Utility Maximization with Habit Formation: Dynamic Programming and Stochastic PDEs

Englezos, Nikolaos; Karatzas, Ioannis

doi:10.1137/070686998

Cited by 78 publications

(66 citation statements)

References 26 publications

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“…[1,13,14,37,43,44]). In the dynamic programming theory, some nonlinear BSPDEs as the backward stochastic Hamilton-Jacobi-Bellman equations, are also introduced in the investigation of non-Markovian control problems (see e.g [9,26]). Recently, there are many papers studying backward stochastic partial differential equations (see [6,28,29,36,38,42] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Liu

Zhu

2020

Forum Mathematicum

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In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDE where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.

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Section: Introductionmentioning

confidence: 99%

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Liu

Zhu

2020

Forum Mathematicum

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“…We will be interested in L p -solutions (U, V ) with values in X. BSEEs, as infinite dimensional extensions of backward stochastic differential equations, arise in many applications related to stochastic control. For instance, the Duncan-Mortensen-Zakai filtration equation for the optimal control problem of partially observed stochastic differential equations is a linear BSEE (see, e.g., [4]); in order to establish the maximum principle for the optimal control problem of stochastic evolution equations one needs to introduce a linear BSEE as the adjoint equation (see, e.g., [21,37]); in the study of controlled non-Markovian SDEs the stochastic Hamilton-Jacobi-Bellman equation is a class of fully nonlinear BSEEs (see, e.g., [11,31]); and when the coefficients of the stochastic differential equation describing the stock price are random processes, the stochastic version of the Black-Scholes formula for option pricing is a BSEE (see, e.g., [23]).…”

Section: Introductionmentioning

confidence: 99%

Backward Stochastic Evolution Equations in UMD Banach Spaces

Lü¹,

Neerven

2019

Trends in Mathematics

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“…where both the random fields u(t, x) and ψ(t, x) are unknown. Along this line, a specific fully nonlinear stochastic HJB equation was formulated by Englezos and Karatzas [12] for the utility maximization with habit formation, and more applications are referred to [1,4,14] among many others. The stochastic HJB equations are a class of backward stochastic partial differential equations (BSPDEs).…”

Section: Introductionmentioning

confidence: 99%

Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations

Qiu¹

2018

SIAM J. Control Optim.

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In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman (HJB) equation for the optimal stochastic control problem of stochastic differential equations with random coefficients. The notion of viscosity solution is introduced, and we prove that the value function of the optimal stochastic control problem is the maximal viscosity solution of the associated stochastic HJB equation. For the superparabolic cases when the diffusion coefficients are deterministic functions of time, states and controls, the uniqueness is addressed as well. (2010): 49L20, 49L25, 93E20, 35D40, 60H15 Keywords: stochastic Hamilton-Jacobi-Bellman equation, optimal stochastic control, backward stochastic partial differential equation, viscosity solution Mathematics Subject Classificationwhere T ∈ (0, ∞) is a fixed deterministic terminal time. Let U ⊂ R n be a nonempty compact set and U the set of all the U -valued and F t -adapted processes. The process (X t ) t∈[0,T ] is the state process. It is governed by the control θ ∈ U . We sometimes write X r,x;θ t for 0 ≤ r ≤ t ≤ T

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Utility Maximization with Habit Formation: Dynamic Programming and Stochastic PDEs

Cited by 78 publications

References 26 publications

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Well-posedness of backward stochastic partial differential equations with Lyapunov condition

Backward Stochastic Evolution Equations in UMD Banach Spaces

Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations

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