Stochastic Differential Utility Research Articles

Consumption and portfolio optimization with generalized stochastic differential utility in incomplete markets

This paper examines continuous time intertemporal consumption and portfolio choice problems of an investor in a generalized stochastic differential utility preference of Epstein–Zin type with subjective beliefs and with ambiguity, respectively. In these two situations, the aggregators of the utility both contain the “variability” process. Based on the generalized verification theorem which formulates Hamilton–Jacobi-Bellman equations under a weak non-Lipschitz condition, we show the explicit closed-form optimal consumption and portfolio solutions with subjective beliefs and the numerical solutions with ambiguity for Heston model in incomplete market, respectively.

Green Bond Pricing and Optimization Based on Carbon Emission Trading and Subsidies: From the Perspective of Externalities

In this paper, we establish a model based on real options theory and fractional Brownian motion (FBM) with jumps to price green bonds, and thus alleviate the externalities of green bonds. We assume that the floating value of green bonds is linked to the carbon price. The carbon emission trading mechanism and government subsidy policy are introduced into this model, and the expression is derived from the stochastic differential utility framework based on the fast Fourier transform method. Based on the numerical analysis and the simulations, this paper analyzes when governments are facing financial and carbon emission constraints and how policymakers balance the allocation between carbon allowances and government subsidies to help green bonds reach the exogenous equilibrium price. Our results have implications in terms of optimizing the distribution of economic resources by the reasonable pricing of green bonds. It is in line with the current theme of global energy conservation and emission reduction, and also has certain guiding significance for the development of the carbon emission trading market.

Optimal investment, consumption and life insurance strategies under stochastic differential utility with habit formation

<p style='text-indent:20px;'>This paper studies the optimal investment, consumption and life insurance decisions of an agent under stochastic differential utility. The optimal choice is obtained through dynamic programming method. We state a verification theorem using the Hamilton-Jacobi-Bellman equation. For the special case of Epstein-Zin preferences, we derive the analytical solution to the problem. Moreover, we explore the effects of habit formation and of the elasticity of the utility function on the optimal decision through a numerical simulation based on Chinese mortality rates. We show that habit formation does not change the basic shape of the consumption and bequest curves. With habit formation, the optimal consumption curve moves up with lower initial consumption, while the bequest curve moves down. Increasing the value of initial habit formation slightly decreases both optimal consumption and bequests. The changes in the habit formation parameters have a greater impact on the curves than does a change in the initial habit formation.</p>

The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. II: Existence, uniqueness and verification for vartheta in (0,1)

In this article, we consider the optimal investment–consumption problem for an agent with preferences governed by Epstein–Zin (EZ) stochastic differential utility (SDU) over an infinite horizon. In a companion paper Herdegen et al. (Finance Stoch. 27:127–158, 2023), we argued that it is best to work with an aggregator in discounted form and that the coefficients R of relative risk aversion and S of elasticity of intertemporal complementarity (the reciprocal of the coefficient of elasticity of intertemporal substitution) must lie on the same side of unity for the problem to be well founded. This can be equivalently expressed as vartheta := frac{1-R}{1-S} >0.In this paper, we focus on the case vartheta in (0,1). The paper has three main contributions: first, to prove existence of infinite-horizon EZ SDU for a wide class of consumption streams and then (by generalising the definition of SDU) to extend this existence result to any consumption stream; second, to prove uniqueness of infinite-horizon EZ SDU for all consumption streams; and third, to verify the optimality of an explicit candidate solution to the investment–consumption problem in the setting of a Black–Scholes–Merton financial market.

The infinite-horizon investment–consumption problem for Epstein–Zin stochastic differential utility. I: Foundations

The goal of this article is to provide a detailed introduction to infinite-horizon investment–consumption problems for agents with preferences described by Epstein–Zin (EZ) stochastic differential utility (SDU). In the setting of a Black–Scholes–Merton market, we seek to describe all parameter combinations that lead to a well-founded problem in the sense that the problem is not just mathematically well posed, but the solution is also economically meaningful. The key idea is to consider a novel and slightly different description of EZ SDU under which the aggregator has only one sign. This new formulation clearly highlights the necessity for the coefficients of relative risk aversion and of elasticity of intertemporal complementarity (the reciprocal of the coefficient of intertemporal substitution) to lie on the same side of unity.

