Paper: arXiv 2502.19213

Abstract

We consider an optimal investment-consumption problem for a utility-maximizing investor who has access to assets with different liquidity and whose consumption rate as well as terminal wealth are subject to lower-bound constraints. Assuming utility functions that satisfy standard conditions, we develop a methodology for deriving the optimal strategies in semi-closed form. Our methodology is based on the generalized martingale approach and the decomposition of the problem into subproblems. We illustrate our approach by deriving explicit formulas for agents with power-utility functions and discuss potential extensions of the proposed framework. In numerical studies, we substantiate how the parameters of our framework impact the optimal proportion of initial capital allocated to the illiquid asset, the monetary value that the investor subjectively assigns to the fixed-term asset, and the potential of the illiquid asset to increase terminal the terminal value of liabilities without loss in the investor’s expected utility.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper involves advanced stochastic calculus (martingale approach, decomposition into subproblems) and derives semi-closed-form solutions for constrained utility maximization problems, indicating high mathematical density. However, the empirical portion is limited to illustrative numerical studies and conceptual discussion of parameters without mention of backtesting or real-world data analysis.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Optimal investment-consumption<br>with illiquid assets & constraints"] --> B["Methodology<br>Generalized Martingale Approach &<br>Problem Decomposition"]
  B --> C["Data/Inputs<br>Utility function parameters,<br>liquidity constraints, asset returns"]
  C --> D["Computational Process<br>Solve subproblems & derive<br>semi-closed form optimal strategies"]
  D --> E["Key Outcomes<br>Explicit formulas for power utility<br>Optimal allocation to illiquid asset<br>Value of fixed-term securities"]

Cited References (259)

  1. Less is more: Increasing retirement gains by using an upside terminal wealth constraint (2015) — link
  2. High-Water Marks and Hedge Fund Management Contracts (2003) — link
  3. Robust reinsurance contract with asymmetric information in a stochastic Stackelberg differential game (2021)
  4. Optimal excess-of-loss reinsurance contract with ambiguity aversion in the principal-agent model (2020)
  5. Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model (2016)
  6. Robust Stochastic Stackelberg Differential Reinsurance and Investment Games for an Insurer and a Reinsurer with Delay (2021)
  7. A hybrid stochastic differential reinsurance and investment game with bounded memory (2022)
  8. A stochastic Stackelberg differential reinsurance and investment game with delay in a defaultable market (2021)
and 251 more
  1. Mixed dynamic and static risk-minimization with an application to survivor swaps (2011)
  2. Worst-case-optimal dynamic reinsurance for large claims (2012)
  3. Utility maximization with a given pricing measure when the utility is not necessarily concave (2013) — link
  4. The core of an economy with nonconvex preferences (1963)
  5. A variational problem arising in economics (1965) — link
  6. Optimal investment and premium policies under risk shifting and solvency regulation (2015) — link
  7. Dynamic programming and stochastic control processes (1958) — link
  8. Portfolio Optimization with Stochastic Volatilities and Constraints: An Application in High Dimension (2007) — link
  9. Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints (2002) — link
  10. The Implicit Function Theorem (2003)
  11. Intermediate Calculus (1985) — link
  12. Basic Elements of Real Analysis (1998) — link
  13. Probability: Theory and Examples (2019)
  14. Nichtlineare Optimierung (2012)
  15. Optimal Portfolios (1997) — link
  16. Convex Analysis (1972)
  17. Does option compensation increase managerial risk appetite ? (2000) — link
  18. Risk Aversion in the Small and in the Large (1964)
  19. Essays in the Theory of Risk-Bearing (1970)
  20. Dynamic Programming (1957)
  21. Continuous-Time Finance (1992)
  22. Mathematical methods and models for economists (2000)
  23. Value-at-Risk-Based Risk Management: Optimal Policies and Asset Prices (2001) — link
  24. Optimal Portfolio Choice under Loss Aversion (2004) — link
  25. Martingale and duality methods for utility maximization in an incomplete market (1991) — link
  26. Optimal portfolio and consumption decisions for a small ‘‘investor’’ on a finite horizon (1987) — link
  27. Optimal consumption and portfolio policies when asset prices follow a diffusion process (1989) — link
  28. Lecture notes in Investment Strategies (2013)
  29. Lecture notes in Portfolio Analysis (2016)
