arXiv subjects

Thaleia Zariphopoulou

Publications and source records attributed to Thaleia Zariphopoulou.

At least 19 recordsLinked to original sources

arXiv2026

Forward recursive aggregator systems and the forward Epstein-Zin recursive paradigm

We introduce forward recursive aggregator systems in It{ô}-diffusion markets, bridging the theories of recursive utilities and of forward performance criteria. The resulting criteria are constructed forward in time and accommodate arbitrary or rolling horizons. We derive the associated HJB SPDE and sufficient conditions for consistency and admissibility. We also develop a convex duality theory based on state price densities and identify the density generated by the optimal wealth process. For homothetic aggregators, we characterize a class of forward Epstein--Zin recursive utility processes, derive the optimal investment and consumption policies, and obtain explicit solutions in the Black and Scholes and Heston models.

math.PR↗

arXiv2026

High-Order Expansions of the Optimizer Map via Bell Polynomials

Completely monotonic inverse marginal (CMIM) utilities, introduced in [MSZ24], constitute a tractable class of preferences that includes many of the most important utility functions used in mathematical finance, such as power and exponential utilities. In stochastically dominant markets, their Bernstein representation induces a hidden linear structure in the dual optimization problem that serves as the foundation for the present analysis. In this paper, we investigate the sensitivity of optimal investment with respect to perturbations of investor preferences within the CMIM class. Exploiting Bernstein's representation theorem, we show that, under stochastic dominance, affine perturbations of Bernstein measures induce an affine representation of the dual value function. As a result, the dependence of the optimization problem on preferences can be analyzed through a scalar budget equation, allowing us to prove analyticity of the associated Lagrange multiplier with respect to the perturbation parameter and to derive convergent analytic expansions of arbitrary order for the primal value function and the optimal terminal wealth, with explicit recursive formulas expressed through Bell polynomials.

q-fin.MF↗

arXiv2026

On the optimal portfolio problem with partial information and related mean field games with relative performance criteria

We study optimal portfolio choice models in markets with partial information about the stock's drift. We solve the single agent problem for general utilities using a new approach that yields regularity of the value function and closed form expressions for the optimal processes. We consider a N player game in which players interact through the law of peer's wealth and study its mean field limit. This leads to a a game with field equilibrium. We analyze the cases of separable couplings and general utilities, and represent the value of the game as a compilation of the single player problem and a function solving a non local quasilinear pde in the space of measures. We interpret the findings using elements from indifference valuation and arbitrage free pricing, Finally, when the couplings depend only on the average of peer's wealth, we derive explicit solutions and various regularity results for representative cases.

math.OC↗

arXiv2026

Decision Making under Costly Sequential Information Acquisition: the Paradigm of Reversible and Irreversible Decisions

Decision making in modern stochastic systems, including e-commerce platforms, financial markets and healthcare systems, has evolved into a multifaceted process that combines information acquisition and adaptive information sources. This paper initiates a study on such integrated settings, where these elements are not only fundamental but, also, interact in a complex and stochastically intertwined manner. We introduce a relatively simple model, which, however, captures the involved novel elements. A decision maker (DM) may choose between an established product $A$ of known value and a new product $B$ whose value is unknown. In parallel, the DM observes signals about the unknown value of product $B$ and can, also, opt to exchange it for product $A$ if $B$ is initially chosen. Mathematically, the model gives rise to sequential optimal stopping problems with distinct informational regimes (before and after buying product $B$), differentiated by the initial, coarser signal and the subsequent, more accurate one. We analyze in detail the underlying problems using predominantly viscosity solution techniques, departing from the existing literature on information acquisition which is based on traditional optimal stopping arguments. More broadly, the modeling approach introduced herein offers a novel framework for developing more complex interactions among decisions, information sources and information costs in stochastic environments, through a sequence of nested obstacle problems.

math.OC↗

arXiv2025

Representation of forward performance criteria with random endowment via FBSDE and its application to forward optimized certainty equivalent

We extend the notion of forward performance criteria to settings with random endowment in incomplete markets. Building on these results, we introduce and develop the novel concept of \textit{forward optimized certainty equivalent (forward OCE)}, which offers a genuinely dynamic valuation mechanism that accommodates progressively adaptive market model updates, stochastic risk preferences, and incoming claims with arbitrary maturities. In parallel, we develop a new methodology to analyze the emerging stochastic optimization problems by directly studying the candidate optimal control processes for both the primal and dual problems. Specifically, we derive two new systems of forward-backward stochastic differential equations (FBSDEs) and establish necessary and sufficient conditions for optimality, and various equivalences between the two problems. This new approach is general and complements the existing one for forward performance criteria with random endowment based on backward stochastic partial differential equations (backward SPDEs) for the related value functions. We, also, consider representative examples for both forward performance criteria with random endowment and for forward OCE. Furthermore, for the case of exponential criteria, we investigate the connection between forward OCE and forward entropic risk measures.

