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Invariant representation for stochastic differential operator by BSDEs with uniformly continuous coefficients and its applications  [PDF]
Na Zhang,Guangyan Jia
Mathematics , 2012,
Abstract: In this paper, we prove that a kind of second order stochastic differential operator can be represented by the limit of solutions of BSDEs with uniformly continuous coefficients. This result is a generalization of the representation for the uniformly continuous generator. With the help of this representation, we obtain the corresponding converse comparison theorem for the BSDEs with uniformly continuous coefficients, and get some equivalent relationships between the properties of the generator $g$ and the associated solutions of BSDEs. Moreover, we give a new proof about $g$-convexity.
Multidimensional BSDEs with uniformly continuous coefficients: the general result  [PDF]
Min Li,Yufeng Shi
Mathematics , 2015,
Abstract: In this paper, by introducing a new notion of envelope of the stochastic process, we construct a family of random differential equations whose solutions can be viewed as solutions of a family of ordinary differential equations and prove that the multidimensional backward stochastic differential equations (BSDEs for short) with the general uniformly continuous coefficients are uniquely solvable. As a result, we solve the open problem of multidimensional BSDEs with uniformly continuous coefficients.
A linear numerical scheme for nonlinear BSDEs with uniformly continuous coefficients  [PDF]
Omid. S. Fard,Ali V. Kamyad
Journal of Applied Mathematics , 2004, DOI: 10.1155/s1110757x04401168
Abstract: We attempt to present a new numerical approach to solve nonlinear backward stochastic differential equations. First, we present some definitions and theorems to obtain the condition, from which we can approximate the nonlinear term of the backward stochastic differential equation (BSDE) and we get a continuous piecewise linear BSDE corresponding to the original BSDE. We use the relationship between backward stochastic differential equations and stochastic controls by interpreting BSDEs as some stochastic optimal control problems to solve the approximated BSDE and we prove that the approximated solution converges to the exact solution of the original nonlinear BSDE.
Representation theorems for generators of BSDEs with monotonic and convex growth generators  [PDF]
Shiqiu Zheng,Shoumei Li
Mathematics , 2014,
Abstract: In this paper, we establish representation theorems for generators of backward stochastic differential equations (BSDEs in short), whose generators are monotonic and convex growth in $y$ and quadratic growth in $z$. We also obtain a converse comparison theorem for such BSDEs.
A Note on BSDEs with singular coefficients  [PDF]
Monique Jeanblanc,Anthony Reveillac
Mathematics , 2013,
Abstract: In this Note we study a class of BSDEs which admits a particular singularity in their driver. More precisely, we assume that the driver is not integrable and degenerates when approaching to the terminal time of the equation.
On Trigonometric Series with Monotonic Coefficients in $L^1_{mu}$  [PDF]
Sergo A. Episkoposian,Tigran M. Grigorian
Journal of Nonlinear Analysis and Application , 2011, DOI: 10.5899/2011/jnaa-00075
Abstract: In this paper we consider the question of a representations of functions from weighed class $L^1_{mu}[0,2pi]$ by series with monotonic coefficients concerning trigonometric systems .
Limit points of the monotonic schemes  [PDF]
Julien Salomon
Physics , 2005,
Abstract: Many numerical simulations in quantum (bilinear) control use the monotonically convergent algorithms of Krotov (introduced by Tannor), Zhu & Rabitz or the general form of Maday & Turinici. This paper presents an analysis of the limit set of controls provided by these algorithms and a proof of convergence in a particular case.
Differentiability of quadratic BSDEs generated by continuous martingales  [PDF]
Peter Imkeller,Anthony Réveillac,Anja Richter
Mathematics , 2009, DOI: 10.1214/11-AAP769
Abstract: In this paper we consider a class of BSDEs with drivers of quadratic growth, on a stochastic basis generated by continuous local martingales. We first derive the Markov property of a forward--backward system (FBSDE) if the generating martingale is a strong Markov process. Then we establish the differentiability of a FBSDE with respect to the initial value of its forward component. This enables us to obtain the main result of this article, namely a representation formula for the control component of its solution. The latter is relevant in the context of securitization of random liabilities arising from exogenous risk, which are optimally hedged by investment in a given financial market with respect to exponential preferences. In a purely stochastic formulation, the control process of the backward component of the FBSDE steers the system into the random liability and describes its optimal derivative hedge by investment in the capital market, the dynamics of which is given by the forward component.
On the mixing coefficients of piecewise monotonic maps  [PDF]
Jon Aaronson,Hitoshi Nakada
Mathematics , 2004,
Abstract: We investigate the mixing coefficients of interval maps satisfying Rychlik's conditions. A mixing Lasota-Yorke map is reverse $\phi$-mixing. If its invariant density is uniformly bounded away from 0, it is $\phi$-mixing iff all images of all orders are big in which case it is $\psi$-mixing. Among $\b$-transformations, non-$\phi$-mixing is generic. In this sense, the asymmetry of $\phi$-mixing is natural.
Weak martingale representation for continuous Markov processes and application to quadratic growth BSDEs  [PDF]
Anthony Reveillac
Mathematics , 2011,
Abstract: In this paper we prove that every random variable of the form $F(M_T)$ with $F:\real^d \to\real$ a Borelian map and $M$ a $d$-dimensional continuous Markov martingale with respect to a Markov filtration $\mathcal{F}$ admits an exact integral representation with respect to $M$, that is, without any orthogonal component. This representation holds true regardless any regularity assumption on $F$. We extend this result to Markovian quadratic growth BSDEs driven by $M$ and show they can be solved without an orthogonal component. To this end, we extend first existence results for such BSDEs under a general filtration and then obtain regularity properties such as differentiability for the solution process.
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