Optimization Methods

We study a general robust utility maximization problem in a discrete-time frictionless market. The investor is assumed to have a possibly infinite, random, nonconcave, and nondecreasing utility function defined on the whole real line. She also faces model ambiguity on her beliefs about the market, w

In this paper we analyze the resilience of a network of banks to joint price fluctuations of the external assets in which they have shared exposures, and evaluate the worst-case effects of the possible default contagion. Indeed, when the prices of certain external assets either decrease or increase,

The key result of this paper is to characterize all the multivariate symmetric Bernoulli distributions whose sum is minimal under convex order. In doing so, we automatically characterize extremal negative dependence among Bernoulli random vectors, since multivariate distributions with minimal convex

In this article, we consider the problem of a bank’s loan portfolio in the context of liquidity risk, while allowing for the limited liability protection enjoyed by the bank. Accordingly, we construct a novel loan portfolio model with limited liability, while maintaining a threshold level of haircut

We study a robust utility maximization problem in a general discrete-time frictionless market under quasi-sure no-arbitrage. The investor is assumed to have a random and concave utility function defined on the whole real-line. She also faces model ambiguity on her beliefs about the market, which is

In portfolio compression, market participants (banks, organizations, companies, financial agents) sign contracts, creating liabilities between each other, which increases the systemic risk. Large, dense markets commonly can be compressed by reducing obligations without lowering the net notional of e

The geometric approach to financial markets with proportional transaction cost prescribes to imbed a specific model (of stock market, of currency market etc.), usually given in a parametric form, into a natural framework defined by the two random processes, S and K. The first one, d-dimensional, mod

For simultaneous independent events with finitely many outcomes, consider the expected-utility problem with nonnegative wagers and an endogenous cash position. We prove a short support theorem for a broad class of strictly increasing strictly concave utilities. On any fixed support family and at any

We study a problem of optimal allocation in a discrete-time multi-period pure-exchange economy, where agents have preferences over stochastic endowment processes that are represented by strongly time-consistent dynamic risk measures. We introduce the notion of dynamic Pareto-optimal allocation proce

This paper develops a unified framework for the robustification of risk measures beyond the classical convex and cash-additive setting. We consider general risk measures on Lp spaces and construct their robust counterparts through families of uncertainty sets that capture ambiguity. Two complementar

We study the optimal routing problem in decentralized exchanges built on Constant Function Market Makers when trades can be split across multiple heterogeneous pools and execution incurs fixed on-chain costs (gas fees). While prior routing formulations typically abstract from fixed activation costs,

We study entropy-regularized mean-variance portfolio optimization under Bayesian drift uncertainty. Gaussian policies remain optimal under partial information, the value function is quadratic in wealth, and belief-dependent coefficients admit closed-form solutions. The mean control is identical to d

We generalize classical results on the existence of optimal portfolios in discrete time frictionless market models to models with capital gains taxes. We consider the realistic but mathematically challenging rule that losses do not trigger negative taxes but can only be offset against potential gain

This paper assumes a robust stochastic model where a set $\mathcal{P}$ of probability measures replaces the single probability measure of dominated models. We introduce and study $\mathcal{P}$-sensitive functions defined on robust function spaces of random variables. We show that $\mathcal{P}$-sensi

The Lambda Value-at-Risk (Lambda-VaR) is a generalization of the Value-at-Risk (VaR), which has been actively studied in quantitative finance. Over the past two decades, the Expected Shortfall (ES) has become one of the most important risk measures alongside VaR because of its various desirable prop

In this paper, we provide extended convolution bounds for the Fréchet problem and discuss related implications in quantitative risk management. First, we establish a new form of inequality for the Range-Value-at-Risk (RVaR). Based on this inequality, we obtain bounds for robust risk aggregation with

In recent years, stochastic dominance for independent and identically distributed (iid) infinite-mean random variables has received considerable attention. The literature has identified several classes of distributions of nonnegative random variables that encompass many common heavy-tailed distribut

This paper studies the time-inconsistent MV optimal stopping problem via a game-theoretic approach to find equilibrium strategies. To overcome the mathematical intractability of direct equilibrium analysis, we propose a vanishing regularization method: first, we introduce an entropy-based regulariza

We study a consumption-investment problem in a multi-asset market where the returns follow a generic rank-based model. Our main result derives an HJB equation with Neumann boundary conditions for the value function and proves a corresponding verification theorem. The control problem is nonstandard d