q-fin.PM↗

arXiv2021

Entropy Regularization for Mean Field Games with Learning

Entropy regularization has been extensively adopted to improve the efficiency, the stability, and the convergence of algorithms in reinforcement learning. This paper analyzes both quantitatively and qualitatively the impact of entropy regularization for Mean Field Game (MFG) with learning in a finite time horizon. Our study provides a theoretical justification that entropy regularization yields time-dependent policies and, furthermore, helps stabilizing and accelerating convergence to the game equilibrium. In addition, this study leads to a policy-gradient algorithm for exploration in MFG. Under this algorithm, agents are able to learn the optimal exploration scheduling, with stable and fast convergence to the game equilibrium.

math.OC↗

arXiv2021

$N$-player and Mean-field Games in Itô-diffusion Markets with Competitive or Homophilous Interaction

In Itô-diffusion environments, we introduce and analyze $N$-player and common-noise mean-field games in the context of optimal portfolio choice in a common market. The players invest in a finite horizon and also interact, driven either by competition or homophily. We study an incomplete market model in which the players have constant individual risk tolerance coefficients (CARA utilities). We also consider the general case of random individual risk tolerances and analyze the related games in a complete market setting. This randomness makes the problem substantially more complex as it leads to ($N$ or a continuum of) auxiliary ''individual'' Itô-diffusion markets. For all cases, we derive explicit or closed-form solutions for the equilibrium stochastic processes, the optimal state processes, and the values of the games.

q-fin.MF↗

arXiv2021

On the analyticity of the value function in optimal investment and stochastically dominant markets

We study the analyticity of the value function in optimal investment with expected utility from terminal wealth and the relation to stochastically dominant financial models. We identify both a class of utilities and a class of semi-martingale models for which we establish analyticity. Specifically, these utilities have completely monotonic inverse marginals, while the market models have a maximal element in the sense of infinite-order stochastic dominance. We construct two counterexamples, themselves of independent interest, which show that analyticity fails if either the utility or the market model does not belong to the respective special class. We also provide explicit formulas for the derivatives, of all orders, of the value functions as well as their optimizers. Finally, we show that for the set of supermartingale deflators, stochastic dominance of infinite order is equivalent to the apparently stronger dominance of second order.

math.PR↗

arXiv2020

Personalized Robo-Advising: Enhancing Investment through Client Interaction

Automated investment managers, or robo-advisors, have emerged as an alternative to traditional financial advisors. The viability of robo-advisors crucially depends on their ability to offer personalized financial advice. We introduce a novel framework, in which a robo-advisor interacts with a client to solve an adaptive mean-variance portfolio optimization problem. The risk-return tradeoff adapts to the client's risk profile, which depends on idiosyncratic characteristics, market returns, and economic conditions. We show that the optimal investment strategy includes both myopic and intertemporal hedging terms which are impacted by the dynamics of the client's risk profile. We characterize the optimal portfolio personalization via a tradeoff faced by the robo-advisor between receiving client information in a timely manner and mitigating behavioral biases in the risk profile communicated by the client. We argue that the optimal portfolio's Sharpe ratio and return distribution improve if the robo-advisor counters the client's tendency to reduce market exposure during economic contractions when the market risk-return tradeoff is more favorable.

q-fin.PM↗

arXiv2020

Competition in Fund Management and Forward Relative Performance Criteria

In an Ito-diffusion market, two fund managers trade under relative performance concerns. For both the asset specialization and diversification settings, we analyze the passive and competitive cases. We measure the performance of the managers' strategies via forward relative performance criteria, leading to the respective notions of forward best-response criterion and forward Nash equilibrium. The motivation to develop such criteria comes from the need to relax various crucial, but quite stringent, existing assumptions -- such as, the a priori choices of both the market model and the investment horizon, the commonality of the latter for both managers as well as the full a priori knowledge of the competitor's policies for the best-response case. We focus on locally riskless criteria and deduce the random forward equations. We solve the CRRA cases, thus also extending the related results in the classical setting. An important by-product of the work herein is the development of forward performance criteria for investment problems in Ito-diffusion markets under the presence of correlated random endowment process for both the perfectly and the incomplete market cases.

q-fin.PM↗

arXiv2019

An approximation scheme for semilinear parabolic PDEs with convex and coercive Hamiltonians

We propose an approximation scheme for a class of semilinear parabolic equations that are convex and coercive in their gradients. Such equations arise often in pricing and portfolio management in incomplete markets and, more broadly, are directly connected to the representation of solutions to backward stochastic differential equations. The proposed scheme is based on splitting the equation in two parts, the first corresponding to a linear parabolic equation and the second to a Hamilton-Jacobi equation. The solutions of these two equations are approximated using, respectively, the Feynman-Kac and the Hopf-Lax formulae. We establish the convergence of the scheme and determine the convergence rate, combining Krylov's shaking coefficients technique and Barles-Jakobsen's optimal switching approximation.

math.OC↗

arXiv2019

Forward Rank-Dependent Performance Criteria: Time-Consistent Investment Under Probability Distortion

