OP-0709OpenMathematical economics and game theory

Stochastic modeling and methods in optimal portfolio construction · 2014-econ-17

Decision, Utility, Risk and Preference Theory · ICM 2014: Stochastic modeling and methods in optimal portfolio construction

MSC 91B · Mathematical economics

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Problem statement

Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge 0},\mathbb P)$ be a filtered probability space supporting a $d$ -dimensional Brownian motion $W$ . Consider a frictionless market consisting of a riskless asset and $k$ risky assets. After discounting by the riskless asset, the wealth $X^\pi$ generated by a self-financing portfolio $\pi=(\pi^1,\ldots,\pi^k)^{\mathsf T}$ satisfies

dX_t^\pi=(\sigma_t\pi_t)\mathbin{\cdot}(\lambda_t\,dt+dW_t),

where $\sigma_t$ is an adapted $d\times k$ volatility matrix, $\mu_t\in\mathbb R^k$ is the vector of risky-asset appreciation rates, $r_t$ is the riskless rate, $\mathbf 1\in\mathbb R^k$ is the all-ones vector, and

\lambda_t=(\sigma_t^{\mathsf T})^+(\mu_t-r_t\mathbf 1)

is the $d$ -dimensional market price of risk; here $(\sigma_t^{\mathsf T})^+$ denotes the Moore--Penrose pseudoinverse. Assume that $\mu_t-r_t\mathbf 1$ lies in the range of $\sigma_t^{\mathsf T}$ and that

\mathbb E\!\left[\int_0^T|\lambda_t|^2\,dt\right]<\infty \qquad\text{for every }T>0.

Fix an interval $\mathbb D\subseteq\mathbb R$ of admissible discounted wealth values. Let $\mathcal A$ be the class of self-financing adapted portfolios for which the stochastic integral above is well defined, $X_t^\pi\in\mathbb D$ almost surely for every $t\ge0$ , and

\mathbb E\!\left[\int_0^T|\sigma_t\pi_t|^2\,dt\right]<\infty \qquad\text{for every }T>0.

For $0\le t\le s$ , $x\in\mathbb D$ , and $\pi\in\mathcal A$ , set

X_s^{t,x,\pi} =x+\int_t^s(\sigma_u\pi_u)\mathbin{\cdot}(\lambda_u\,du+dW_u),

whenever the resulting wealth remains in $\mathbb D$ , and let $\mathcal X_{t,s}(x)$ be the set of all such $\mathcal F_s$ -measurable terminal wealths, identified up to almost-sure equality. Let $\mathcal X_{t,s}=\bigcup_{x\in\mathbb D}\mathcal X_{t,s}(x)$ . For every $0\le t\le s<\infty$ , let $\succeq_{t,s}$ be a conditional preference relation, evaluated at time $t$ , on $\mathcal X_{t,s}$ .

Construct economically meaningful axioms on the entire family $(\succeq_{t,s})_{0\le t\le s<\infty}$ , including the required cross-horizon consistency, and prove that these axioms hold if and only if there exists a jointly measurable random field

U:\Omega\times\mathbb D\times[0,\infty)\longrightarrow\mathbb R

such that the following properties hold:

For every $x\in\mathbb D$ and $t\ge0$ , $U(x,t)$ is $\mathcal F_t$ -measurable, and for every $t\ge0$ , the map $x\mapsto U(x,t)$ is almost surely strictly increasing and strictly concave.
The relevant utility values are integrable, and for all $0\le t\le s$ and $X,Y\in\mathcal X_{t,s}$ ,

X\succeq_{t,s}Y \quad\Longleftrightarrow\quad \mathbb E[U(X,s)\mid\mathcal F_t] \ge \mathbb E[U(Y,s)\mid\mathcal F_t] \quad\text{almost surely}.

For every $x\in\mathbb D$ and every $0\le t\le s<\infty$ , the self-generation identity

U(x,t)=\operatorname*{ess\,sup}_{\pi\in\mathcal A:\,X_s^{t,x,\pi}\in\mathcal X_{t,s}(x)} \mathbb E\!\left[U\!\left(X_s^{t,x,\pi},s\right)\middle|\mathcal F_t\right] \quad\text{almost surely}

holds.

The axioms and representation theorem must apply to an explicitly stated general class of strictly increasing, strictly concave initial utilities $U(\cdot,0)$ , rather than imposing an exponential form or constant absolute risk aversion.

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Question 1

Let $(\Omega,\mathcal F,(\mathcal F_t)_{t\ge0},\mathbb P)$ support a $d$ -dimensional Brownian motion $W$ , and consider the discounted wealth dynamics

dX_t^\pi=(\sigma_t\pi_t)\mathbin{\cdot}(\lambda_t\,dt+dW_t)

on a wealth interval $\mathbb D\subseteq\mathbb R$ , where $\sigma_t$ is an adapted $d\times k$ volatility matrix and $\lambda_t\in\operatorname{range}(\sigma_t)$ is the adapted market price of risk. Assume that $\mathbb E\int_0^T|\lambda_t|^2\,dt<\infty$ for every $T>0$ . Call a self-financing adapted portfolio $\pi$ admissible if its wealth remains in $\mathbb D$ and $\mathbb E\int_0^T|\sigma_t\pi_t|^2\,dt<\infty$ for every $T>0$ .