This paper considers two investors who perform mean-variance portfolio selection with asymmetric information: one knows the true stock dynamics, while the other has to infer the true dynamics from observed stock evolution. Their portfolio selection is interconnected through relative performance conc

This paper studies an optimal dividend problem for a company that aims to maximize the mean-variance (MV) objective of the accumulated discounted dividend payments up to its ruin time. The MV objective involves an integral form over a random horizon that depends endogenously on the company’s dividen

We present a novel approach for the bank’s decision problem, incorporating Limited Liability in the objective function. Accordingly, we consider continuous time models, with and without Limited Liability. We compare the solutions of these two models to demonstrate the effect of inclusion of Limited

The extreme cases of risk measures, when considered within the context of distributional ambiguity, provide significant guidance for practitioners specializing in risk management of quantitative finance and insurance. In contrast to the findings of preceding studies, we focus on the study of extreme

This paper studies the dividend and capital injection problem under a diffusion risk model with general discount functions. A proportional cost is imposed when injecting capitals. For exponential discounting as time-consistent benchmark, we obtain the closed-form solutions and show that the optimal

We propose a quantum unstructured search algorithm to find the extrema or roots of discrete functions, $f(\mathbf{x})$, such as the objective functions in combinatorial and other discrete optimisation problems. The first step of the Quantum Search for Extrema and Roots Algorithm (QSERA) is to transl

The paper investigates the robust distortion risk measure with linear penalty function under distribution uncertainty. The distribution uncertainties are characterized by predetermined moment conditions or constraints on the Wasserstein distance. The optimal quantile distribution and the optimal val

We provide a constructive way of defining new elicitable risk measures that are characterised by a multiplicative scoring function. We show that depending on the choice of the scoring function’s components, the resulting risk measure possesses properties such as monotonicity, translation invariance,

This paper provides a dual formulation of the optimal consumption problem with internal multiplicative habit formation. In this problem, the agent derives utility from the ratio of consumption to the internal habit component. Due to this multiplicative specification of the habit model, the optimal c

This article studies the problem of utility maximization in an incomplete market under a class of nonlinear expectations and general constraints on trading strategies. Using a $g$-martingale method, we provide an explicit solution to our optimization problem for different utility functions and chara

This paper discusses a nonlinear integral equation arising from portfolio selection with a class of time-inconsistent preferences. We propose a unified framework requiring minimal assumptions, such as right-continuity of market coefficients and square-integrability of the market price of risk. Our m

Decentralized Finance (DeFi), a financial ecosystem without centralized controlling organization, has introduced a new paradigm for lending and borrowing. However, its capital efficiency remains constrained by the inability to effectively assess the risk associated with each user/wallet. This paper

We study the anticipating version of the classical portfolio optimization problem in a financial market with the presence of a trader who possesses privileged information about the future (insider information), but who is also subjected to a delay in the information flow about the market conditions;

This paper studies robust forward investment and consumption preferences and optimal strategies for a risk-averse and ambiguity-averse agent in an incomplete financial market with drift and volatility uncertainties. We focus on non-zero volatility and constant relative risk aversion forward preferen

We investigate the portfolio selection problem for an agent with rank-dependent utility in an incomplete financial market. For a constant-coefficient market and CRRA utilities, we characterize the deterministic strict equilibrium strategies. In the case of time-invariant probability weighting functi

This paper studies a robust stochastic control problem with a monotone mean-variance cost functional and random coefficients. The main technique is to find the saddle point through two backward stochastic differential equations (BSDEs) with unbounded coefficients. We further show that the robust sto

The paper investigates the consumption-investment problem for an investor with Epstein-Zin utility in an incomplete market. A non-Markovian environment with unbounded parameters is considered, which is more realistic in practical financial scenarios compared to the Markovian setting. The optimal con

In risk-sharing markets with aggregate uncertainty, characterizing Pareto-optimal allocations when agents might not be risk averse is a challenging task, and the literature has only provided limited explicit results thus far. In particular, Pareto optima in such a setting may not necessarily be como

This paper investigates the optimal investment problem in a market with two types of illiquidity: transaction costs and search frictions. Extending the framework established by arXiv:2101.09936, we analyze a power-utility maximization problem where an investor encounters proportional transaction cos

We expose a theoretical hedging optimization framework with variational preferences under convex risk measures. We explore a general dual representation for the composition between risk measures and utilities. We study the properties of the optimization problem as a convex and monotone map per se. W

Geometrically convex functions constitute an interesting class of functions obtained by replacing the arithmetic mean with the geometric mean in the definition of convexity. As recently suggested, geometric convexity may be a sensible property for financial risk measures ([7,13,4]). We introduce a n

