Paper: arXiv 2502.13678

Authors: Thijs Kamma, Antoon Pelsser

Abstract

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 consumption problem is not strictly concave and incorporates irremovable path-dependency. As a consequence, standard Lagrangian techniques fail to supply a candidate for the corresponding dual formulation. Using Fenchel’s Duality Theorem, we manage to identify a candidate formulation and prove that it satisfies strong duality. On the basis of this strong duality result, we are able to derive duality relations that stipulate how the optimal primal controls depend on the optimal dual controls and vice versa. {Moreover, using the dual formulation, we develop an analytical evaluation mechanism to bound the accuracy of approximations to the optimal solutions.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical contribution by applying Fenchel’s Duality Theorem to solve a non-concave optimal consumption problem with multiplicative habits. However, it is purely theoretical mathematical finance with no empirical data analysis or backtesting, making empirical rigor very low.

Research Flowchart

  flowchart TD
    A[Goal: Solve Optimal Consumption with<br>Multiplicative Habit Formation] --> B[Identify Challenges:<br>Non-concavity & Path-dependency]
    B --> C[Apply Fenchel's Duality Theorem]
    C --> D[Establish Strong Duality Result]
    D --> E[Derive Primal-Dual Control<br>Duality Relations]
    D --> F[Develop Analytical Evaluation Mechanism<br>for Approximation Bounds]

Cited References (309)

  1. M-quantiles (1988)
  2. Kernel quantile estimators (1990)
  3. Term structure movements and pricing interest rate contingent claims (1986)
  4. Pricing treasury inflation protected securities and related derivatives using an HJM model (2003)
  5. Dynamic asset allocation under mean-reverting returns, stochastic interest rates, and inflation uncertainty: Are popular recommendations consistent with rational behavior? (2004)
  6. Axiomatic foundations of multiplier preferences (2011)
  7. Robust portfolio choice and indifference valuation (2014)
  8. Entropy coherent and entropy convex measures of risk (2013)
and 301 more
  1. Entropy coherent and entropy convex measures of risk. Tilburg University CentER discussion paper 2011–2031 (2011)
  2. A dual theory for decision under risk and ambiguity (2015)
  3. Convex duality in constrained portfolio optimization (1992)
  4. Hedging and portfolio optimization under transaction costs: a martingale approach (1996)
  5. Backward stochastic differential equations with reflection and Dynkin games (1996)
  6. On portfolio optimization under" drawdown" constraints (1994)
  7. Contingent claim valuation and hedging with constrained portfolio (1996)
  8. Robust utility maximization for complete and incomplete market models (2005)
  9. Stochastic Differential Equations (1974)
  10. A M onte C arlo method for optimal portfolios (2003)
  11. The Malliavin Calculus and Related Topics (2006)
  12. Dynamic asset allocation under inflation (2002)
  13. Optimal consumption and investment strategies in dynamic stochastic economies (2007)
  14. Portfolio selection: a review (2014)
  15. Modeling the optimal strategy in an incomplete market (2003)
  16. When can life cycle investors benefit from time-varying bond risk premia? (2009)
  17. Why should older people invest less in stocks than younger people (1996)
  18. Optimal portfolio choice under loss aversion (2004)
  19. The pricing of options and corporate liabilities (1973)
  20. The sharpe ratio (1994)
  21. Optimal investment under var-regulation and minimum insurance (2018)
  22. Portfolio selection (1952)
  23. Delegation of Investment Management and Finanical Fairness (2018)
  24. Optimal collective investment: how costly are guarantees? (2018)
  25. The mathematics of arbitrage (2006)
  26. Methods of Mathematical Finance (1998)
  27. On the history of the Growth Optimal Portfolio (2005)
  28. A generalized C lark representation formula, with application to optimal portfolios (1991)
  29. Deflators: An Introduction (2004)
  30. Risk-neutral valuation: Pricing and hedging of financial derivatives (2013)
  31. A stochastic calculus model of continuous trading: optimal portfolios (1986)
  32. Martingale and duality methods for utility maximization in an incomplete market (1991)
