Nonlinear Expectations, Nonlinear Evaluations and Risk Measures

Péng, Shigē

doi:10.1007/978-3-540-44644-6_4

Cited by 272 publications

(345 citation statements)

References 140 publications

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“…This result is an extension of the decomposition obtained in theorem 4.3 in [BBHPS03] in the case of a dominated g-expectation (in the paper [BBHPS03], the notion of domination corresponds to the case of a driver having at most linear growth in z). In the sequel, Y stands for a given g-submartingale with terminal value Y T = B.…”

Section: Proof Of Theoremsupporting

confidence: 61%

“…Besides, this construction extends to the non subadditive case the pricing rule introduced in [EQ97]. Some other major references are [KS07], [BEK06], [RG06] and [BBHPS03]. In [KS07], the authors study some properties of these DMCUF, especially the inf-convolution procedure of these functionals and they provide links with utility indifference valuation.…”

Section: Motivationmentioning

confidence: 99%

“…This process corresponds to the conditional non linear expectation (defined in [BBHPS03]) which has been introduced for a driver g such that g = g(t, z) and g is lipschitz w.r.t the variable z.…”

Section: Notations and Preliminariesmentioning

confidence: 99%

“…The proof of the existence of the decomposition is divided in three mains steps: those steps consist in following the same scheme than in the proof of Theorem 4.3 in [BBHPS03] or also in [BCHM02] (for dominated g expectations). Many computations are standard and, for sake of completeness, we provide the outline of the proofs.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…To justify that: y n ≥ Y , for all n, we also refer to the proof given in Lemma 4.11 in [BBHPS03], which holds for dominated g-expectations 2 : the key idea of this proof consists in using both the g-submartingale property of Y and the construction of (y n ) to show that for any positive δ and for each n, {y n ≤ Y −δ} is a P-null set. Then, the monotonicity property of (y n ) results from the comparison theorem applied here for the BSDEs given by parameters (g n , Y T ) with quadratic drivers g n := g n (s, y, z) (for these kind of drivers having linear growth w.r.t.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 4 more Smart Citations

Reflected backward stochastic differential equations and a class of non-linear dynamic pricing rule

Morlais

2012

Stochastics

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In that paper, we provide a new characterization of the solutions of specific reflected backward stochastic differential equations (or RBSDEs) whose driver g is convex and has quadratic growth in its second variable: this is done by introducing the extended notion of g-Snell enveloppe. Then, in a second step, we relate this representation to a specific class of dynamic monetary concave functionals already introduced in a discrete time setting. This connection implies that the solution, characterized by means of non linear expectations, has again the time consistency property.

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Section: Proof Of Theoremsupporting

confidence: 61%

Section: Motivationmentioning

confidence: 99%

Section: Notations and Preliminariesmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

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Reflected backward stochastic differential equations and a class of non-linear dynamic pricing rule

Morlais

2012

Stochastics

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Mixed optimal control of forward‐backward stochastic system

Zhang

2021

Optim Control Appl Methods

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This paper is concerned with a mixed optimal control problem driven by forward‐backward stochastic differential equation, where the term “mixed” means that there are two controllers, one is a deterministic controller and the other is a random controller. We derive a necessary condition and a sufficient condition for the mixed optimal control problem. A linear‐quadratic mixed optimal control problem is discussed, and the corresponding optimal control is expressed by the solution of mean‐field forward‐backward stochastic differential equation. A feature here is that we obtain a mean‐field forward‐backward stochastic differential equation, which naturally arises from the study of mixed optimal control problem driven by forward‐backward stochastic differential equation without mean‐field term. As an application of theoretical results obtained here, we solve a problem of information security investment and cyber insurance.

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A BSDE approach to stochastic linear quadratic control problem

Zhang

2021

Optim Control Appl Methods

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In this article, we study a kind of linear quadratic optimal control problem driven by forward-backward stochastic differential equations (FBSDEs in short) with deterministic coefficients. The cost functional is defined by the solution of FBSDEs. By means of the Girsanov transformation, the original issue is turned equivalently into the classical LQ problem. By functional analysis approach, some necessary and sufficient conditions for the existence of optimal controls have been obtained. Moreover, we investigate the relationship between two groups of first-order and second-order adjoint equations. A new stochastic Riccati equation is derived, which leads to the state feedback form of optimal control. By introducing a new Hamiltonian function with an exponential factor, we establish the stochastic maximum principle to deal with the stochastic linear quadratic problem for forward-backward stochastic system with nonconvex control domain using first-order adjoint equation. An illustrative example is given as well. K E Y W O R D Sforward-backward stochastic differential equations, Hilbert method, maximum principle, stochastic linear quadratic optimal control the literature called a linear quadratic optimal control Problem (LQ problem, in short).

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Nonlinear Expectations, Nonlinear Evaluations and Risk Measures

Cited by 272 publications

References 140 publications

Reflected backward stochastic differential equations and a class of non-linear dynamic pricing rule

Reflected backward stochastic differential equations and a class of non-linear dynamic pricing rule

Mixed optimal control of forward‐backward stochastic system

A BSDE approach to stochastic linear quadratic control problem

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