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Stochastic modeling and methods in optimal portfolio construction · 2014-econ-18

Actuarial Science and Mathematical Finance · ICM 2014: Stochastic modeling and methods in optimal portfolio construction

MSC 91B · Mathematical economics

Inferred from the community field; finer classification has not been reviewed.

Source assessment · not reviewedAs of 2026-08-18Source & evidence ↓
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Problem statement

Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal F,(\mathcal F_t)_{t\geq0},\mathbb P) be a filtered probability space carrying a discrete- or continuous-time financial market, and let A\mathcal A be the class of admissible self-financing portfolio strategies. For π∈A\pi\in\mathcal A, write Xπ=(Xtπ)t≥0X^\pi=(X_t^\pi)_{t\geq0} for its wealth process. Construct a genuinely dynamic family of conditional investment-performance functionals (Jt)t≥0(J_t)_{t\geq0}, where JtJ_t is Ft\mathcal F_t-measurable and ranks admissible continuations of portfolio strategies using the information available at time tt, with the following properties:

  1. The ranking at time tt directly incorporates conditional investment quantities such as the conditional expected return and conditional variance of future wealth,
E[Xuπ−Xtπ∣Ft]andVar⁡(Xuπ∣Ft),u>t, \mathbb E[X_u^\pi-X_t^\pi\mid\mathcal F_t] \quad\text{and}\quad \operatorname{Var}(X_u^\pi\mid\mathcal F_t),\qquad u>t,

or comparably direct targets and constraints, such as a prescribed conditional expected return, a volatility bound, a reserve constraint, or a risk limit.

  1. The family is time-consistent: if π∗\pi^* is optimal according to JtJ_t at time tt, then for every s≥ts\geq t, conditional on the wealth and information generated by π∗\pi^* up to time ss, the continuation π∗∣[s,∞)\pi^*|_{[s,\infty)} is optimal according to JsJ_s among all admissible continuations from that state.

  2. The construction applies in a multiperiod setting and admits a genuine continuous-time formulation; it is not obtained by fixing one terminal date TT and maximizing a one-period quadratic-utility functional of XTπX_T^\pi.

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Question 1

Let (Ω,F,(Ft)t≥0,P)(\Omega,\mathcal F,(\mathcal F_t)_{t\geq0},\mathbb P) support an NN-dimensional Brownian motion WW, and consider a market, discounted by its riskless asset, whose cumulative excess-return process satisfies

dRt=μt dt+σt dWt, dR_t=\mu_t\,dt+\sigma_t\,dW_t,

where μt∈RN\mu_t\in\mathbb R^N, σt∈RN×N\sigma_t\in\mathbb R^{N\times N} is invertible, and μ\mu, σ\sigma, and (σσT)−1(\sigma\sigma^{\mathsf T})^{-1} are adapted, bounded, and Hölder-continuous in time. Let U(t,x)U(t,x) be a forward investment-performance process that is not assumed to be time-monotone: for every tt, x↦U(t,x)x\mapsto U(t,x) is increasing and strictly concave; U(t,Xtπ)U(t,X_t^\pi) is a supermartingale for every admissible self-financing wealth process XπX^\pi; and it is a martingale for an optimal wealth process Xπ∗X^{\pi^*}.

For a time step δ>0\delta>0, let an investor solve at each date ti=iδt_i=i\delta the consecutive one-period conditional mean–variance problem

min⁡πti{Var⁡ ⁣(Xti+1π∣Fti)−η(ti,Xtiπ)E ⁣[Xti+1π∣Fti]}, \min_{\pi_{t_i}}\left\{ \operatorname{Var}\!\left(X_{t_{i+1}}^\pi\mid\mathcal F_{t_i}\right) -\eta(t_i,X_{t_i}^\pi) \mathbb E\!\left[X_{t_{i+1}}^\pi\mid\mathcal F_{t_i}\right] \right\},

where the trade-off coefficient η(ti,Xtiπ)\eta(t_i,X_{t_i}^\pi) is Fti\mathcal F_{t_i}-measurable. Determine which non-time-monotone forward performance processes admit such an infinitesimal mean–variance representation, and, for every process that does, construct the adapted trade-off rule η\eta and prove that the wealth and portfolio processes generated by the consecutive mean–variance problems converge, as δ↓0\delta\downarrow0, to Xπ∗X^{\pi^*} and π∗\pi^*, respectively.