  30. Optimal investment in hedge funds under loss aversion (2017) — link
  31. The incentives of hedge fund fees and high-water marks (2016) — link
  32. Incentive Contracts and Hedge Fund Management (2007) — link
  33. Risk aversion and incentive effects (2002) — link
  34. Portfolio optimization under convex incentive schemes (2014) — link
  35. INVESTING WITH LIQUID AND ILLIQUID ASSETS (2018) — link
  36. Illiquid Assets and Optimal Portfolio Choice (2006) — link
  37. Portfolio choice and pricing in illiquid markets (2009) — link
  38. Pricing Shared-Loss Hedge Fund Fee Structures (2016) — link
  39. Profit sharing in hedge funds (2018) — link
  40. Methods of mathematical finance (1998) — link
  41. Risk limitation and risk sharing in investment and insurance (2023) — link
  42. Non-concave utility maximization: optimal investment, stability and applications (2012)
  43. Optimal portfolio under state-dependent expected utility (2018) — link
  44. Optimal portfolio delegation when parties have different coefficients of risk aversion (2005) — link
  45. A benchmarking approach to optimal asset allocation for insurers and pension funds (2010) — link
  46. Optimal asset allocation with fixed-term securities (2016) — link
  47. Dynamic surplus optimization with performance- and index-linked liabilities (2022) — link
  48. Risk Management with Benchmarking (2006) — link
  49. Minimum return guarantees, investment caps, and investment flexibility (2016) — link
  50. Optimal Asset Allocation and Risk Shifting in Money Management (2007) — link
  51. Portfolio management with constraints (2007) — link
  52. Optimal consumption/investment policies with undiversifiable income risk and liquidity constraints (2000) — link
  53. Optimal portfolios with a positive lower bound on final wealth (2005) — link
  54. Optimal investment with minimum performance constraints (2001) — link
  55. A Dynamic Programming Approach to Constrained Portfolios (2013) — link
  56. Evaluating the Financial Performance of Pension Funds (2010) — link
  57. Reinsurance: Actuarial and Statistical Aspects (2017) — link
  58. Investment policies and reinsurance for pension funds (1985) — link
  59. Optimal management of DC pension plan under loss aversion and Value-at-Risk constraints (2016) — link
  60. Paying for Minimum Interest Rate Guarantees: Who Should Compensate Who? (2001) — link
  61. Asset allocation under loss aversion and minimum performance constraint in a DC pension plan with inflation risk (2017) — link
  62. Guarantee valuation in notional defined contribution pension systems (2016) — link
  63. The role of guarantees in defined contribution pensions (2011) — link
  64. Optimal design of the guarantee for defined contribution funds (2004) — link
  65. Optimal investment strategies and performance sharing rules for pension schemes with minimum guarantee (2011) — link
  66. Optimal management under stochastic interest rates: the case of a protected defined contribution pension fund (2001) — link
  67. Optimal investment strategies for participating contracts (2017) — link
  68. Funding Life Insurance Contracts with Guarantees: How Can We Optimally Respond to the Policyholder’s Needs? (2017)
  69. Nonconcave Optimal Investment with Value-at-Risk Constraint: An Application to Life Insurance Contracts (2020) — link
  70. Exotic Unit-Linked Life Insurance Contracts (1996) — link
  71. Optimal design of equity-linked products with a probabilistic constraint (2009) — link
  72. The design of equity-indexed annuities (2008) — link
  73. The Representation of Functionals of Brownian Motion by Stochastic Integrals (1970) — link
  74. Optimal portfolios with sustainable assets: aspects for life insurers (2023) — link
  75. Optimal portfolio strategies benchmarking the stock market (2006) — link
  76. Portfolio optimization with downside constraints (2006) — link
  77. Constrained nonsmooth utility maximization on the positive real line (2015) — link
  78. O ptimal C ontrol of D C P ension P lan M anagement under T wo I ncentive S chemes (2019) — link
  79. Optimal dynamic excess-of-loss reinsurance and multidimensional portfolio selection (2010) — link
  80. On reinsurance and investment for large insurance portfolios (2008) — link
  81. Optimal control of excess-of-loss reinsurance and investment for insurers under a CEV model (2012) — link
  82. Optimal proportional reinsurance and investment in a stock market with Ornstein–Uhlenbeck process (2011) — link
  83. Optimal reinsurance–investment problem for maximizing the product of the insurer’s and the reinsurer’s utilities under a CEV model (2014) — link
  84. Optimal investment strategies: A short survey of classical and recent results (2006)
  85. Constant proportion portfolio insurance in defined contribution pension plan management (2018) — link
