Les Aspects Probabilistes Du Controle Stochastique

Karoui, Nicole El

doi:10.1007/bfb0097499

Cited by 330 publications

(270 citation statements)

References 47 publications

Supporting

Mentioning

258

Contrasting

Unclassified

Order By: Relevance

“…Recently, using the method of Snell envelope (see, e.g. El Karoui [8]) combined with the theory of scalar valued RBSDEs, Hamadene and Jeanblanc [11] studied the switching problem with two modes (i.e., m = 2) in the non-Markovian context. Djehiche, Hamadene and Popier [6] generalized their result to the above switching problem with multi modes.…”

Section: Rbsde (11) Evolves In the Closureq Of Domain Qmentioning

confidence: 99%

Multi-dimensional BSDE with oblique reflection and optimal switching

Tang

2009

Probab. Theory Relat. Fields

115

165

View full text Add to dashboard Cite

In this paper, we study a multi-dimensional backward stochastic differential equation (BSDE) with oblique reflection, which is a BSDE reflected on the boundary of a special unbounded convex domain along an oblique direction, and which arises naturally in the study of optimal switching problem. The existence of the adapted solution is obtained by the penalization method, the monotone convergence, and the a priori estimations. The uniqueness is obtained by a verification method (the first component of any adapted solution is shown to be the vector value of a switching problem for BSDEs). As applications, we apply the above results to solve the optimal switching problem for stochastic differential equations of functional type, and we give also a probabilistic interpretation of the viscosity solution to a system of variational inequalities.

show abstract

Section: Rbsde (11) Evolves In the Closureq Of Domain Qmentioning

confidence: 99%

Multi-dimensional BSDE with oblique reflection and optimal switching

Tang

2009

Probab. Theory Relat. Fields

115

165

View full text Add to dashboard Cite

show abstract

“…From Theorems 2.28 and 2.29 of [5] (El Karoui, 1979), the Snell envelope has the following properties: …”

Section: Some Remarks About the Snell Envelopementioning

confidence: 99%

Reflected backward stochastic differential equations with two RCLL barriers

Lepeltier

2007

ESAIM: PS

View full text Add to dashboard Cite

show abstract

“…For more details on the definitions of Y 1 and Y 2 (and more generally on the Snell envelope of processes) one can see [1]. 2…”

Section: Verification Theoremmentioning

confidence: 99%

The stopping and starting problem in the model with jumps

Hamadène

Hdhiri

2007

Proc Appl Math and Mech

View full text Add to dashboard Cite

We consider the problem of optimal two modes switching in finite horizon in the model with jumps. We first give a verification theorem to shape the problem and later we show that it is satisfied. Finally when randomness comes from a diffusion process we study the the associated PDI variational inequalities with inter-connected obstacles. Setting of the problemLet (Ω, F , (F t ) t≤T , P ) be a stochastic basis such that F 0 contains all the P -null sets of F and for any t < T , F t+ := ∩ >0 F t+ = F t . We assume that the filtration (F t ) t≤T is generated by a standard d-dimensional Brownian motion B = (B t ) t≤T and a Poisson random measure μ on R + × U where U = R l \ {0}.A problem which underlies the one we consider here is related to management of a power plant. Actually assume we have a power station (or plant) which produces electricity whose selling price in the market is an F t -adapted stochastic process (X t ) t≤T . Electricity is non-storable, then once produced it should be almost immediately consumed. As a result, the station will produce electricity only when the production is profitable in relation with the level of X. Otherwise the production is stopped until the market selling price reaches a level which makes it profitable again and it will be resumed. Therefore a management strategy δ of this power plant is an increasing sequence of F t -stopping times (τ n ) n≥1 where lim n→∞ τ n = T , P − a.s.. For any n ≥ 1, τ 2n−1 (resp. τ 2n ) are the times when the manager of the plant makes the decision to stop (resp. resume) the production. We suppose that at t = 0 the production is open.When a strategy δ := (τ n ) n≥1 is implemented its yield is given by:the process u is bind to δ by u t = 1 if the production is open at time t and 0 otherwise ; Ψ(t, x, 1) = ψ 1 (t, x) and Ψ(t, x, 0) = ψ 2 (t, x) where ψ 1 (t, X t )dt (resp. ψ 2 (t, X t )dt) is the yield provided by the power station in a short time dt when open (resp. closed) ; a 12 > 0 (resp. a 21 > 0) is the cost when the plant is switched from the operating mode to the mothballed one (resp. conversely). The problem is to find a strategy δ * such that J(δ * ) = sup δ J(δ). The quantity J(δ * ) is the price of the power plant in the energy market. 2 Verification Theoremwhere T t is the set of F t -stopping times τ ≥ t, P − a.s.. Then Y 1 0 = sup δ J(δ) and the strategy δ * = (τ * n ) n≥1 defined as follows: τ * 0 = 0 and for n ≥ 1,

show abstract

Les Aspects Probabilistes Du Controle Stochastique

Cited by 330 publications

References 47 publications

Multi-dimensional BSDE with oblique reflection and optimal switching

Multi-dimensional BSDE with oblique reflection and optimal switching

Reflected backward stochastic differential equations with two RCLL barriers

The stopping and starting problem in the model with jumps

Contact Info

Product

Resources

About