Volatility-reducing biodiversity conservation under strategic interactions

How can decentralized individual decisions inefficiently reduce the ability of biodiversity to mitigate ecological and environmental variability and then its “natural insurance” role? In this article we present a simple theoretical set-up to address this question and to evaluate some policy options.We study a model of strategic competition among farmers for the conversion of a natural forest to agricultural land. Unconverted forest land allows to conserve biodiversity, which contributes to reducing the volatility of agricultural production. Agents' utility is given in terms of a Kreps Porteus stochastic differential utility capable of disentangling risk aversion and aversion to fluctuations.We characterize the land used by each farmer and her welfare at the Nash equilibrium, we evaluate the over-exploitation of the land and the agents' welfare loss compared to the socially optimal solution and we study the drivers of the inefficiencies of the decentralized equilibrium.After characterizing the value of biodiversity in the model, we use it to obtain a decomposition which helps to study the policy implications of the model by identifying in which cases the allocation of property rights is preferable to the introduction of a tax on land conversion. Our results suggest that enforcing property rights is more relevant in case of stagnant economies while taxing land conversion may be more suited for rapidly developing economies.

Robust consumption portfolio optimization with stochastic differential utility

This paper examines a continuous time intertemporal consumption and portfolio choice problem with a stochastic differential utility preference of Epstein–Zin type for a robust investor, who worries about model misspecification and seeks robust decision rules. We provide a verification theorem which formulates the Hamilton–Jacobi–Bellman–Isaacs equation under a non-Lipschitz condition. Then, with the verification theorem, the explicit closed-form optimal robust consumption and portfolio solutions to a Heston model are given. Also we compare our robust solutions with the non-robust ones, and the comparisons shown in a few figures coincide with our common sense.

Proper solutions for Epstein–Zin Stochastic Differential Utility

In this article, we consider the optimal investment-consumption problem for an agent with preferences governed by Epstein--Zin stochastic differential utility (EZ-SDU) who invests in a constant-parameter Black-Scholes-Merton market over the infinite horizon. The parameter combinations that we consider in this paper are such that the risk aversion parameter $R$ and the elasticity of intertemporal complementarity $S$ satisfy $\theta=\frac{1-R}{1-S}>1$. In this sense, this paper is complementary to Herdegen, Hobson and Jerome [arXiv:2107.06593]. The main novelty of the case $\theta>1$ (as opposed to $\theta\in(0,1)$) is that there is an infinite family of utility processes associated to every nonzero consumption stream. To deal with this issue, we introduce the economically motivated notion of a proper utility process, where, roughly speaking, a utility process is proper if it is nonzero whenever future consumption is nonzero. We then proceed to show that for a very wide class of consumption streams $C$, there exists a proper utility process $V$ associated to $C$. Furthermore, for a wide class of consumption streams $C$, the proper utility process $V$ is unique. Finally, we solve the optimal investment-consumption problem in a constant parameter financial market, where we optimise over the right-continuous attainable consumption streams that have a unique proper utility process associated to them.

Gain/loss asymmetric stochastic differential utility

This study examines a gain/loss asymmetric utility in continuous time in which the investor discounts their utility gain by more than the utility loss. By employing the theory of stochastic differential utility, the model allows an endogenously time-varying subjective discount rate. In addition, the model can express various forms of utility functions including a version of the Epstein–Zin utility. Under the model, even if the state variables do not have any jump in their paths, the optimal consumption/wealth ratio and portfolio weight can change non-smoothly.

Gestão de Carteiras sob Múltiplos Regimes: Estratégias que Performam Acima do Mercado

