In a 1999 Comptes Rendus note, Lasry and Lions announced a program for partial-observation stochastic control in finance, deferring all detailed proofs to a later publication. The program has two parts. The first model converts an exponential-utility pricing and hedging problem with an unobserved drift factor into a full-information control problem whose state is the pair of observed asset price and the unnormalized conditional density governed by the controlled Zakai equation, leading to a formal infinite-dimensional Hamilton–Jacobi–Bellman (HJB) equation, an announced decomposition of the transformed value function into a payoff term and a nonpositive filtering term, and semi-explicit solutions with a feedback hedging rule in solvable cases. The second model makes information acquisition itself a control: the agent pays a running cost to decide whether to observe a payoff-relevant diffusion, and the note announces a threshold rule for this observation decision. The promised full paper with the deferred proofs never appeared, so the announced results have remained unproven, and the program has been realized only partially and independently of the note: control of partially observable systems via the Zakai equation was already treated by Bensoussan (1992); exponential-utility maximization under partial information was reduced to observable-process backward stochastic differential equations (BSDEs) by Mania and Santacroce (2010); and continuous-time control with costly, controllable information acquisition — the distinctive second model, whose observation filtration is endogenous — was only rigorously formulated in 2024–2025 by Knochenhauer, Merkel and Zhang and by Liang, Wang and Xia, in different models that do not cite the note. The surviving open core is a complete rigorous theory for the two announced models: well-posedness (including the endogenous-observation circularity), dynamic programming, viscosity-solution characterization of the infinite-dimensional HJB equations, verification of the announced feedback and threshold rules, and the value decomposition.

Background

Stochastic control problems with partial information (also called partial observation) arise when a decision maker controls a system whose state is only partially observed: the available data are a noise-contaminated signal — in the financial setting typically the asset prices — while economically relevant state variables, such as the instantaneous expected return (drift) of an asset or a factor driving the payoff, remain hidden. The standard route to such problems is filtering: the conditional law of the hidden state given the observations evolves as a Markov state in its own right, and its unnormalized conditional density νt\nu_t (the object whose integrals against test functions give conditional expectations up to a normalizing constant) solves the Zakai equation, a linear stochastic partial differential equation driven by the observations. Replacing the hidden state by its conditional law converts the partial-observation problem into a standard control problem on an infinite-dimensional state (observed price plus conditional density); this is the classical separated problem, developed rigorously in Bensoussan's monograph (1992), where the value function is characterized by a Hamilton–Jacobi–Bellman (HJB) equation — the nonlinear partial differential equation expressing Bellman's dynamic-programming principle — posed on the space of densities, for which viscosity solutions (the standard weak-solution notion for nonlinear PDEs, available also in infinite dimensions) provide existence and uniqueness results. For exponential utility U(z)=1−exp⁡(−λz)U(z)=1-\exp(-\lambda z), whose absolute risk aversion λ\lambda is constant, utility-based pricing and hedging take a multiplicative form: the utility indifference price is the compensation that leaves the agent's maximal expected utility unchanged between selling and not selling a contingent claim (a financial contract paying a random amount at maturity), and the optimal hedging strategy can be read off from the value function. In parallel, a substantial literature on Bayesian adaptive portfolio optimization (Lakner 1995, 1998; Brendle 2006, 2008) solved Merton-type (continuous-time portfolio-choice) investment problems with unobserved drift by duality and PDE methods, treating the filtered posterior mean of the drift as an observable state. In this context, Lasry and Lions (1999) announced, in an eight-page Comptes Rendus note, a program of partial-observation stochastic control with financial applications built on two models. In the first, the asset drift is driven by an unobserved factor and the payoff of the contingent claim also involves an independent random variable; using the controlled Zakai equation for νt\nu_t, the problem is rewritten as a full-information control problem in the state (St,νt)(S_t,\nu_t), leading to a formal infinite-dimensional HJB equation; a transform of the exponential-utility value function is announced to decompose into a payoff-randomness term and a filtering term (the latter nonpositive), with semi-explicit solutions and an optimal feedback hedge in a constant-hidden-drift case. In the second model, the payoff depends on a diffusion that the agent may or may not observe at a running cost, so that information acquisition itself is a control (a binary decision at each time), and an explicit threshold rule for when to buy information is announced in a solvable case. The note stated that the detailed proofs would appear in a later publication and that the general setting could be handled by adapting viscosity-solution arguments in infinite dimensions, without carrying out the technical implementation. The promised full paper never appeared, so the rigorous status of these announced results has remained open ever since.

Problem Statement

Give a complete rigorous treatment of the two partial-observation stochastic control problems announced by Lasry and Lions (1999), either proving the announced statements as formulated (or under explicitly identified stronger assumptions), or refuting them and establishing corrected versions. Concretely: (1) First model — for the exponential-utility hedging problem in which the asset drift is driven by an unobserved factor and the claim depends on the asset price and an independent random variable, justify the reduction of the partial-observation problem to a well-posed full-information control problem whose state is the observed price together with the unnormalized conditional density governed by the controlled Zakai equation; prove a dynamic programming principle and characterize the value function as the viscosity solution of the resulting infinite-dimensional Hamilton–Jacobi–Bellman equation (existence and uniqueness under stated hypotheses); prove the announced decomposition of the transformed value function into a payoff-randomness term and a nonpositive filtering term — the transform here being the exponential-utility transform V = 1 − exp(−λ[P − u]) (risk-aversion parameter λ > 0, initial wealth P, scalar u), so that the announced decomposition reads u = u₁ + u₂ with u₁ the payoff-randomness term and u₂ ≤ 0 the filtering term, both to be characterized; and rigorously justify the announced optimal feedback hedging rule, including its semi-explicit form in the constant-hidden-drift solvable case. (2) Second model — for the problem in which the agent chooses, at a running cost, whether to observe a payoff-relevant diffusion, construct a well-posed formulation resolving the circular dependence between admissible controls and the observation filtration (which is itself determined by the observation decisions); derive the dynamic programming principle and the infinite-dimensional HJB equation in the viscosity sense; and prove or refute the announced optimal observation rule, which in the solvable case — constant model coefficients with the payoff equal to a function of the asset price times the payoff-relevant diffusion — asserts that the optimal observation policy β ∈ {0,1} is a threshold rule: the agent observes (β = 1) exactly when a risk-aversion-weighted information gain, namely the reduction in the conditional variance of the payoff-relevant diffusion purchased by observing, is at least the running observation cost c; the exact functional form of the information gain and of the threshold, and the precise solvable-case conditions, must themselves be established, or corrected if the announced rule fails. The two headline announced results — the part (1) value decomposition u = u₁ + u₂ with u₂ ≤ 0 and the part (2) threshold observation rule — are the central claims to be established or refuted, each as formulated or under explicitly identified stronger assumptions, within a well-posed formulation of its model; the other enumerated components belong to the treatment wherever those claims rely on them, and the two models are jointly the target. The answer must, in addition, state precisely which of the note's announced results survive unchanged, which require additional assumptions, and which are false. The statements to be established or refuted are the two models and their announced results exactly as formulated in this Problem Statement, which is the operative and self-contained target; the underlying note is C. R. Acad. Sci. Paris Sér. I 328(11):1003–1010, 1999, DOI 10.1016/S0764-4442(99)80314-0.

References

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