We introduce the concept of forward rank-dependent performance processes, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time-consistent nature of forward performance criteria with the time-inconsistency stemming from probability distortions. For this, we first propose two distinct definitions, one based on the preservation of performance value and the other on the time-consistency of policies and, in turn, establish their equivalence. We then fully characterize the viable class of probability distortion processes, providing a bifurcation-type result. Specifically, it is either the case that the probability distortions are degenerate in the sense that the investor would never invest in the risky assets, or the marginal probability distortion equals to a normalized power of the quantile function of the pricing kernel. We also characterize the optimal wealth process, whose structure motivates the introduction of a new, distorted measure and a related market. We then build a striking correspondence between the forward rank-dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection also provides a direct construction method for forward rank-dependent criteria. A byproduct of our work are some new results on the so-called dynamic utilities and on time-inconsistent problems in the classical (backward) setting.

q-fin.MF↗

arXiv2019

Predictable Forward Performance Processes: The Binomial Case

We introduce a new class of forward performance processes that are endogenous and predictable with regards to an underlying market information set and, furthermore, are updated at discrete times. We analyze in detail a binomial model whose parameters are random and updated dynamically as the market evolves. We show that the key step in the construction of the associated predictable forward performance process is to solve a single-period inverse investment problem, namely, to determine, period-by-period and conditionally on the current market information, the end-time utility function from a given initial-time value function. We reduce this inverse problem to solving a functional equation and establish conditions for the existence and uniqueness of its solutions in the class of inverse marginal functions.

q-fin.MF↗

arXiv2019

Exploration versus exploitation in reinforcement learning: a stochastic control approach

We consider reinforcement learning (RL) in continuous time and study the problem of achieving the best trade-off between exploration of a black box environment and exploitation of current knowledge. We propose an entropy-regularized reward function involving the differential entropy of the distributions of actions, and motivate and devise an exploratory formulation for the feature dynamics that captures repetitive learning under exploration. The resulting optimization problem is a revitalization of the classical relaxed stochastic control. We carry out a complete analysis of the problem in the linear--quadratic (LQ) setting and deduce that the optimal feedback control distribution for balancing exploitation and exploration is Gaussian. This in turn interprets and justifies the widely adopted Gaussian exploration in RL, beyond its simplicity for sampling. Moreover, the exploitation and exploration are captured, respectively and mutual-exclusively, by the mean and variance of the Gaussian distribution. We also find that a more random environment contains more learning opportunities in the sense that less exploration is needed. We characterize the cost of exploration, which, for the LQ case, is shown to be proportional to the entropy regularization weight and inversely proportional to the discount rate. Finally, as the weight of exploration decays to zero, we prove the convergence of the solution of the entropy-regularized LQ problem to the one of the classical LQ problem.

math.OC↗

arXiv2018

Mean field and n-agent games for optimal investment under relative performance criteria

We analyze a family of portfolio management problems under relative performance criteria, for fund managers having CARA or CRRA utilities and trading in a common investment horizon in log-normal markets. We construct explicit constant equilibrium strategies for both the finite population games and the corresponding mean field games, which we show are unique in the class of constant equilibria. In the CARA case, competition drives agents to invest more in the risky asset than they would otherwise, while in the CRRA case competitive agents may over- or under-invest, depending on their levels of risk tolerance.

q-fin.MF↗

arXiv2018

Optimal contract for a fund manager, with capital injections and endogenous trading constraints

In this paper, we construct a solution to the optimal contract problem for delegated portfolio management of the fist-best (risk-sharing) type. The novelty of our result is (i) in the robustness of the optimal contract with respect to perturbations of the wealth process (interpreted as capital injections), and (ii) in the more general form of principals objective function, which is allowed to depend directly on the agents strategy, as opposed to being a function of the generated wealth only. In particular, the latter feature allows us to incorporate endogenous trading constraints in the contract. We reduce the optimal contract problem to the following inverse problem: for a given portfolio (defined in a feedback form, as a random field), construct a stochastic utility whose optimal portfolio coincides with the given one. We characterize the solution to this problem through a Stochastic Partial Differential Equation (SPDE), prove its well-posedness, and compute the solution explicitly in the Black-Scholes model.

q-fin.PM↗

arXiv2017

On the Black's equation for the risk tolerance function

We analyze a nonlinear equation proposed by F. Black (1968) for the optimal portfolio function in a log-normal model. We cast it in terms of the risk tolerance function and provide, for general utility functions, existence, uniqueness and regularity results, and we also examine various monotonicity, concavity/convexity and S-shape properties. Stronger results are derived for utilities whose inverse marginal belongs to a class of completely monotonic functions.

q-fin.PM↗

arXiv2017

An ergodic BSDE approach to forward entropic risk measures: representation and large-maturity behavior

Using elements from the theory of ergodic backward stochastic differential equations (BSDE), we study the behavior of forward entropic risk measures. We provide their general representation results (via both BSDE and convex duality) and examine their behavior for risk positions of long maturities. We show that forward entropic risk measures converge to some constant exponentially fast. We also compare them with their classical counterparts and derive a parity result.

q-fin.MF↗