  33. Closed-Form Solutions for Optimal Portfolio Selection with Stochastic Interest Rate and Investment Constraints (2005)
  34. Evaluating portfolio policies: a duality approach (2006)
  35. Optimal consumption–portfolio choices and retirement planning (2004)
  36. Consumption and portfolio decisions when expected returns are time varying (1999)
  37. Optimal consumption/investment policies with undiversifiable income risk and liquidity constraints (2000)
  38. The role of learning in dynamic portfolio decisions (1998)
  39. Approximation, perturbation, and projection methods in economic analysis (1996)
  40. Optimal pension management in a stochastic framework (2004)
  41. Stochastic lifestyling: optimal dynamic asset allocation for defined contribution pension plans (2006)
  42. Malliavin calculus for L 'e vy processes with applications to finance (2009)
  43. Dynamic portfolio optimization with transaction costs: Heuristics and dual bounds (2011)
  44. Information relaxations and duality in stochastic dynamic programs (2010)
  45. Prospect theory: An analysis of decision under risk (2013)
  46. Dynamic portfolio optimization with bounded shortfall risks (2005)
  47. Utility maximization under bounded expected loss (2009)
  48. A comparative study of portfolio insurance (2002)
  49. Dynamic asset allocation: Portfolio decomposition formula and applications (2009)
  50. Does the failure of the expectations hypothesis matter for long-term investors? (2005)
  51. Lifetime Portfolio Selection by Dynamic Stochastic Programming (1969)
  52. Monte C arlo computation of optimal portfolios in complete markets (2003)
  53. Dynamic nonmyopic portfolio behavior (1996)
  54. Portfolio selection in stochastic environments (2006)
  55. Portfolio and consumption decisions under mean-reverting returns: An exact solution for complete markets (2002)
  56. Hedging Options: The Malliavin Calculus Approach versus the $\Delta$-Hedging Approach (2003)
  57. Applications of Malliavin calculus to Monte Carlo methods in finance (1999)
  58. Option hedging for semimartingales (1991)
  59. An extension of C lark’s formula (1991)
  60. A duality method for optimal consumption and investment under short-selling prohibition. I. General market coefficients (1992)
  61. Malliavin Monte Carlo greeks for jump diffusions (2006)
  62. Dynamic consumption and portfolio choice with stochastic volatility in incomplete markets (2005)
  63. Strategic asset allocation: portfolio choice for long-term investors (2002)
  64. Learning about predictability: The effects of parameter uncertainty on dynamic asset allocation (2001)
  65. Numerical dynamic programming in economics (1996)
  66. Strategic asset allocation (1997)
  67. Transaction costs and predictability: Some utility cost calculations (1999)
  68. Optimal consumption and investment with capital gains taxes (2001)
  69. Asset prices in an exchange economy with habit formation (1991)
  70. Consumption and portfolio choice under loss aversion and endogenous updating of the reference level (2017)
  71. Labor income, borrowing constraints, and equilibrium asset prices (1993)
  72. Optimal consumption and equilibrium prices with portfolio constraints and stochastic income (1997)
  73. Utility maximization in incomplete markets with random endowment (2001)
  74. Utility based optimal hedging in incomplete markets (2002)
  75. Optimal hedging and valuation of nontraded assets (2000)
  76. Pricing derivative securities using cross-entropy: an economic analysis (2004)
  77. On the optimality of portfolio insurance (1985)
  78. Who should buy portfolio insurance? (1980)
  79. Optimal portfolio management with American capital guarantee (2005)
  80. Constrained optimization with respect to stochastic dominance: application to portfolio insurance (2006)
  81. Prospect Theory: An Analysis of Decision under Risk (1979)
  82. A computational scheme for the optimal strategy in an incomplete market (2007)
  83. Solving Constrained Consumption–Investment Problems by Simulation of Artificial Market Strategies (2013)
  84. Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursive preferences (2003)
  85. Utilitarianism and fairness in portfolio positioning (2008)
  86. Standardized versus customized portfolio: a compensating variation approach (2009)
