SIAM Journal on Control and Optimization· 2026Q1
Uniqueness of Adapted Solutions to Scalar BSDEs with Peano-Type Generators
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- 2026year
Short summary
This paper proves the uniqueness of adapted solutions for scalar backward stochastic differential equations (BSDEs) with Peano-type generators when the terminal value is almost surely positive.
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Key points
- Proves uniqueness of adapted solutions for scalar BSDEs with Peano-type generators and a positive terminal value.
- Develops a method connecting the BSDE to an optimal stochastic control problem, identifying the solution's first component as the value process.
- Applies a change of variables to reduce the BSDE to a convex quadratic form for analytical treatment.
- Utilizes the [Formula: see text]-difference method to achieve a sharp result in a specific case, such as the Kreps–Porteus utility BSDE.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract. A backward stochastic differential equation (BSDE) with a Peano-type generator is known to have infinitely many solutions when the terminal value is vanishing, and it is shown to have possibly multiple solutions even when the terminal value is not vanishing but nonnegative. In this paper, we study the uniqueness of adapted solutions of such a BSDE when the terminal value is almost surely positive. Two methods are developed. The first one is to connect the BSDE to an optimal stochastic control problem: under suitable integrability of the terminal value, with a verification argument, we prove that the first component of the adapted solution pair is the value process for the optimal stochastic control problem. The second one appeals to a change of variables and is more inclined to analysis: by a change of variables, the original BSDE is reduced to a convex quadratic BSDE, and then using the [Formula: see text]-difference method, we give a sharp result in some special case, which includes the BSDE governing the well-known Kreps–Porteus utility.
The authors' abstract, as published at the source. SIAM Journal on Control and Optimization, 2026 · DOI ↗
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Field: Finance
FinanceEconomics, Econometrics and Finance