Source-linked AI summary
Ambiguous Volatility, Possibility and Utility in Continuous Time
Larry Epstein, Shaolin Ji
TL;DR
The paper addresses how to model continuous-time utility when decision-makers are ambiguous about both drift and volatility, a setting where volatility ambiguity prevents reliance on a single probability space. It constructs a quasisure, dynamically conditioned utility framework for this setting and establishes existence and uniqueness of the utility process, while noting limits to strict dynamic consistency.
Problem
Continuous-time utility models typically restrict priors to equivalent measures and ambiguity to drift, whereas volatility ambiguity yields nonequivalent priors that disagree about possible scenarios.
Method
The paper constructs utility using the entire set of priors for almost-sure qualification, combining quasisure analysis with ambiguity about drift and volatility.
Results
Theorem 2.7 establishes a unique utility process solving the model’s defining equation and characterizes it as the unique solution satisfying the stated terminal condition.
Takeaways & Limitations
The framework provides a mathematically rigorous foundation for utility and equilibrium analysis in asset markets with ambiguous volatility.
Takeaways & Limitations
The recursive utility is not strictly dynamically consistent, and a general analysis of whether an optimal plan will be implemented remains to be done.
Abstract
from arXiv · showhide
This paper formulates a model of utility for a continuous time framework that captures the decision-maker's concern with ambiguity about both the drift and volatility of the driving process. At a technical level, the analysis requires a significant departure from existing continuous time modeling because it cannot be done within a probability space framework. This is because ambiguity about volatility leads invariably to a set of nonequivalent priors, that is, to priors that disagree about which scenarios are possible.
1. Introduction
The paper develops a continuous-time utility model for ambiguity about both drift and volatility. Because volatility ambiguity generates nonequivalent priors, it replaces probability-space methods with quasisure analysis and a broader framework for dynamic utility.
- The model extends continuous-time multiple-priors utility to ambiguity about both the drift and volatility of the driving process.
- Volatility ambiguity produces an undominated set of priors that may disagree about which scenarios are possible.
- The paper provides a mathematically rigorous treatment that supplies foundations for equilibrium analysis of asset markets with ambiguous volatility.
- The motivation includes complicated volatility dynamics, difficult empirical identification, and the possibility that confidence in one parametric specification is unwarranted.
- The analysis defines stochastic-process domains using almost-sure qualification under every prior, implementing quasisure stochastic analysis.
- The framework combines quasisure analysis with G-expectation and adapts existing approaches to conditioning undominated measures.
2. Utility
The paper constructs continuous-time recursive utility under ambiguity about drift and volatility, where nonequivalent priors require quasisure analysis rather than a single probability-space framework. It defines and conditions expectations over these priors, establishes well-posed recursive utility, and characterizes limits on dynamic consistency.
- Framework: Continuous paths on [0,T] form the canonical state space, with the coordinate process generating information through its filtration.
- Drift and volatility: The model allows multidimensional driving processes, ambiguity about drift, and unrestricted time variation in ambiguity about volatility.
- Drift and volatility: Robust stochastic volatility represents uncertainty between empirically comparable volatility models whose implications for choice or derivative pricing differ.
- Priors: When volatility is ambiguous, priors can be mutually singular; when both drift and volatility are ambiguous, nonequivalence generally prevails.
- Conditioning: Because no single prior defines null events, the paper uses quasisure analysis, treating equality as almost-sure equality under every prior.
- Conditioning: The conditional-expectation domain UCb(Ω) is insufficient because conditioning may fail to preserve uniform continuity, obstructing stochastic-process and recursive modeling.
- Conditioning: Conditional expectation extends uniquely and 1-Lipschitz continuously to c L2(Ω), satisfies the law of iterated expectations, and supports recursive utility.The extension also preserves monotonicity, measurability, and the stated nonlinear expectation properties.
- Utility: The utility process exists uniquely and is recursive, but nonequivalent priors reduce dynamic consistency to a weaker form.Strict dynamic consistency can fail because the meaning of a non-negligible set is unclear across nonequivalent priors.
A. Appendix: Uniform Continuity
The appendix fixes the canonical and shifted path-space framework, constructs shifted processes and priors after concatenation, and imposes uniform continuity on the correspondence process.
- Canonical and shifted spaces: The appendix defines a shifted canonical space with a canonical process and a Brownian probability measure on that shifted space.The shifted construction supports analysis after conditioning at an intermediate time.
- Concatenation and shifting: For paths up to time s and continuation paths, concatenation at time t provides the path operation used to form shifted objects.Random variables are correspondingly shifted using the history ω at time t.
- Shifted correspondences: Given a process of correspondences Θ, the appendix defines a shifted correspondence process Θ^{t,ω} for each time-path pair.The shifted process inherits the required conditions, including the condition addressed next.