  86. Constant proportion portfolio insurance in defined contribution pension plan management under discrete-time trading (2018) — link
  87. Dynamic hybrid products with guarantees — An optimal portfolio framework (2019) — link
  88. Optimal consumption, investment and life insurance with surrender option guarantee (2015) — link
  89. Optimal life-cycle consumption and investment decisions under age-dependent risk preferences (2021) — link
  90. Utility maximisation in a factor model with constant and proportional transaction costs (2019) — link
  91. Changes of num 'e raire, changes of probability measure and option pricing (1995) — link
  92. Strategic Asset Allocation in Money Management (2014) — link
  93. Convex Duality in Constrained Portfolio Optimization (1992) — link
  94. Optimal investment with S-shaped utility and trading and Value at Risk constraints: An application to defined contribution pension plan (2020) — link
  95. A Stackelberg Game of Backward Stochastic Differential Equations with Applications (2019) — link
  96. O n a new paradigm of optimal reinsurance: a stochastic S tackelberg differential game between an insurer and a reinsurer (2018) — link
  97. Stochastic differential portfolio games for an insurer in a jump-diffusion risk process (2012) — link
  98. A Leader-Follower Stochastic Linear Quadratic Differential Game (2002) — link
  99. A leader-follower stochastic linear quadratic differential game with time delay (2018) — link
  100. S tochastic differential games between two insurers with generalized mean-variance premium principle (2018) — link
  101. A class of non-zero-sum stochastic differential investment and reinsurance games (2014) — link
  102. Stackelberg Equilibrium Premium Strategies for Push-Pull Competition in a Non-Life Insurance Market with Product Differentiation (2019) — link
  103. The Maximum Principle for Global Solutions of Stochastic Stackelberg Differential Games (2015) — link
  104. Maximum Principles for Optimal Control of Forward-Backward Stochastic Differential Equations with Jumps (2010) — link
  105. Optimal investment under VaR-Regulation and Minimum Insurance (2018) — link
  106. Risk management with multiple VaR constraints (2018) — link
  107. A utility-based comparison of pension funds and life insurance companies under regulatory constraints (2011) — link
  108. Accurate and Robust Numerical Methods for the Dynamic Portfolio Management Problem (2017) — link
  109. ESG: a new dimension in portfolio allocation (2021) — link
  110. The Cross-Section of Managerial Ability, Incentives, and Risk Preferences (2014) — link
  111. Solving Consumption and Portfolio Choice Problems: The State Variable Decomposition Method (2010) — link
  112. The evolution of insurer portfolio investment strategies for long-term investing (2016) — link
  113. The relaxed investor and parameter uncertainty (2001) — link
  114. Optimal control of option portfolios and applications (1999) — link
  115. The costs of suboptimal dynamic asset allocation: General results and applications to interest rate risk, stock volatility risk, and growth/value tilts (2012) — link
  116. Constrained non-concave utility maximization: An application to life insurance contracts with guarantees (2019) — link
  117. Optimal life insurance, consumption and portfolio: A dynamic programming approach (2008) — link
  118. Optimum consumption and portfolio rules in a continuous-time model (1971) — link
  119. Portfolio optimization under S olvency II (2019) — link
  120. Portfolio optimization with wealth-dependent risk constraints (2022) — link
  121. Consumption and investment under constraints (1994) — link
  122. Optimal investment of DC pension plan under short-selling constraints and portfolio insurance (2019) — link
  123. Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems (2011)
  124. Extreme Events in Finance (2016)
  125. Volker Paper (2017)
  126. Portfolio Benchmarking Under Drawdown Constraint and Stochastic Sharpe Ratio (2018) — link
  127. Worst-Case Approach to Strategic Optimal Portfolio Selection Under Transaction Costs and Trading Limits (2015) — link
  128. The numeraire portfolio: a new perspective on financial theory (1997) — link
  129. A general equilibrium model of portfolio insurance (1995)
  130. Post-‘87 crash fears in the S & P 500 futures option market (2000) — link
  131. Robust multivariate portfolio choice with stochastic covariance in the presence of ambiguity (2018) — link
  132. Portfolio Insurance Strategies: OBPI versus CPPI (2005) — link
  133. Portfolio Insurance: The Extreme Value Approach Applied to the CPPI Method: 18 (2016) — link
  134. Arbitrage theory in continuous time (2009) — link
  135. Simplifying portfolio insurance (1987) — link