In this paper, we introduce a new class of set-valued risk measures, named set-valued star-shaped risk measures. Motivated by the results of scalar monetary and star-shaped risk measures, this paper investigates the representation theorems in the set-valued framework. It is demonstrated that set-val

We study a reinsurance Stackelberg game in which both the insurer and the reinsurer adopt the mean-variance (abbr. MV) criterion in their decision-making and the reinsurance is irreversible. We apply a unified singular control framework where irreversible reinsurance contracts can be signed in both

This paper studies a type of periodic utility maximization for portfolio management in an incomplete market model, where the underlying price diffusion process depends on some external stochastic factors. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealt

We construct Nash equilibria in feedback form for a class of two-person stochastic games of singular control with absorption, arising from a stylized model for corporate finance. More precisely, the paper focusses on a strategic dynamic game in which two financially-constrained firms operate in the

We consider an optimal stochastic control problem in which a firm’s cash/surplus process is controlled by dividend payments and capital injections. Stockholders aim to maximize their dividend stream minus the cost of injecting capital, if needed. We consider absolutely continuous dividend policies s

Constant function market makers (CFMMs) are the most popular type of decentralized trading venue for cryptocurrency tokens. In this paper, we give a very general geometric framework (or ‘axioms’) which encompass and generalize many of the known results for CFMMs in the literature, without requiring

The Hamilton-Jacobi-Bellman equation arising from the optimal portfolio selection problem is studied by means of the maximal monotone operator method. The existence and uniqueness of a solution to the Cauchy problem for the nonlinear parabolic partial integral differential equation in an abstract se

In this paper, we study the portfolio optimization problem formulated by Lacker and Soret. They formulate a finite time horizon model that allows agents to be competitive, measuring their utility not only by their absolute wealth but also relative performance compared to the average of other agents.

This paper provides an introductory overview of how one may employ importance sampling effectively as a tool for solving stochastic optimization formulations incorporating tail risk measures such as Conditional Value-at-Risk. Approximating the tail risk measure by its sample average approximation, w

The aim of this short note is to present a solution to the discrete time exponential utility maximization problem in a case where the underlying asset has a multivariate normal distribution. In addition to the usual setting considered in Mathematical Finance, we also consider an investor who is info

The classical mean-variance portfolio selection problem induces time-inconsistent (precommited) strategies (see Zhou and Li (2000)). To overcome this time-inconsistency, Basak and Chabakauri (2010) introduce the game theoretical approach and look for (sub-game perfect Nash) equilibrium strategies, w

We study a continuous-time expected utility maximization problem in which the investor at maturity receives the value of a contingent claim in addition to the investment payoff from the financial market. The investor knows nothing about the claim other than its probability distribution, hence an ``i

We consider an investor who is dynamically informed about the future evolution of one of the independent Brownian motions driving a stock’s price fluctuations. With linear temporary price impact the resulting optimal investment problem with exponential utility turns out to be not only well posed, bu

In this paper, we consider the portfolio optimization problem in a financial market where the underlying stochastic volatility model is driven by n-dimensional Brownian motions. At first, we derive a Hamilton-Jacobi-Bellman equation including the correlations among the standard Brownian motions. We

In this article, we study the generalized modern portfolio theory, with utility functions admitting higher-order cumulants. We establish that under certain genericity conditions, the utility function has a constant number of complex critical points. We study the discriminant locus of complex critica

In actuarial science and quantitative finance, convex order provides a natural way to compare risks with the same mean. In dimension one, convex order is well understood through several characterisations. In higher dimensions, a natural approach is to compare all one-dimensional projections, but, al

We propose a decomposition method for solving a general class of linear-quadratic (LQ) McKean-Vlasov control problems involving conditional expectations and random coefficients, where the system dynamics are driven by two independent Wiener processes. Unlike existing approaches in the literature for

In this paper we introduce a generalization of classical risk measures in which the risk is represented by a step function taking two values, corresponding to two endogenously determined market regimes. This extends the traditional framework where risk measures map random variables to single real nu

Reinforcement learning (RL) is an innovative approach to financial decision making, offering specialized solutions to complex investment problems where traditional methods fail. This review analyzes 167 articles from 2017–2025, focusing on market making, portfolio optimization, and algorithmic tradi

We investigate a portfolio selection problem involving multi competitive agents, each exhibiting mean-variance preferences. Unlike classical models, each agent’s utility is determined by their relative wealth compared to the average wealth of all agents, introducing a competitive dynamic into the op

