Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market

Karoui, Nicole El; Quenez, Marie‐Claire

doi:10.1137/s0363012992232579

Cited by 535 publications

(316 citation statements)

References 12 publications

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“…Application in the pricing of contingent claims in incomplete markets It is well known that, in incomplete security markets, for a given contingent claim £, the maximum price of the contingent claim at time t is given by El Karoui and Quenez [9] and Kramkov [17] showed that under the assumption £ > 0 and s u p^ £ e »£ < oo for each v € D {V,} is a (2"-supermartingale and has the following optional decomposition theorem:…”

Section: Some Applications In Security Markets and Economic Theorymentioning

confidence: 99%

Infinite time interval BSDEs and the convergence of g-martingales

Chen¹,

Wang

2000

J Aust Math Soc A

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In this paper, we first give a sufficient condition on the coefficients of a class of infinite time interval backward stochastic differential equations (BSDEs) under which the infinite time interval BSDEs have a unique solution for any given square integrable terminal value, and then, using the infinite time interval BSDEs, we study the convergence of g-martingales introduced by Peng via a kind of BSDEs. Finally, we study the applications of g-expectations and g-martingales in both finance and economics.2000 Mathematics subject classification: primary 60H10, 60G48. Keywords and phrases: infinite time interval BSDEs, g-expectation, g-martingale, upcrossing inequality of g-martingale, convergence of g-martingale.The adapted solution for a linear BSDE which appears as the adjoint process for a stochastic control problem was first introduced by Bismut in 1973, then by Bensoussan and others, while the first result for the existence and uniqueness of an adapted solution to a nonlinear BSDE with finite time interval and Lipschitzian coefficient was obtained by Pardoux and Peng [20]. Later many researchers developed the theory and its applications in a series of papers (see for example Darling [5] Karoui, Peng and Quenez [8] and the references therein) under some other assumptions on coefficients but for fixed terminal time. From these papers, the basic theorem is that, for a fixed terminal time T > 0, under the suitable assumptions on terminal value £, coefficient g and driving process M, the following BSDE has a solution pair (y,, z,)

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Section: Some Applications In Security Markets and Economic Theorymentioning

confidence: 99%

Infinite time interval BSDEs and the convergence of g-martingales

Chen¹,

Wang

2000

J Aust Math Soc A

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“…In a frictionless financial market, this set coincides -under suitable regularity assumptions -with the set of all initial investments greater than every expectation of the contingent claim calculated with respect to all equivalent (local) martingale measures (see e.g. [9], [1] and [6]). …”

Section: Introductionmentioning

confidence: 99%

A super-replication theorem in Kabanov’s model of transaction costs

2006

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We prove a general version of the super-replication theorem, which applies to Kabanov's model of foreign exchange markets under proportional transaction costs. The market is described by a matrix-valued càdlàg bid-ask process (Π t ) t∈[0,T ] evolving in continuous time.We propose a new definition of admissible portfolio processes as predictable (not necessarily right or left continuous) processes of finite variation related to the bid-ask process by economically meaningful relations. Under the assumption of existence of a Strictly Consistent Price System (SCPS), we prove a closure property for the set of attainable vector-valued contingent claims. We then obtain the super-replication theorem as a consequence of that property, thus generalizing to possibly discontinuous bid-ask processes analogous results obtained by Kabanov [11], Kabanov and Last [12] and Kabanov and Stricker [15]. Rásonyi's counter-example [16] served as an important motivation for our approach.

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“…See Rosazza Gianin [43] or Peng [35], El Karoui & Barrieu [15], [16] for dynamic risk measures using g-expectations. Super-hedging and super pricing (see [17] and [18]) are also closely related to this formulation. For any X ∈ H, the mappings…”

Section: Nonlinear Expectation: a General Frameworkmentioning

confidence: 99%

A Girsanov Type Theorem Under G-Framework

Hao

Zhang

2011

Stochastic Analysis and Applications

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Summary. We introduce a notion of nonlinear expectation -G-expectationgenerated by a nonlinear heat equation with a given infinitesimal generator G. We first discuss the notion of G-standard normal distribution. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a G-Brownian motion. We then establish the related stochastic calculus, especially stochastic integrals of Itô's type with respect to our G-Brownian motion and derive the related Itô's formula. We have also given the existence and uniqueness of stochastic differential equation under our G-expectation. As compared with our previous framework of g-expectations, the theory of G-expectation is intrinsic in the sense that it is not based on a given (linear) probability space.

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Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market

Cited by 535 publications

References 12 publications

Infinite time interval BSDEs and the convergence of g-martingales

Infinite time interval BSDEs and the convergence of g-martingales

A super-replication theorem in Kabanov’s model of transaction costs

A Girsanov Type Theorem Under G-Framework

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