  87. Dual control M onte- C arlo method for tight bounds of value function in regime switching utility maximization (2017)
  88. Portfolio optimization with downside constraints (2006)
  89. Asset prices under habit formation and reference-dependent preferences (2008)
  90. Optimal portfolio and consumption decisions for a small investor on a finite horizon (1987)
  91. A variational problem arising in financial economics (1991)
  92. Asymptotic properties of M onte C arlo estimators of diffusion processes (2006)
  93. Optimal portfolio policies with borrowing and shortsale constraints (2000)
  94. Optimal I nvestment (2013)
  95. Investing for State-dependent Utility (2019)
  96. Near-Optimal Dynamic Asset Allocation in Financial Markets with Trading Constraints (2019)
  97. Portfolio selection with transaction costs (1990)
  98. Dual formulation of the utility maximization problem under transaction costs (2001)
  99. Hedging and liquidation under transaction costs in currency markets (1999)
  100. Optimal portfolio selection with transaction costs and finite horizons (2002)
  101. On using shadow prices in portfolio optimization with transaction costs (2010)
  102. Risk management with benchmarking (2006)
  103. Stochastic optimal control of DC pension funds (2008)
  104. Investing for Retirement with an Explicit Benchmark (2020)
  105. Optimal portfolio, partial information and Malliavin calculus (2009)
  106. Duality in constrained optimal investment and consumption problems: a synthesis (2003)
  107. Utility maximization on the real line under proportional transaction costs (2002)
  108. Portfolio optimization under small transaction costs: a convex duality approach (2013) — link
  109. Duality theory for portfolio optimisation under transaction costs (2016)
  110. On optimal terminal wealth under transaction costs (2001)
  111. Finite horizon optimal investment and consumption with transaction costs (2009)
  112. Optimal consumption and investment with fixed and proportional transaction costs (2017)
  113. Consumption and investment under constraints (1994)
  114. Hedging contingent claims with constrained portfolios (1993)
  115. On the pricing of contingent claims under constraints (1996)
  116. Hedging American contingent claims with constrained portfolios (1998)
  117. Optimal portfolios of a small investor in a limit order market: a shadow price approach (2010)
  118. On the existence of shadow prices (2013)
  119. The general structure of optimal investment and consumption with small transaction costs (2017)
  120. Optimal investment with random endowments and transaction costs: Duality theory and shadow prices (2019)
  121. Shadow prices for continuous processes (2017)
  122. Transaction costs, shadow prices, and duality in discrete time (2014)
  123. Shadow price in the power utility case (2015)
  124. Existence of shadow prices in finite probability spaces (2011)
  125. On the existence of shadow prices for optimal investment with random endowment (2017)
  126. Transaction costs, trading volume, and the liquidity premium (2014)
  127. The dual optimizer for the growth-optimal portfolio under transaction costs (2013)
  128. Optimal consumption and investment with transaction costs and multiple risky assets (2004)
  129. Optimal portfolio selection with transaction costs for a CARA investor with finite horizon (2002)
  130. Optimal investment and consumption with transaction costs (1994)
  131. There is no nontrivial hedging portfolio for option pricing with transaction costs (1995)
  132. Optimal proportional reinsurance policies for diffusion models with transaction costs (1998)
  133. Modeling non-monotone risk aversion using SAHARA utility functions (2011)
  134. Consumption and portfolio choice under internal multiplicative habit formation (2019)
  135. Investment strategies under transaction costs: the finite horizon case (1994)
  136. Capital market equilibrium with transaction costs (1986)
  137. Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs (2013)
  138. On the game interpretation of a shadow price process in utility maximization problems under transaction costs (2013)
  139. Option pricing and hedging with small transaction costs (2015)