  136. Expected performance and risks of various portfolio insurance strategies (2018)
  137. Correlation Risk and Optimal Portfolio Choice (2010) — link
  138. Financial mathematics: A comprehensive treatment (2014)
  139. Mean–variance hedging and optimal investment in Heston’s model with correlation (2008)
  140. Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets (2005) — link
  141. Dynamic proportion portfolio insurance using genetic programming with principal component analysis (2008) — link
  142. Dynamic Asset Allocation (2017)
  143. Financial modelling with jump processes (2004) — link
  144. Constant proportion portfolio insurance in the presence of jumps in asset prices (2009) — link
  145. The valuation of options for alternative stochastic processes (1976) — link
  146. Optimal hedging and equilibrium in a dynamic futures market (1990)
  147. Risk appetite and endogenous risk (2010)
  148. Robust portfolio choice with derivative trading under stochastic volatility (2015) — link
  149. HARA utility maximization in a Markov-switching bond–stock market (2017) — link
  150. Theory of constant proportion portfolio insurance (1992) — link
  151. Robust portfolio choice with stochastic interest rates (2014) — link
  152. Expected stock returns and volatility (1987) — link
  153. Risk Management of CPPI Funds in Switching Regime Markets (2011)
  154. Equity portfolio insurance against a benchmark: S etting, replication and optimality (2014) — link
  155. Martingales and arbitrage in multiperiod securities markets (1979) — link
  156. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993) — link
  157. A new choice of dynamic asset management: the variable proportion portfolio insurance (2008) — link
  158. Portfolio insurance and prospect theory investors: Popularity and optimal design of capital protected financial products: rand (2011) — link
  159. Optimization methods in portfolio management and option hedging (2008)
  160. Performance evaluation of portfolio insurance strategies using stochastic dominance criteria (2009) — link
  161. Ambiguity and optimal portfolio choice with Value-at-Risk constraint (2016) — link
  162. Portfolio Optimisation with a Value at Risk Constraint in the Presence of Unhedgeable Risks (2009) — link
  163. First European Congress of Mathematics Paris, July 6-10, 1992: Vol. II: Invited Lectures (Part 2) (1994) — link
  164. Introduction to Option Pricing Theory (2000) — link
  165. Utility Maximization In Affine Stochastic Volatility Model (2010) — link
  166. Dynamic Nonmyopic Portfolio Behavior (1996) — link
  167. Optimal portfolios and Heston’s stochastic volatility model: an explicit solution for power utility (2005) — link
  168. Dynamic derivative strategies (2003) — link
  169. Portfolio Selection in Stochastic Environments (2007) — link
  170. Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case (1969) — link
  171. Stochastic Control~Problems, Viscocity Solutions, and Applications to Finance (2002)
  172. Alternative models for conditional stock volatility (1990) — link
  173. Recent developments in portfolio insurance (2008)
  174. The jump-risk premia implicit in options: evidence from an integrated time-series study (2002) — link
  175. Constant Proportion Portfolio Insurance (1986)
  176. Omega performance measure and portfolio insurance (2011) — link
  177. A stochastic calculus model of continuous trading: optimal portfolios (1986)
  178. CPPI with Cushion Insurance (2005)
  179. Contingent Claims and Market Completeness in a Stochastic Volatility Model (1997) — link
  180. Short-horizon regulation for long-term investors (2012) — link
  181. Stochastic Calculus and Financial Applications (2001) — link
  182. Infinite Dimensional Analysis: A Hitchhiker’s Guide (2006) — link
  183. Stock Price Distributions with Stochastic Volatility: An Analytic Approach (1991) — link
  184. Modeling stochastic volatility: A review and comparative study (1994)
  185. Portfolio and Consumption Decisions under Mean-Reverting Returns: An Exact Solution for Complete Markets (2002) — link
  186. Option values under stochastic volatility: Theory and empirical estimates (1987)
  187. Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model (2013) — link
  188. Optimal portfolios with regime switching and value-at-risk constraint (2010) — link
  189. Stochastic dominance of portfolio insurance strategies (2011) — link
  190. A solution approach to valuation with unhedgeable risks (2001) — link
  191. A stochastic volatility model and optimal portfolio selection (2013) — link
  192. Portfolio Selection and Dynamic Behavior in Heston’s Stochastic Volatility Model Using a Contingent Claim (2017) — link
  193. Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework (2000) — link
  194. Dynamic mean–VaR portfolio selection in continuous time (2017) — link
