[Paper Review] Choquet expectations and g-expectations with multi-dimensional Brownian motion

Mingshang Hu|ArXiv.org|Oct 14, 2009

Stochastic processes and financial applications13 references3 citations

TL;DR

This paper establishes that a g-expectation coincides with a Choquet expectation if and only if the generator g is independent of y and linear in z, i.e., g(t,z) = ∑ g(t,e_i)z_i, even in multi-dimensional Brownian motion settings without requiring continuity of g in t or assuming one-dimensional Brownian motion. The proof uses comonotonic indicator functions and properties of backward stochastic differential equations (BSDEs).

ABSTRACT

We prove that a g-expectation is a Choquet expectation if and only if g is independent of y and is linear in z, i.e., classical linear expectation, without the assumptions that the deterministic generator g is continuous in t and the dimension of the Brownian motion is one.

Motivation & Objective

• To determine the exact conditions under which a g-expectation coincides with a Choquet expectation in the context of multi-dimensional Brownian motion.
• To extend previous results—previously limited to one-dimensional Brownian motion and continuous g—by removing these restrictive assumptions.
• To resolve the challenge that components of multi-dimensional Brownian motion are not comonotonic, which invalidates direct extension of prior methods.
• To establish that g-expectation and Choquet expectation coincide precisely when g is linear in z and independent of y, regardless of dimension or continuity of g.

Proposed method

• Uses comonotonic indicator functions as test random variables to exploit the comonotonic additivity property of Choquet expectations.
• Applies the solution theory of backward stochastic differential equations (BSDEs) with Lipschitz generator g satisfying (H1)–(H3).
• Employs a limiting argument via sequences of BSDE solutions corresponding to indicator functions of events like {W_T^1 ≥ n} and {W_T^2 ≥ 0}.
• Uses the stability of BSDE solutions under L² convergence and the Lipschitz continuity of g to pass to limits and derive functional equations for g.
• Applies symmetry and transformation arguments (e.g., replacing W with (W^1, -W^2)) to extend linearity results to all orthant regions of z-space.
• Uses a recursive decomposition argument over dimensions, reducing the d-dimensional case to lower-dimensional cases via orthogonal projections.

Experimental results

Research questions

• RQ1 Under what conditions on the generator g does a g-expectation coincide with a Choquet expectation in multi-dimensional Brownian motion?
• RQ2 Can the result from the one-dimensional case—where g must be independent of y and linear in z—be extended to higher dimensions without continuity assumptions on g?
• RQ3 Why does the standard method based on comonotonicity fail in multi-dimensional settings, and how can it be adapted?
• RQ4 Is the linearity of g in z and independence from y necessary and sufficient for g-expectation to be a Choquet expectation in general d-dimensional settings?
• RQ5 Can the functional form of g be fully characterized when the g-expectation satisfies the comonotonic additivity of Choquet expectations?

Key findings

• A g-expectation is a Choquet expectation if and only if the generator g is independent of y and linear in z, i.e., g(t,z) = ∑_{i=1}^d g(t,e_i) z_i.
• This characterization holds without requiring continuity of g in t, even for multi-dimensional Brownian motion with d ≥ 2.
• The proof establishes that g must be additive and homogeneous in z over all orthants of R^d, leading to full linearity in z.
• The result is derived by constructing sequences of BSDE solutions corresponding to comonotonic indicator functions and using convergence and Lipschitz properties.
• The method overcomes the non-comonotonicity of components of multi-dimensional Brownian motion by using carefully chosen test functions and symmetry arguments.
• The final characterization shows that g(t,z) is a linear functional in z, with coefficients g(t,e_i) for each standard basis vector e_i.