RESUMO Contexto: modelos de múltiplos regimes econômicos para alocação de ativos sob a função utilidade estocástica diferencial não constavam na literatura até o modelo de Campani, Garcia e Lewin (2020). Esta função está na fronteira do conhecimento pois, mais realisticamente que as demais, separa os principais parâmetros de risco do investidor. Tal configuração, porém, torna-se complexa a ponto de não admitir solução exata na literatura e a simulação de Monte Carlo não é ágil para solucionar o problema em tempo real. O modelo resolve esta questão com uma aproximação precisa o suficiente para a alocação e nosso artigo oferece a primeira aplicação fora do cenário estadunidense. Objetivo: avaliar estratégias de portfólio montadas a partir do modelo em tela, e comparar suas performances aos retornos dos principais benchmarks do mercado. Métodos: propomos estratégias com e sem vendas a descoberto sob quatro regimes econômicos latentes a partir dos retornos dos ativos: cash (CDI), renda fixa (IMA-G), ações domésticas (IBrX-100) e ações internacionais (S&P 500) em reais. Com estes parâmetros, estimamos as probabilidades de ocorrência dos regimes e definimos os pesos de cada ativo nas carteiras (estratégias de múltiplos regimes). Comparamos as performances destas carteiras com índices de mercado e modelos míopes (estratégias de regime único), calculando a significância estatística dos resultados através do teste de Wilcox. Resultados: com esta pesquisa, tornamo-nos pioneiros ao identificar pela primeira vez quatro regimes econômicos no Brasil para otimização de carteiras. Os resultados indicam que (i) a política de portfólio depende fortemente do estado corrente da economia; e (ii) as estratégias propostas superam o mercado em termos de retornos e índice de Sharpe. Conclusão: os modelos de múltiplos regimes mostram-se relevantes para a gestão de carteiras e estratégias baseadas nestes modelos, por sua vez, podem implicar em soluções que beneficiem gestores de investimentos.

Mean‐field backward stochastic differential equations driven by G‐Brownian motion and related partial differential equations

In this paper, we study mean‐field backward stochastic differential equations driven by G‐Brownian motion (G‐BSDEs). We first obtain the existence and uniqueness theorem of these equations. In fact, we can obtain local solutions by constructing Picard contraction mapping for Y term on small interval, and the global solution can be obtained through backward iteration of local solutions. Then, a comparison theorem for this type of mean‐field G‐BSDE is derived. Furthermore, we establish the connection of this mean‐field G‐BSDE and a nonlocal partial differential equation. Finally, we give an application of mean‐field G‐BSDE in stochastic differential utility model.

A generalized stochastic differential utility driven by G-Brownian motion

This paper introduces a class of generalized stochastic differential utility (GSDU) models in a continuous-time framework to capture ambiguity aversion on the financial market. This class of GSDU models encompasses several classical approaches to ambiguity aversion and includes new models about ambiguity aversion. For a general GSDU model, we demonstrate its continuity, monotonicity, time consistency, concavity, and homotheticity. We investigate its comparative ambiguity aversion and direction aversion under sufficient conditions. We further solve an optimal portfolio choice problem in one GSDU model as an application.

Convex duality for Epstein–Zin stochastic differential utility

AbstractThis paper introduces a dual problem to study a continuous‐time consumption and investment problem with incomplete markets and Epstein–Zin stochastic differential utilities. Duality between the primal and dual problems is established. Consequently, the optimal strategy of this consumption and investment problem is identified without assuming several technical conditions on market models, utility specifications, and agent's admissible strategies. Meanwhile, the minimizer of the dual problem is identified as the utility gradient of the primal value and is economically interpreted as the “least favorable” completion of the market.

Continuous-time smooth ambiguity preferences

This study extends the smooth ambiguity preferences model proposed by Klibanoff et al. (2005) to a continuous-time dynamic setting. It is known that the original smooth ambiguity preferences converge to the subjective expected utility as the time interval shortens so that decision makers do not exhibit any ambiguity-sensitive behavior in the continuous-time limit. Accordingly, this study proposes an alternative model of these preferences that interchanges the role of the second-order utility function with that of the second-order probability to prevent the smooth ambiguity attitude of decision maker from evaporating in the continuous-time limit. By utilizing the utility convergence results established by Kraft and Seifried (2014), our model is eventually represented by the stochastic differential utility with distorted beliefs so that most existing techniques in economics and financial studies can be made applicable together with these distorted beliefs. We give an asset-pricing example to demonstrate the applicability of our model.