  140. A closed-form solution to the problem of super-replication under transaction costs (1999)
  141. Markets with transaction costs: Mathematical Theory (2009)
  142. Portfolio selection with transactions costs (1976)
  143. Martingales and arbitrage in securities markets with transaction costs (1995)
  144. Local risk-minimization under transaction costs (1998)
  145. Asymptotics and duality for the Davis and Norman problem (2012)
  146. Utility maximisation in incomplete markets (2004)
  147. Option pricing in incomplete markets (1997)
  148. Efficient analytic approximation of the optimal hedging strategy for a European call option with transaction costs (2006)
  149. Optimal portfolio hedging with nonlinear derivatives and transaction costs (1999)
  150. Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion (2017)
  151. Optimal investment with an unbounded random endowment and utility-based pricing (2009)
  152. Consumption and portfolio selection with labor income: a continuous time approach (1998)
  153. Optimal investment in incomplete markets when wealth may become negative (2001)
  154. Optimal investment and consumption when allowing terminal debt (2017)
  155. Optimal portfolio choice in the bond market (2006)
  156. Optimal investment with minimum performance constraints (2001)
  157. Offsetting the implicit incentives: Benefits of benchmarking in money management (2008)
  158. Optimal asset allocation and risk shifting in money management (2007)
  159. A benchmarking approach to optimal asset allocation for insurers and pension funds (2010)
  160. Optimal management under stochastic interest rates: the case of a protected defined contribution pension fund (2001)
  161. Optimal asset allocation for DC pension plans under inflation (2012)
  162. Optimal investment with random endowments in incomplete markets (2004)
  163. On utility maximization under convex portfolio constraints (2013)
  164. Maximizing survival, growth and goal reaching under borrowing constraints (2015)
  165. Optimal consumption and portfolio selection with portfolio constraints (2009)
  166. Optimal trading under constraints (1997)
  167. Optimal consumption choices for a ‘large’investor (1998)
  168. Optimal investment in incomplete financial markets (2002)
  169. An introduction to Malliavin calculus with applications to economics (1997)
  170. A M onte C arlo method for optimal portfolios (2003)
  171. Representation formulas for Malliavin derivatives of diffusion processes (2005)
  172. Decomposition of Optimal Dynamic Portfolio Choice with Wealth-Dependent Utilities in Incomplete Markets (2020)
  173. The instantaneous capital market line (2006)
  174. Optimal investment: bounds and heuristics (2015)
  175. Duality theory and simulation in financial engineering (2003)
  176. Duality theory and approximate dynamic programming for pricing American options and portfolio optimization (2007)
  177. Consumption and portfolio choice over the life cycle (2005)
  178. Portfolio choice with transaction costs: a user’s guide (2013)
  179. A general equilibrium model of portfolio insurance (1995)
  180. Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark (2013)
  181. Optimal housing, consumption, and investment decisions over the life cycle (2011)
  182. Backward stochastic differential equations in finance (1997)
  183. Dynamic asset allocation: Portfolio decomposition formula and applications (2010)
  184. Optimal consumption from investment and random endowment in incomplete semimartingale markets (2003)
  185. Convex duality in mean-variance hedging under convex trading constraints (2012)
  186. Convex duality in stochastic optimization and mathematical finance (2011)
  187. Minimizing expected loss of hedging in incomplete and constrained markets (2000)
  188. Dynamic hedging in incomplete markets: a simple solution (2012)
  189. Optimal portfolio choice and the valuation of illiquid securities (2001)
  190. Portfolio choice and liquidity constraints (2003)
  191. A numerical approach to solve consumption-portfolio problems with predictability in income, stock prices, and house prices (2020)
  192. Dimension-wise decompositions and their efficient parallelization (2013)
  193. Dual control Monte-Carlo method for tight bounds of value function under Heston stochastic volatility model (2020)