  195. Dynamic Asset Allocation And Consumption Choice In Incomplete Markets (2005) — link
  196. Portfolio Optimization with Risk Constraints (2005)
  197. Portfolio Optimization with Risk Constraints in the View of Stochastic Interest Rates (2017) — link
  198. Optimal fees in hedge funds with first-loss compensation using non-concave utility maximization (2018)
  199. The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets (1999) — link
  200. Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case (1991) — link
  201. Robust utility maximization with limited downside risk in incomplete markets (2007) — link
  202. A Simulation Approach to Dynamic Portfolio Choice with an Application to Learning About Return Predictability (2005) — link
  203. Optimum Consumption and Portfolio Rules in a Continuous-Time Model**I would like to thank P. A. Samuelson, R. M. Solow, P. A. Diamond, J. A. Mirrlees, J. A. Flemming, and D. T. Scheffman for their helpful discussions. Of course, all errors are mine. Aid from the National Science Foundation is gratefully acknowledged. An earlier version of the paper was presented at the second World Congress of the Econometric Society, Cambridge, England (2014) — link
  204. Decrease of capital guarantees in life insurance products: Can reinsurance stop it? (2022) — link
  205. Risk sharing in equity-linked insurance products: Stackelberg equilibrium between an insurer and a reinsurer (2022) — link
  206. Optimal fees in hedge funds with first-loss compensation — → 2310.19023
  207. A continuous-time theory of reinsurance chains (2020) — link
  208. Optimal reinsurance with model uncertainty and Stackelberg game (2022) — link
  209. Bilateral Monopoly (1928) — link
  210. Marktform und Gleichgewicht (1934)
  211. Stochastic controls: Hamiltonian systems and HJB equations (1999)
  212. Moderne Finanzmathematik – Theorie und praktische Anwendung (2014) — link
  213. Moderne Finanzmathematik – Theorie und praktische Anwendung (2018) — link
  214. A class of approximations of ruin probabilities (1977)
  215. Aspects of risk theory (1991)
  216. Lineare Algebra (2020)
  217. A course in game theory (1994)
  218. Interest-Rate Management (2002) — link
  219. Game theory (1991)
  220. Probability theory: a comprehensive course (2013)
  221. Analysis (1976)
  222. Solving ill-posed bilevel programs (2016)
  223. Pessimistic bilevel optimization (2013)
  224. Pessimistic bilevel optimization: a survey (2018)
  225. Noncooperative differential games (2011)
  226. Weak stackelberg problem and mixed solutions under data perturbations 1this paper is based upon a communication presented at the 14th symposium on mathematical programming with data perturbation, Washington DC 1992. This research has been supported by MURST Roma, Italia. $ ef (1995)
  227. A perfect calibration! Now what? (2004)
  228. Parameters Recovery via Calibration in the Heston Model: A Comprehensive Review (2016)
  229. Option valuation using the fast Fourier transform (1999)
  230. Differentiation Under the Integral Sign (1973) — link
  231. The integrated square-root process (2001)
  232. Optimal fee structures in hedge funds (2018) — link
  233. Incentives and risk taking in hedge funds (2007) — link
  234. Numerical Optimization (2006) — link
  235. Detection of Interacting Variables for Generalized Linear Models via Neural Networks (2022) — link
  236. Minimum capital requirements for market risk (2019) — link
  237. Advances in prospect theory: C umulative representation of uncertainty (1992) — link
  238. Solvency II solvency capital requirement for life insurance companies based on expected shortfall (2017) — link
  239. Bowley vs. Pareto optima in reinsurance contracting (2023) — link
  240. On the pricing of forward starting options in Heston’s model on stochastic volatility (2005) — link
  241. Constrained portfolios in incomplete markets: a dynamic programming approach to Heston’s model (2022) — link
  242. Mean field and n-agent games for optimal investment under relative performance criteria (2019) — link
  243. OPTIMAL INVESTMENT UNDER RELATIVE PERFORMANCE CONCERNS (2015) — link
  244. On time-inconsistent stochastic control in continuous time (2017) — link
  245. Time-Inconsistent Control Theory with Finance Applications (2021) — link
  246. Dynamic Mean-Variance Asset Allocation (2010)
  247. A framework for optimal portfolios with sustainable assets and climate scenarios (2024)
  248. Portfolio choice with illiquid asset for a loss-averse pension fund investor (2023) — link
  249. Portfolio Choice with Illiquid Assets (2014)
  250. Asset-liability management for long-term insurance business (2018) — link
  251. Nachhaltigkeitsratings in der Optimierung der Strategischen Asset Allokation (2021) — link

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