Optimal consumption, portfolio, and life insurance policies under interest rate and inflation risks

This paper solves the optimal life insurance, consumption, and portfolio decisions of a wage earner before retirement under interest rate and inflation risks. The wage earner’s preferences are represented by the stochastic differential utility, which separates the coefficient of relative risk aversion from the elasticity of intertemporal substitution (EIS). The wage earner’s life insurance demand is affected by the volatile interest rates and inflation. The optimal life insurance demand decreases with the level of nominal interest rates. Under an assumption of deterministic nominal income, the demand for life insurance would not be affected by the level of inflation. However, if the wage earner’s income is indexed to inflation, the life insurance demand would increase with the level of inflation. Furthermore, under investment opportunities with greater volatilities, wage earners who optimally allocate their wealth to the financial market benefit more from financial investments and cut their demand for life insurance. An analysis of EIS and risk aversion on life insurance demand shows that the demand for life insurance over the planning horizon increases with the measure of relative risk aversion but decreases with EIS. Optimal consumption is affected by the insurance premium load and the direction depends on the size of EIS relative to unity.

Consumption optimization for recursive utility in a jump-diffusion model

In this paper, we consider a market model with prices and consumption following a jump-diffusion dynamics. In this setting, we first characterize the optimal consumption plan for an investor with recursive stochastic differential utility on the basis of his/her own beliefs, then we solve the inverse problem to find what beliefs make a given consumption plan optimal. The problem is viewed in general for a class of homogeneous recursive utility, and later we choose a logarithmic model for the utility aggregator as an explicitly computable example. When beliefs, represented via Girsanov’s theorem, get incorporated into the model, the change of measure gives rise, up to a transformation, to a backward stochastic differential equation whose generator exhibits a quadratic behavior in the Brownian component and a locally Lipschitz one in the jump component, which is solvable on the basis of some recent results.

Decisiones de consumo y portafolio con Utilidad Diferencial Recursiva Estocástica (UDRE): Modelos alternativos

The intertemporal elasticity of substitution and the risk aversion coefficient are regularly obtained from models for the valuation of assets based on consumption as in Merton (1973), Lucas (1978), Breeden (1979) and Maenhout (2004). However, this approach is criticized for two reasons: the first is related to the empirical issue, since in practical research the models do not fit the data, and the second one does not distinguish between intertemporal elasticity of substitution and risk aversion coefficient due the functional form of utility assumed in this approach. Evidently, these two concepts are useful in the economic theory, since they are related to different relevant aspects of consumer preferences. This research focuses on the second problem, being the goal of this paper to use the concept of Recursive Stochastic Differential Utility to obtain the parameters in question, separately, but in the same model. Under this framework, several continuous-time decision making models of a rational consumer having access to different assets are developed, this allows a better interpretation and explanation of the mentioned parameters. Finally, comparative static exercises are performed to explain the dynamics of the decision variable to changes in the independent variables with Mexican data.

A Continuous-Time Asset Pricing Model with Smooth Ambiguity Preferences

This study extends the smooth ambiguity preferences model proposed by Klibanoff et al. (2005, 2009) to a continuous-time dynamic setting. It is known that these preferences converge to the subjective expected utility as the time interval shortens so that decision makers do not exhibit any ambiguity-sensitive behavior in the continuous-time limit. Accordingly, this study proposes an alternative model of decision making that applies Yaari’s (1987) dual theory to the original preferences and interchanges the role of the second-order utility function with that of the second-order probability. We then formulate a recursive utility of smooth ambiguity-sensitive decision makers in continuous-time. Our model is represented by the stochastic differential utility with distorted beliefs so that most existing techniques in financial studies can be made applicable together with these distorted beliefs. We give an asset-pricing example to demonstrate the applicability of our model.

Optimistic about the future? How uncertainty and expectations about future consumption prospects affect optimal consumer behavior

AbstractThis paper analyzes a continuous time stochastic growth model with a Duffie and Epstein (Duffie, D., and L. Epstein. 1992. “Stochastic Differential Utility.”

Convex Duality for Stochastic Differential Utility

This paper introduces a dual problem to study a continuous-time consumption and investment problem with incomplete markets and Epstein-Zin stochastic differential utility. Duality between the primal and dual problems is established. Consequently the optimal strategy of this consumption and investment problem is identified without assuming several technical conditions on market model, utility specification, and agent's admissible strategy. Meanwhile the minimizer of the dual problem is identified as the utility gradient of the primal value and is economically interpreted as the least favorable completion of the market.