We consider the optimal risk sharing problem with a continuum of agents, modeled via a non-atomic measure space. Individual preferences are not assumed to be convex. We show the multiplicity of agents induces the value function to be convex, allowing for the application of convex duality techniques

Price benchmarks are used to incorporate market price trends into contracts, but their use can create opportunities for manipulation by parties involved in the contract. This paper examines this issue using a realistic and tractable model inspired by smart contracts on blockchains like Ethereum. In

We investigate the well-posedness of a general class of singular stochastic control problems in which controls are processes of finite variation. We develop an abstract framework, which we then apply to storage management and portfolio investment problems under proportional transaction costs. Within

Decentralised exchanges (DEXs) have transformed trading by enabling trustless, permissionless transactions, yet they face significant challenges such as impermanent loss and slippage, which undermine profitability for liquidity providers and traders. In this paper, we introduce QubitSwap, an innovat

We study the existence of equilibrium when agents’ preferences may not beconvex. For some specific utility functions, we provide a necessary and sufficientcondition under which there exists an equilibrium. The standard approach cannot be directly applied to our examples because the demand correspond

We present a novel approach for parameter calibration of the Heston model for pricing an Asian put option, namely space mapping. Since few parameters of the Heston model can be directly extracted from real market data, calibration to real market data is implicit and therefore a challenging task. In

The domain of hedge fund investments is undergoing significant transformation, influenced by the rapid expansion of data availability and the advancement of analytical technologies. This study explores the enhancement of hedge fund investment performance through the integration of machine learning t

We consider the Merton problem of optimizing expected power utility of terminal wealth in the case of an unobservable Markov-modulated drift. What makes the model special is that the agent is allowed to purchase costly expert opinions of varying quality on the current state of the drift, leading to

We introduce the concept of set risk measures (SRMs), which are real-valued maps defined on the space of all non-empty, closed, and bounded sets of almost surely bounded random variables. Traditional risk measures typically operate on random variables, but SRMs extend this framework to sets of rando

Automated market makers with concentrated liquidity capabilities are programmable at the tick level. The maximization of earned fees, plus depreciated reserves, is a convex optimization problem whose vector solution gives the best provision of liquidity at each tick under a given set of parameter es

This paper studies an optimal investment-reinsurance problem for an insurer (she) under the Cramér–Lundberg model with monotone mean–variance (MMV) criterion. At any time, the insurer can purchase reinsurance (or acquire new business) and invest in a security market consisting of a risk-free asset a

The paper provides a mathematical model and a tool for the focused investing strategy as advocated by Buffett, Munger, and others from this investment community. The approach presented here assumes that the investor’s role is to think about probabilities of different outcomes for a set of businesses

We provide a new characterization of second-order stochastic dominance, also known as increasing concave order. The result has an intuitive interpretation that adding a risk with negative expected value in adverse scenarios makes the resulting position generally less desirable for risk-averse agents

Shadow prices simplify the derivation of optimal trading strategies in markets with transaction costs by transferring optimization into a more tractable, frictionless market. This paper establishes that a naïve shadow price Ansatz for maximizing long term returns given average volatility yields a st

We consider the problem of being a cross-chain wealth management platform with deposits, redemptions and investment assets across multiple networks. We discuss the need for blockchain bridges to facilitates fund flows across platforms. We point out several issues with existing bridges. We develop an

We present explicit oracles designed to be used in Grover’s algorithm to match investor preferences. Specifically, the oracles select portfolios with returns and standard deviations exceeding and falling below certain thresholds, respectively. One potential use case for the oracles is selecting port

Portfolio construction traditionally relies on separately estimating expected returns and covariance matrices using historical statistics, often leading to suboptimal allocation under time-varying market conditions. This paper proposes a joint return and risk modeling framework based on deep neural

We study the Merton portfolio management problem within a complete market, non constant time discount rate and general utility framework. The non constant discount rate introduces time inconsistency which can be solved by introducing sub game perfect strategies. Under some asymptotic assumptions on

In high-risk environments, traditional indemnity insurance is often unaffordable or ineffective, despite its well-known optimality under expected utility. We compare excess-of-loss indemnity insurance with parametric insurance within a common mean-variance framework, allowing for fixed costs, hetero

The Kelly criterion provides a general framework for optimizing the growth rate of an investment portfolio over time by maximizing the expected logarithmic utility of wealth. However, the optimality condition of the Kelly criterion is highly sensitive to accurate estimates of the probabilities and i

