[Paper Review] Backward stochastic differential equations under super linear G-expectation and associated Hamilton-Jacobi-Bellman equations

Yuhong Xu|arXiv (Cornell University)|Sep 6, 2010

Stochastic processes and financial applications27 references3 citations

TL;DR

This paper establishes the existence and uniqueness of backward stochastic differential equations (BSDEs) under super linear G-expectation, providing a probabilistic representation for viscosity solutions of Hamilton-Jacobi-Bellman (HJB) equations, including the Black-Scholes-Barenblatt equation. It further links G-BSDEs to stochastic control problems via mutually singular probability measures, offering a new Feynman-Kac-type connection in uncertainty-driven models.

ABSTRACT

This paper first studies super linear G-expectation. Uniqueness and existence theorem for backward stochastic differential equations (BSDEs) under super linear expectation is established to provide probabilistic interpretation for the viscosity solution of a class of Hamilton-Jacobi-Bellman equations, including the well known Black-Scholes-Barrenblett equation, arising in the uncertainty volatility model in mathematical finance. We also show that BSDEs under super linear expectation could characterize a class of stochastic control problems. A direct connection between recursive super (sub) strategies with mutually singular probability measures and classical stochastic control problems is provided. By this result we give representation for solutions of Black-Scholes-Barrenblett equations and G-heat equations.

Motivation & Objective

• To develop a probabilistic framework for solving Hamilton-Jacobi-Bellman (HJB) equations under model uncertainty using backward stochastic differential equations (BSDEs) under super linear G-expectation.
• To clarify the connection between super linear G-expectation and stochastic control problems, particularly through mutually singular probability measures.
• To provide a new Feynman-Kac representation for solutions of the Black-Scholes-Barenblatt equation and G-heat equations using G-BSDEs.
• To investigate the properties of G-Brownian motion and G-normal distribution under super linear expectation, especially regarding quasi-sure differentiability of quadratic variation.
• To demonstrate that super linear G-expectation serves as a complementary tool to sublinear G-expectation, enhancing understanding of nonlinear expectations and uncertainty quantification.

Proposed method

• Introduces super linear G-expectation as the pointwise infimum over a set of risk-neutral probabilities: $\mathbb{E}_* [\cdot] = \inf_{P \in \mathcal{P}} \mathbb{E}_P [\cdot]$.
• Defines backward stochastic differential equations (G-BSDEs) driven by a forward diffusion process $X_t^{t,x}$ with drift, diffusion, and quadratic variation components.
• Establishes existence and uniqueness of solutions to G-BSDEs under super linear expectation using capacity-based convergence and dominated convergence theorems.
• Applies the dynamic programming principle to link G-BSDEs to the value function of stochastic control problems.
• Uses the representation $\mathbb{E}_* [X] = \inf_P \mathbb{E}_P [X]$ to interpret super strategies as optimal control problems over mutually singular measures.
• Applies the Borel-Cantelli lemma and capacity convergence to prove convergence of expectations under super linear G-expectation.

Experimental results

Research questions

• RQ1 Can backward stochastic differential equations under super linear G-expectation provide a probabilistic solution for a class of second-order HJB equations?
• RQ2 How is the super linear G-expectation related to stochastic control problems with mutually singular probability measures?
• RQ3 What is the role of G-Brownian motion and its quadratic variation in the context of super linear expectation?
• RQ4 Does the dominated convergence theorem hold under super linear G-expectation when convergence is quasi-sure but not in capacity?
• RQ5 Can the Feynman-Kac formula be extended to non-dominated nonlinear expectations, particularly for the Black-Scholes-Barenblatt equation?

Key findings

• Existence and uniqueness of solutions to backward stochastic differential equations (G-BSDEs) are established under super linear G-expectation, extending classical probabilistic methods to nonlinear, model-uncertain settings.
• The solution of the G-BSDE provides a viscosity solution to a class of Hamilton-Jacobi-Bellman equations, including the Black-Scholes-Barenblatt equation, under uncertainty volatility models.
• The quadratic variation $\langle B \rangle_t$ of G-Brownian motion is shown to be differentiable quasi-surely for each $t$, a key property for stochastic calculus under uncertainty.
• A direct equivalence is established between super linear G-expectation, the value function of a stochastic control problem, and the solution of a HJB equation, revealing a new probabilistic interpretation.
• The counterexample shows that quasi-sure convergence does not imply convergence of expectations under $\mathbb{E}^*$, and that $\lim \mathbb{E}^*[X_\delta^t] \neq \mathbb{E}^*[\lim X_\delta^t]$, invalidating the dominated convergence theorem in this context.
• The paper proves that $\mathbb{E}^*$-Brownian motion is a standard Brownian motion under $\mathbb{E}_*$, and that $\mathbb{E}_*$ is not dominated by itself but by the associated sublinear expectation $\mathbb{E}^* = -\mathbb{E}_*(-\cdot)$.