Source reference: Pedro M. Almeida Serra Costa Vitória, Topics on Forward Investment Theory, DPhil thesis, University of Oxford, 2015, Chapter 1, Remark 5, DOI: https://doi.org/10.5287/ora-8g8vyegqe.

Progress & source assessment

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

The problem is unresolved. No verified source constructs a horizon-free family of conditional portfolio-performance functionals that simultaneously (i) uses conditional mean and variance, or equally direct investment targets or limits, as its criterion, (ii) makes every optimal continuation globally optimal under the later member of the same family, and (iii) has both multiperiod and genuine continuous-time formulations. The literature establishes each of these features only after weakening or restricting another one.

Marek Musiela and Thaleia Zariphopoulou, “Portfolio Choice under Dynamic Investment Performance Criteria,” Quantitative Finance 9(2) (2009), 161–170, DOI 10.1080/14697680802624997, constructs forward performance processes on [0,∞)[0,\infty). Their supermartingale/martingale optimality principle gives the required horizon-free continuation consistency in an incomplete Itô market, but the criterion is an adapted utility random field, not a functional directly specified by conditional return and variance or operational targets and limits.

Christoph Czichowsky, “Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time,” Finance and Stochastics 17(2) (2013), 227–271, DOI 10.1007/s00780-012-0189-9, arXiv 1205.4748, gives a discrete- and continuous-time formulation in a general square-integrable semimartingale market. Definitions 3.1 and 4.1 replace global continuation optimality by local mean-variance efficiency; Theorems 4.6–4.7 characterize existence through the structure condition and a Föllmer–Schweizer decomposition, and Theorem 5.9 proves convergence of the discrete formulations. The model still has a fixed finite terminal horizon, and the solution is locally optimal against infinitesimal or one-step deviations rather than globally optimal among all later continuations as required here.

Pedro M. Almeida Serra Costa Vitória, Topics on Forward Investment Theory, DPhil thesis, University of Oxford (2015), DOI 10.5287/ora-8g8vyegqe, gives the closest direct bridge. Chapter 1, especially Theorem 3, chooses the trade-off coefficient in consecutive conditional mean–variance problems from the local risk-tolerance process and proves that the resulting wealth and portfolio processes converge, as the trading interval tends to zero, to those of a time-monotone forward performance process. The thesis expressly limits the conclusion to the infinitesimal time scale, notes that the construction is not purely mean–variance based, and treats only time-monotone forward processes. It therefore supplies substantive partial progress, not the requested criterion in full generality.

Gabriela Kováčová and Birgit Rudloff, “Time Consistency of the Mean-Risk Problem,” Operations Research 69(4) (2021), 1100–1117, DOI 10.1287/opre.2020.2002, arXiv 1806.10981, proves in Theorem 1 that a nonscalarized, bi-objective mean-risk problem satisfies a set-valued Bellman principle with respect to weak minimizers under mild assumptions. This is an exact discrete-time recursion for efficient frontiers, but it uses a fixed terminal problem and does not provide the demanded continuous-time, scalar conditional evaluation family with global continuation optimality.

Felix Fießinger and Mitja Stadje, “Time-Consistent Asset Allocation for Risk Measures in a Lévy Market,” European Journal of Operational Research 321(2) (2025), 676–695, DOI 10.1016/j.ejor.2024.09.049, solves continuous-time mean-risk problems, including variance, in a multidimensional Lévy market via an extended Hamilton–Jacobi–Bellman equation. Its solution is a Nash subgame equilibrium for gains and losses relative to a benchmark over a fixed horizon, not a horizon-free family whose optimal continuations are globally optimal under later evaluations.

Thus the remaining gap is precisely the simultaneous construction demanded in the statement: a direct, operational mean–variance (or target/constraint) criterion that is horizon-free, genuinely continuous in time, and time-consistent in the full continuation-optimality sense rather than only through local equilibrium, infinitesimal convergence, set-valued efficiency, or a terminal-date formulation.

References
  1. [1]ICM Conjectures · 2014-econ-18 link
  2. [2]ICM 2014: Stochastic modeling and methods in optimal portfolio construction link

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