Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

Ekren, Ibrahim; Zhang, Jianfeng

doi:10.1186/s41546-016-0010-3

Cited by 24 publications

(15 citation statements)

References 29 publications

(118 reference statements)

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“…We shall remark though, the above PPDE is degenerate, and thus the uniqueness result of [13] does not apply here. We refer to the more recent works [28,14], in which it was shown that W is indeed the unique viscosity solution. We also refer to [27,33] for numerical methods for PPDEs.…”

Section: Characterization Of W By Ppdesmentioning

confidence: 99%

Dynamic approaches for some time-inconsistent optimization problems

Karnam¹,

Ma²,

Zhang³

2017

Ann. Appl. Probab.

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In this paper we investigate possible approaches to study general time-inconsistent optimization problems without assuming the existence of optimal strategy. This leads immediately to the need to refine the concept of time-consistency as well as any method that is based on Pontryagin's Maximum Principle. The fundamental obstacle is the dilemma of having to invoke the Dynamic Programming Principle (DPP) in a timeinconsistent setting, which is contradictory in nature. The main contribution of this work is the introduction of the idea of the "dynamic utility" under which the original time inconsistent problem (under the fixed utility) becomes a time consistent one. As a benchmark model, we shall consider a stochastic controlled problem with multidimensional backward SDE dynamics, which covers many existing time-inconsistent problems in the literature as special cases; and we argue that the time inconsistency is essentially equivalent to the lack of comparison principle. We shall propose three approaches aiming at reviving the DPP in this setting: the duality approach, the dynamic utility approach, and the master equation approach. Unlike the game approach in many existing works in continuous time models, all our approaches produce the same value as the original static problem.In this paper we propose some possible approaches to tackle the general time-inconsistent optimization problems in continuous time setting. These approaches are different from all the existing ones in the literature, and are based on our new understanding of the time inconsistency. We note that the time inconsistency appears naturally and frequently in economics and finance, see e.g. and Kahneman-Tversky [20,21]. We refer to the frequently cited survey Strotz [30] for the fundamentals of this problem, and Zhou [34] for some recent development on continuous time models. We should point out that it was [34] that brought the time inconsistency issue to our attention. I. Time inconsistency. We begin by briefly describing the time-inconsistency in an optimization problem that has been understood so far. Consider an optimization problem over a time interval [0, T ]: V 0 := sup u∈U [0,T ] J(u).( 1.1) where U [0,T ] is an appropriate set of admissible controls u defined on [0, T ], and J(u) is a certain utility functional associated to u. Clearly, the problem (1.1) is static. Its dynamic counterpart is the following optimization problem over [t, T ], for any t ∈ [0, T ]:Here U [t,T ] is the corresponding set of admissible controls on [t, T ] and utility functional J t usually involves some conditional expectation, and thus could be random.An admissible control u * ∈ U [0,T ] is called "optimal" for the problem (1.1) if J(u * ) = V 0 .Defining optimal control u t, * for the problem (1.2) similarly and assuming their existence, we say the problem (1.2) is time-consistent if, for any t ∈ [0, T ], it holds thatThe relation (1.3) amounts to saying that a (temporally) global optimum must be a local one.The optimization problem (1.2) is called time-inconsis...

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Section: Characterization Of W By Ppdesmentioning

confidence: 99%

Dynamic approaches for some time-inconsistent optimization problems

Karnam¹,

Ma²,

Zhang³

2017

Ann. Appl. Probab.

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“…Existence and uniqueness of strong viscosity solutions was proved for a class of parabolic semilinear PPDEs, which can be expected because of their link with BSDEs and the associated stability results. Finally, in [11], the authors introduced a notion of pseudo-Markovian viscosity solutions for fully nonlinear PPDEs by considering a sequence of PPDEs with generators F n associated to paths frozen up to the exit time of certain domains. In their case, existence in the fully nonlinear setting relies on the convergence of a suitable sequence of functions, which is a-priori non trivial.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we introduce a novel weak notion of solutions in the same vein as the strong viscosity solutions of [4] and the pseudo-Markovian viscosity solutions of [11], in the sense that it is defined by an approximation argument. Similar to [11], we consider PPDEs with fully nonlinear, and possibly degenerate, generator F , which can exhibit a non-uniformly continuous linear part. As in [11], our construction is based on a finite dimensional approximation of paths, that are frozen on some time intervals.…”

Section: Introductionmentioning

confidence: 99%

“…Similar to [11], we consider PPDEs with fully nonlinear, and possibly degenerate, generator F , which can exhibit a non-uniformly continuous linear part. As in [11], our construction is based on a finite dimensional approximation of paths, that are frozen on some time intervals. The difference is that we do not consider a sequence of (possible infinitely many) hitting times to define the frozen-path PPDE, but use a simple sequence π = (π n ) n≥1 of discrete time grids on [0, T ], with π n = (0 = t n 0 < • • • < t n n = T ), and then freeze the path argument on each time interval [t n i , t n i+1 ) to obtain our sequence of generators F n , see (2.7) below.…”

Section: Introductionmentioning

confidence: 99%

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Approximate viscosity solutions of path-dependent PDEs and Dupire's vertical differentiability

Bouchard,

Loeper,

Tan

2021

Preprint

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We introduce a notion of approximate viscosity solution for a class of nonlinear path-dependent PDEs (PPDEs), including the Hamilton-Jacobi-Bellman type equations. Existence, comparaison and stability results have been established under fairly general conditions. It is also consistent with smooth solutions when the dimension is less or equal to two, or the non-linearity is concave in the second order space derivative. We finally investigate the regularity (in the sense of Dupire) of the solution to the PPDE.

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“…When all the coefficients are just possibly random but with state-dependence, the resulting Hamilton-Jacobi equation is just a BSPDE (see [3,28,29]); for related research on general BSPDEs, we refer to [2,6,12,17,25] to mention but a few. When the coefficients are both random and path-dependent, some discussions may be found in [14,15,30] where, nevertheless, all the coefficients are required to be continuous and deterministic in ω ∈ Ω and the resulting value function satisfies, instead, a deterministic path-dependet semilinear parabolic PDE. In the present work, all the involved coefficients are just measurable w.r.t.…”

Section: Introductionmentioning

confidence: 99%

Controlled Ordinary Differential Equations with Random Path-Dependent Coefficients and Stochastic Path-Dependent Hamilton-Jacobi Equations

Qiu¹

2020

Preprint

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This paper is devoted to the stochastic optimal control problem of ordinary differential equations allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases, the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.

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Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

Cited by 24 publications

References 29 publications

Dynamic approaches for some time-inconsistent optimization problems

Dynamic approaches for some time-inconsistent optimization problems

Approximate viscosity solutions of path-dependent PDEs and Dupire's vertical differentiability

Controlled Ordinary Differential Equations with Random Path-Dependent Coefficients and Stochastic Path-Dependent Hamilton-Jacobi Equations

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