  194. A super-replication theorem in Kabanov’s model of transaction costs (2006)
  195. Robust hedging in incomplete markets (2014)
  196. Efficient hedging: cost versus shortfall risk (2000)
  197. Closed-form Approximations to Optimal Investment Policies in Markets with Frictions (2020)
  198. Asset prices under habit formation and catching up with the J oneses (1990)
  199. By force of habit: A consumption-based explanation of aggregate stock market behavior (1999)
  200. Habit formation, catching up with the J oneses, and economic growth (2004)
  201. Envelope theorems in Banach lattices and asset pricing (2015)
  202. Intertemporal asset pricing and the marginal utility of wealth (2011)
  203. Kim and Omberg revisited: the duality approach (2015)
  204. Consumption in incomplete markets (2020)
  205. Investing for Retirement with an Explicit Benchmark (2020)
  206. Near-optimal asset allocation in financial markets with trading constraints (2022)
  207. Robust consumption and portfolio choices with habit formation (2021)
  208. Habit formation: A resolution of the equity premium puzzle (1990)
  209. Consumption and portfolio choice under loss aversion and endogenous updating of the reference level (2020)
  210. Consumption and portfolio choice under internal multiplicative habit formation (2020)
  211. An isomorphism between asset pricing models with and without linear habit formation (2002)
  212. Investing with liquid and illiquid assets (2018)
  213. Asymptotic analysis for optimal investment in finite time with transaction costs (2012)
  214. Optimal investment with transaction costs and stochastic volatility part I: Infinite horizon (2017)
  215. Essays on intertemporal consumption and portfolio choice (2015)
  216. Homogenization and asymptotics for small transaction costs (2013)
  217. Balancing small transaction costs with loss of optimal allocation in dynamic stock trading strategies (2010)
  218. Penalty methods for continuous-time portfolio selection with proportional transaction costs (2008)
  219. Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation (2015)
  220. Asset pricing with multiplicative habit and power-expo preferences (2007)
  221. Solving consumption models with multiplicative habits (2000)
  222. Habit formation in consumption and its implications for monetary-policy models (2000)
  223. Portfolio choice with internal habit formation: A life-cycle model with uninsurable labor income risk (2003)
  224. Catching up with the J oneses: Heterogeneous preferences and the dynamics of asset prices (2002)
  225. Comparison utility in a growth model (1997)
  226. Implications of Keeping-Up-with-the- J oneses Behavior for the Equilibrium Cross Section of Stock Returns: International Evidence (2009)
  227. Saving and growth with habit formation (2000)
  228. Bequest Motives in Consumption-Portfolio Decisions with Recursive Utility (2020)
  229. Additive habit formation: consumption in incomplete markets with random endowments (2011)
  230. Utility maximization with addictive consumption habit formation in incomplete semimartingale markets (2015)
  231. Portfolio and consumption choice with stochastic investment opportunities and habit formation in preferences (2008)
  232. Financial asset pricing theory (2013)
  233. Solving life-cycle problems with biometric risk by artificial insurance markets (2021)
  234. Applied nonlinear analysis (2006)
  235. Constrained non-concave utility maximization: An application to life insurance contracts with guarantees (2019)
  236. Consumption and portfolio choice under loss aversion and endogenous updating of the reference level (2020)
  237. Non-addictive habits: optimal consumption-portfolio policies (2003)
  238. Life-cycle portfolio choice with additive habit formation preferences and uninsurable labor income risk (2007)
  239. Multiplicative habit formation and consumption: A note (2011)
  240. Dynamic specifications in optimizing trend-deviation macro models (2002)
  241. Techniques of Variational Analysis (2004)
  242. Optimization by Vector Space Methods (1997)
  243. Dual Formulation of the Optimal Consumption Problem with Multiplicative Habit Formation — → 2502.13678
  244. Convex Analysis (2015)
  245. Kalman Filter Estimation of the KNW Model (2019)