Characterize the strictly increasing, strictly concave initial utility functions $u_0\in C^4(\mathbb D)$ for which there exist a nonzero, progressively measurable $\mathbb R^d$ -valued random field $a(x,t)$ , continuously differentiable in $x$ , and a sufficiently regular adapted random field $U(x,t)$ , defined for every $(x,t)\in\mathbb D\times[0,\infty)$ , such that $U(x,0)=u_0(x)$ , $x\mapsto U(x,t)$ remains strictly increasing and strictly concave, and

dU(x,t) =\frac12\frac{\left|U_x(x,t)\lambda_t+\sigma_t\sigma_t^+a_x(x,t)\right|^2}{U_{xx}(x,t)}\,dt +a(x,t)\mathbin{\cdot}dW_t.

Here $\sigma_t^+$ denotes the Moore--Penrose pseudoinverse. Require in addition that $U(X_t^\pi,t)$ is a local supermartingale for every admissible portfolio $\pi$ , and that for every initial wealth there is an admissible portfolio $\pi^*$ for which $U(X_t^{\pi^*},t)$ is a local martingale. Give an economically meaningful financial interpretation of the resulting characterization of admissible initial utilities.

Source reference: Thaleia Zariphopoulou, “Stochastic Modeling and Methods in Optimal Portfolio Construction,” Proceedings of the International Congress of Mathematicians, Seoul 2014, Vol. IV, 2014, pp. 1163–1186, Section 2.2.2 (open question following equation (2.20)), https://web.ma.utexas.edu/users/zariphop/pdfs/TZ-60a.pdf.

Progress & source assessment

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

The problem is unresolved as of August 4, 2026. No located reference proves the requested if-and-only-if theorem for a family of conditional preferences indexed by every pair $(t,s)$ , with one jointly measurable adapted utility field whose representations are aligned across horizons and whose alignment is equivalent to market self-generation, for a general nonexponential class of initial utilities.

The principal results cover separate parts of this problem:

Thaleia Zariphopoulou, “Stochastic Modeling and Methods in Optimal Portfolio Construction,” Proceedings of the International Congress of Mathematicians, Seoul 2014, Vol. IV, 2014, pp. 1163–1186, direct PDF. Section 2.2 defines forward performance and self-generation and explicitly identifies its axiomatic construction as open, with only the exponential case then available.
Gordan Žitković, “A Dual Characterization of Self-Generation and Exponential Forward Performances,” The Annals of Applied Probability 19(6), 2009, pp. 2176–2210, DOI 10.1214/09-AAP607, arXiv:0809.0739. The paper gives equivalent primal and dual conditions for self-generation of an already specified utility random field and necessary and sufficient conditions plus an Itô-market parametrization in the exponential/log-affine case. It does not derive the field from axioms on the conditional relations $\succeq_{t,s}$ , and its explicit classification is exponential.
François P. Y. S. Berrier, L. C. G. Rogers, and Michael R. Tehranchi, “A Characterization of Forward Utility Functions,” Statistical Laboratory, University of Cambridge preprint, 2009, direct URL. Theorem 2.3 characterizes an already given forward utility through its convex conjugate in a general semimartingale market, and Corollary 2.5 specializes this in complete markets. The paper supplies no preference-axiom representation theorem; its decreasing-forward-utility extension criterion also restricts the allowable initial utility.
Sergey Nadtochiy and Michael Tehranchi, “Optimal Investment for All Time Horizons and Martin Boundary of Space-Time Diffusions,” Mathematical Finance 27(2), 2017, pp. 438–470, DOI 10.1111/mafi.12092, arXiv:1308.2254. Section 1.1 gives an economic justification by applying von Neumann–Morgenstern reasoning separately at each horizon and then requiring a common time-consistent optimizer. It does not formulate or prove the requested conditional all- $(t,s)$ representation equivalence. Its construction results concern factor-form Markov models and positive solutions of linearizable, uniformly elliptic time-reversed equations, not the general Brownian market and initial-utility class in the problem.
Samuel Drapeau and Asgar Jamneshan, “Conditional Preference Orders and Their Numerical Representations,” Journal of Mathematical Economics 63, 2016, pp. 106–118, DOI 10.1016/j.jmateco.2015.12.004, arXiv:1410.5466. The paper proves conditional Debreu and conditional von Neumann–Morgenstern representation theorems. These address the conditional representation component, but not cross-horizon normalization, attainable-wealth dynamics, or equivalence with the self-generation identity.
Levon Avanesyan and Ronnie Sircar, “Power Mixture Forward Performance Processes,” SIAM Journal on Financial Mathematics 13(3), 2022, pp. 1040–1062, DOI 10.1137/20M1385500, arXiv:2012.10847. The authors construct a broader nonexponential power-mixture class, but only within that structured family. They also show that self-generation together with an initial utility does not in general determine the forward evolution, underscoring that this construction result does not supply the missing axiomatic theorem.

Thus there is substantive partial progress on each analytic ingredient and on special utility/model classes, but the exact unified representation theorem in the statement remains unproved.

References

[1]ICM Conjectures · 2014-econ-17 link
[2]ICM 2014: Stochastic modeling and methods in optimal portfolio construction link

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Source & evidence

2014-econ-17

ICM snapshot retrieved 8 September 2026. Source review state: unreviewed. The status text and 1 remaining questions are reproduced with attribution; no independent proof audit has been performed here.

Source date: 2026-08-18
Imported: 2026-09-08

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Difficulty: HardResisted sustained effort by specialists.
Posed: 2014
Source: Curated problem listICM ConjecturesICM 2014: Stochastic modeling and methods in optimal portfolio constructionOriginal statement ↗
Citation: ICM 2014: Stochastic modeling and methods in optimal portfolio construction
Added: 5 Nov 2026
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Mathematical economics and game theory