- Uniform continuity: Uniform continuity requires a positive tolerance ε(t,ω,δ) whenever two histories remain sufficiently close through time t.This condition is adapted from Nutz’s definition and applies to the correspondence process Θ.
- Prior-set notation: The appendix keeps Θ fixed throughout and uses P as shorthand for the associated prior set, with P0 defined as a subset of P.The displayed definition specifies P0 through an interior condition on θ relative to Θ.
B. Appendix: Conditioning
The appendix constructs conditional probabilities and continuation-measure sets for each prior, establishing a conditioning framework compatible with nonequivalent priors. It then uses these objects to characterize conditional expectations and support dynamic programming.
- Conditional priors: The family P(t, P) contains priors that agree with P on the history sigma-field Ft.This fixes the set of continuation priors used after conditioning under a given prior.
- Conditional expectation: Conditional expectation applies Bayesian conditioning to each prior and takes the upper envelope of the resulting expectations.This agrees with prior-by-prior updating in the Chen–Epstein framework, with suprema replacing their infima convention.
- Shifted continuation measures: The shifted measure P^{t,¯θ} represents continuation dynamics from state (t,ω) and belongs to the corresponding set P(t,ω).Independent increments under the reference measure connect shifted solutions with conditional continuation laws.
- Regular conditional probabilities: For each prior P, the paper constructs P^ω_t as a version of its regular conditional probability.The construction is verified through the defining measurability and conditional-expectation properties.
- Dynamic programming: The construction supports a dynamic programming principle for the model.The paper states that the proof adapts the technique used for Nutz’s dynamic programming result.
STEP 2
Step 2 establishes that simpler collections of priors are dense in the full prior sets under weak convergence. This density is used to transfer conditional-expectation results from the simpler collections to the full model.
- STEP 2: Weak convergence is the topology on Δ(Ω) induced by bounded continuous functions.The density statements in this step are formulated in this topology.
- STEP 2: P0 is dense in P, and P0(t,ω) is dense in P(t,ω) for every t and ω.These two density results cover both unconditional and conditional prior sets.
- STEP 2: Uniform interiority permits approximating admissible processes while retaining the required local admissibility conditions.The proof combines this property with standard stochastic-differential-equation approximation.
- STEP 2: The density results allow conditional-expectation identities proved for P0 and P0(t,ω) to extend to P and P(t,ω).The subsequent lemmas invoke density to establish the relevant conditional-expectation relations.
STEP 3
Step 3 extends the conditional-expectation operator from uniformly continuous bounded claims to completed L2 spaces. The resulting operator is uniquely defined and 1-Lipschitz, with lower semicontinuity and approximation arguments supplying the extension.
- STEP 3: The mapping ˆE[· | Ft] extends uniquely from UCb(Ω) to a 1-Lipschitz mapping from c L2(Ω) into c L2(tΩ).This is the stated extension result for conditional expectations.
- STEP 3: For ξ in UCb(Ω), the conditional expectation ˆE[ξ | Ft] is lower semicontinuous.The proof uses uniform continuity of ξ and stability of the prior sets under convergent histories.
- STEP 3: Bounded lower semicontinuous functions on tΩ belong to c L2(tΩ) through bounded continuous approximation and weak compactness arguments.The proof uses Polishness of tΩ, relative compactness of P, and convergence of squared approximation errors.
- STEP 3: The extension preserves the conditional-expectation structure through approximation by sequences of priors and L2 norm estimates.The proof derives the required identities and continuity bounds from priorwise conditional expectations.
- STEP 3: The extended operator satisfies the law of iterated expectations in the completed spaces.The proof establishes both inequality directions using classical iterated expectations and approximating continuation priors.
C. Appendix: Proofs for Utility
The appendix proves existence and uniqueness for the utility-related backward stochastic differential equation by showing that its associated mapping is a contraction. A backward-in-time argument extends the solution across the full horizon.
- C. Appendix: Proofs for Utility: The operator Λ maps M2(0,T) into M2(0,T).This regularity property is established from the aggregator’s growth bound and the square-integrability of the terminal and forcing terms.
- C. Appendix: Proofs for Utility: The approximation estimate for Λ uses the aggregator’s Lipschitz constant K and L = TK^2.The estimate combines Jensen’s inequality with the conditional-expectation result from Theorem 2.6.
- C. Appendix: Proofs for Utility: Λ is a contraction on M2(r1,T), yielding a unique solution on that interval.The contraction argument is applied locally before extending the solution backward.
- C. Appendix: Proofs for Utility: Because δ is independent of t, the local contraction argument can be iterated backward to obtain a unique solution in M2(0,T).The same argument is applied on successive time intervals across the full horizon.
- C. Appendix: Proofs for Utility: Uniqueness in part (b) follows from the contraction-mapping property established in part (a).The proof explicitly attributes uniqueness to the contraction result.