  246. Unhedgeable inflation risk within pension schemes (2020)
  247. Investing for retirement: Terminal wealth constraints or a desired wealth target? (2022)
  248. Sums of lognormals (2008)
  249. The sum of log-normal probability distributions in scatter transmission systems (1960)
  250. Dynamic programming and pricing of contingent claims in an incomplete market (1995)
  251. Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets (1996)
  252. Optional decomposition and Lagrange multipliers (1997)
  253. Dual Formulation of the Optimal Consumption Problem with Multiplicative Habit Formation — → 2502.13678
  254. Martingale Method and Applications (2022)
  255. Expectation puzzles, time-varying risk premia, and affine models of the term structure (2002)
  256. A test of the theory of reference-dependent preferences (1997)
  257. On the theory of reference-dependent preferences (2003)
  258. Toward a positive theory of consumer choice (1980)
  259. Reference-dependent preferences (2018)
  260. Loss aversion in riskless choice: A reference-dependent model (1991)
  261. Expectations as endowments: Evidence on reference-dependent preferences from exchange and valuation experiments (2011)
  262. On a model of portfolio selection with benchmark (2002)
  263. Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark (1999)
  264. Asset pricing implications of benchmarking: a two-factor CAPM (2003)
  265. Keeping up with the J oneses: Consumption externalities, portfolio choice, and asset prices (1994)
  266. The impact of keeping up with the Joneses behavior on asset prices and portfolio choice (2007)
  267. Anomalies: The endowment effect, loss aversion, and status quo bias (1991)
  268. The utility of wealth (1952)
  269. The prediction of decisions among bets. (1955)
  270. Probability versus certainty equivalence methods in utility measurement: Are they equivalent? (1985)
  271. On probabilities and loss aversion (2010)
  272. Dynamic consumption and portfolio choice under prospect theory (2020)
  273. Static portfolio choice under cumulative prospect theory (2010)
  274. Downside loss aversion and portfolio management (2006)
  275. Optimal decentralized investment management (2008)
  276. Portfolio choice under cumulative prospect theory: An analytical treatment (2011)
  277. Local expected shortfall-hedging in discrete time (2003)
  278. The Recovery Potential for Underfunded Pension Plans (2021)
  279. Behavioral finance in financial market theory, utility theory, portfolio theory and the necessary statistics: A review (2014)
  280. Dynamic mean- LPM and mean- CV a R portfolio optimization in continuous-time (2017)
  281. Asset pricing in a generalized mean-lower partial moment framework: Theory and evidence (1989)
  282. Hedging Goals (2021) — link
  283. Option valuation using the fast Fourier transform (1999)
  284. Optimal Portfolios With Lower Partial Moment Constraints And Lpm-Risk-Optimal Martingale Measures (2008)
  285. Model uncertainty: the effect on robustness, estimation and stochastic optimisation (2016)
  286. Sustainable investing with ESG rating uncertainty (2022)
  287. Dynamic portfolio allocation in goals-based wealth management (2020)
  288. Maximizing the probability of a perfect hedge (1999)
  289. ESG: a new dimension in portfolio allocation (2021)
  290. Markowitz revisited: Social portfolio engineering (2017)
  291. Sustainability in supplier selection and order allocation: Combining integer variables with Markowitz portfolio theory (2019)
  292. Goal-based portfolio optimization (2016)
  293. Sustainable investing in equilibrium (2021)
  294. Responsible investing: The ESG-efficient frontier (2021)
  295. Convex duality and Orlicz spaces in expected utility maximization (2020)
  296. Admissible strategies in semimartingale portfolio selection (2011)
  297. Shortfall risk through Fenchel duality (2018)
  298. Benchmark-driven investment for DC pension plans (2024)
  299. Pension fund investments and the valuation of liabilities under conditional indexation (2008)
  300. The cross-section of managerial ability, incentives, and risk preferences (2014)
  301. Investment–consumption–insurance optimisation problem with multiple habit formation and non-exponential discounting (2024)

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