We propose \textit{OpenAlpha}, a community-led strategy validation framework for decentralised capital management on a host blockchain network, which integrates game-theoretic validation, adversarial auditing, and market-based belief aggregation. This work formulates treasury deployment as a capital

A complete description of taxation systems that ensure sustainable economic development is given. These tax systems depend on production technologies and gross output volumes. Explicit formulas for such dependencies are found. In a sustainable economy, the value added either exceeds or is strictly l

This paper derives the best- and worst-case GlueVaR distortion risk measure within a unified framework, based on partial information of the underlying distributions and shape information such as symmetry. In addition, we characterize the extremal distributions of GlueVaR with convex envelopes of the

We prove that weak convergence within generalized gamma convolution (GGC) distributions implies convergence in the mean value. We use this fact to show the robustness of the expected utility maximizing optimal portfolio under exponential utility function when return vectors are modelled by hyperboli

The classical theory of efficient allocations of an aggregate endowment in a pure-exchange economy has hitherto primarily focused on the Pareto-efficiency of allocations, under the implicit assumption that transfers between agents are frictionless, and hence costless to the economy. In this paper, w

The paper provides an overview of the theory and applications of risk-sensitive Markov decision processes. The term ‘risk-sensitive’ refers here to the use of the Optimized Certainty Equivalent as a means to measure expectation and risk. This comprises the well-known entropic risk measure and Condit

We present simple general conditions on the acceptance sets under which their induced monetary risk and deviation measures are comonotonic additive. We show that acceptance sets induce comonotonic additive risk measures if and only if the acceptance sets and their complements are stable under convex

We consider a rational agent who at time $0$ enters into a financial contract for which the payout is determined by a quantum measurement at some time $T>0$. The state of the quantum system is given in the Heisenberg representation by a known density matrix $\hat p$. How much will the agent be willi

In this paper we formulate and solve an optimal problem for Stochastic process with a regime absorbing state. The solution for this problem is obtained through a system of partial differential equations. The method is applied to obtain an explicit solution for the Merton portfolio problem when an as

The takeoff point for this paper is the voluminous body of literature addressing recursive betting games with expected logarithmic growth of wealth being the performance criterion. Whereas almost all existing papers involve use of linear feedback, the use of nonlinear control is conspicuously absent

The energy transition is expected to significantly increase the share of renewable energy sources whose production is intermittent in the electricity mix. Apart from key benefits, this development has the major drawback of generating a mismatch between power supply and demand. The innovative dynamic

In this paper we aim to study viability and completeness in finite markets. In order to do that, we characterize the set of equivalent martingale measures of two-period markets as convex combinations of a finite number of martingale measures. We provide an algorithm for finding such measures, that c

We consider a competitive market with risk-averse participants. We assume that agents’ risks are measured by coherent risk measures introduced by Artzner et al. (1999). Fundamental theorems of welfare economics have long established the equivalence of competitive equilibria and system welfare optimi

We show that the Markowitz portfolio is a scalar multiple of another portfolio which replaces the covariance with the second moment matrix, via simple application of the Sherman-Morrison identity. Moreover it is shown that when using conditional estimates of the first two moments, this “Sherman-Morr

The Nobel-price winning Mirrlees’ theory of optimal taxation inspired a long sequence of research on its refinement and enhancement. However, an issue of concern has been always the fact that, as was shown in many publications, the optimal schedule in Mirrlees’ paradigm of maximising the total utili

We give an alternative proof of a fact that a finite continuous non-decreasing submodular set function on a measurable space can be expressed as a supremum of measures dominated by the function, if there exists a class of sets which is totally ordered with respect to inclusion and generates the sigm

We have designed an innovative portfolio rebalancing mechanism termed the Cascading Waterfall Round Robin Mechanism. This algorithmic approach recommends an ideal size and number of trades for each asset during the periodic rebalancing process, factoring in the gas fee and slippage. The essence of t

In this short paper we introduce a new class of performance measures based on certainty equivalents defined via scaled utility functions. We analyse their properties, show that the corresponding portfolio optimization problem is well-posed under generic conditions, and analyse the link between portf

Retirement prediction helps individuals and institutions make informed financial, lifestyle, and workforce decisions based on estimated retirement portfolios. This paper attempts to predict retirement using Monte Carlo simulations, allowing one to probabilistically account for a range of possibiliti

This short note illustrates the theoretical solution to a trader determining how to optimally swap her wealth into a target asset through on-chain operations. It offers the framework to solve optimal slippage parameters and optimal trade size.