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1

One-dimensional reflected bsdes with two barriers under logarithmic growth and applications

El Asri Brahim, Oufdil Khalid, Ourkiya Nacer

Probability and Mathematical Statistics

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2022

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Vol. 42, Fasc. 2

251--282

EN

We deal with the problem of existence and uniqueness of a solution for one-dimensional reflected backward stochastic differential equations with two strictly separated barriers when the generator has logarithmic growth |y| |ln |y|| + |z|√(|ln |z||) in the state variables y and z. The terminal value ξ and the obstacle processes (Lt)0≤t≤T and (Ut)0≤t≤T are Lp-integrable for a suitable p > 2. The main idea is to use the concept of local solution to construct a global one. As applications, we broaden the class of functions for which mixed zero-sum stochastic differential games admit an optimal strategy and the related double-obstacle partial differential equation problem has a unique viscosity solution.

2

The maximum principle for viscosity solutions of elliptic differential functional equations

Karpowicz A.

Opuscula Mathematica

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2013

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Vol. 33, no. 1

99-105

EN

This paper is devoted to the study of the maximum principle for the elliptic equation with a deviated argument. We will consider viscosity solutions of this equation.

3

Generalized RBSDEs with random terminal time and applications to PDEs

Jańczak-Borkowska K.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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Vol. 59, no 1

85-100

EN

Generalized reflected backward stochastic differential equations have been considered so far only in the case of a deterministic interval. In this paper the existence and uniqueness of solution for generalized reflected backward stochastic differential equations in a convex domain with random terminal time is studied. Applications to the obstacle problem with Neumann boundary conditions for partial differential equations of elliptic type are given.

4

A perspective shape-from-shading metod using fast sweeping numerical scheme

Yang L., Han J.

Optica Applicata

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2008

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Vol. 38, nr 2

387-398

EN

Shape-from-shading (SFS) is an approach of 3-D shape reconstruction from only one image. A new perspective SFS method is proposed in this paper. Firstly, a reflectance map equation under perspective projection is introduced. Then, the reflectance map equation turns into a static Hamilton-Jacobi equation. So the SFS problem is formulated as a viscosity solution of the static Hamilton-Jacobi equation. The fast sweeping numerical method is used to solve the Hamilton-Jacobi equation and a new SFS method is gained. At last, experiments on both synthetic and real images are given. Experiments on the synthetic image show that the proposed SFS method is fast and accurate. Results of the real image show the efficiency of the proposed method when dealing with complicated real surface, and new criteria to evaluate the performance of the method are proposed.

5

Optimal synthesis via superdifferentials of value function

Frankowska H.

Control and Cybernetics

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2005

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Vol. 34, no 3

787-803

EN

We derive a differential inclusion governing the evolution of optimal trajectories to the Mayor problem. The value function is allowed to be discontinuous. This inclusion has convex compact right-hand sides.

6

Reflected BSDE with superlinear quadratic coefficient

Kobylański M., Lepeltier J. P., Quenez M. C., Torres S.

Probability and Mathematical Statistics

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2002

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Vol. 22, Fasc. 1

51--83

EN

In this paper, we provide existence of a reflected solution of the one-dimensional backward stochastic differential equation when the coefficient is continuous, has a superlinear growth in y and quadratic growth in z. We also give a characterization of the solution as the value function of an optimal stopping time problem. We also study the links between the solution of the quadratic RBSDE and the corresponding obstacle problem. Then we give an application of quadratic RBSDE’s to the pricing of American contingent claims in an incomplete market.