueness for scalar BSDEs: answers to four open problems of Fan, Hu and Tang

Fan, Hu and Tang listed five open problems on one-dimensional backward stochastic differential equations. We answer four of them in the negative by explicit constructions. For the generator |z|/√log(e+|z|), integrability of the terminal value does not ensure a solution of class (D). For the generator |z| sin|z|, a terminal value with moments of all orders admits two distinct solutions with |Y|ᵖ of class (D) for every p ≥ 1. For a convex generator with 0 ≤ g(z) ≤ |z|²/2, one exponential moment of the terminal value does not ensure uniqueness among solutions with exp(|Y|) of class (D); the same example shows that the strong convexity assumption cannot be removed from the critical-case uniqueness theorem of Delbaen, Hu and Richou. For quadratic generators that are neither convex nor concave in z and satisfy a quadratic Lipschitz bound, terminal values with exponential moments of all orders do not ensure uniqueness among solutions with exp(μ|Y|) of class (D) for every μ > 0, even when the generator does not depend on y. The first example also answers, for solutions of class (D), the case n = 1 of a question of Fan, Hu and Tang on iterated-logarithmic growth conditions.