Backward problems for stochastic differential equations on the Sierpinski gasket

Li, Xuan; Qian, Zhongmin

doi:10.1016/j.spa.2017.11.002

Cited by 4 publications

(19 citation statements)

References 36 publications

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“…We end this section with a review on the representing martingale on the Sierpinski gasket. The following result was first shown in [6,Theorem (5.4)] (see also [7,Theorem 2.6]). Theorem 2.1.…”

Section: Notations and Related Resultsmentioning

confidence: 70%

“…The martingale additive functional W given by (2.1) is called the Brownian martingale on S. The following result on the singularity between the Lebesgue-Stieltjes measure induced by t → W t and the Lebesgue measure on [0, ∞) was proved in [7,Lemma 4.10].…”

Section: Notations and Related Resultsmentioning

confidence: 99%

“…By (3.6), Notice that the last integral term E λ T 0 δ(∂ x σ)(t)q(t)y ǫ (t)d W t in the above is also of order M 1 (E) o(|I ǫ |) + o m 1,λ (I ǫ ; E) . To see this, by [7,Theorem 3.5], E λ T 0 q(t) 2 e t d W t is bounded. Therefore, for any k ≥ 2, in view of supp(δ(∂ x σ)) ⊆ E ǫ and Lemma 2.3,…”

Section: )mentioning

confidence: 99%

“…Recently, to study non-linear analysis on the Sierpinski gasket, [7] developed a theory of backward stochastic differential equations (BSDEs) on the Sierpinski gasket. BSDEs and related stochastic analysis on fractals, though initially considered as efficient tools to treat quasi-linear parabolic PDEs on fractals, also have interests on their own from a mathematical finance point of view.…”

Section: Introductionmentioning

confidence: 99%

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A Stochastic Pontryagin Maximum Principle on the Sierpinski Gasket

Li¹

2018

SIAM J. Control Optim.

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In this paper, we consider stochastic control problems on the Sierpinski gasket. An order comparison lemma is derived using heat kernel estimate for Brownian motion on the gasket. Using the order comparison lemma and techniques of BSDEs, we establish a Pontryagin stochastic maximum principle for these control problems. It turns out that the stochastic maximum principle on the Sierpinski gasket involves two necessity equations in contrast to its counterpart on Euclidean spaces. This effect is due to singularity between the Hausdorff measure and the energy dominant measure on the gasket, which is a common feature shared by many fractal spaces. The linear regulator problems on the gasket is also considered as an example. * Nomura International, 30/FL Two International Finance Centre, Hong Kong.

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Section: Notations and Related Resultsmentioning

confidence: 70%

Section: Notations and Related Resultsmentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Stochastic Pontryagin Maximum Principle on the Sierpinski Gasket

Li¹

2018

SIAM J. Control Optim.

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“…Chen et al [6,7] developed a modified Lie-group shooting method and proposed a highly accurate backward-forward algorithm for multi-dimensional backward heat conduction problems. Liu and Qian [8] considered the non-linear backward problems and proved the existence and uniqueness of solutions of backward stochastic differential equations. Cheng et al [9] established a modified Tikhonov regularization method for the radially symmetric backward heat conduction problem.…”

Section: Introductionmentioning

confidence: 99%

The Solution of Backward Heat Conduction Problem with Piecewise Linear Heat Transfer Coefficient

Luo

Zhang

et al. 2019

Mathematics

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In the fields of continuous casting and the roll stepped cooling, the heat transfer coefficient is piecewise linear. However, few papers discuss the solution of the backward heat conduction problem in this situation. Therefore, the aim of this paper is to solve the backward heat conduction problem, which has the piecewise linear heat transfer coefficient. Firstly, the ill-posed of this problem is discussed and the truncated regularized optimization scheme is introduced to solve this problem. Secondly, because the regularization parameter is the key factor for the regularization method, this paper presents an improved method for choosing the regularization parameter to reduce the iterative number and proves the fourth-order convergence of this method. Furthermore, the numerical simulation experiments show that, compared with other methods, the improved method of fourth-order convergence effectively reduces the iterative number. Finally, the truncated regularized optimization scheme is used to estimate the initial temperature, and the results of numerical simulation experiments illustrate that the inverse values match the exact values very well.

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Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket

Li¹,

Qian

2019

Probab. Theory Relat. Fields

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By using analytic tools from stochastic analysis, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along fractals (which can be considered as models of simplified rough porous media). Unlike the regular space case, such parabolic type equations involving non-linear convection terms must take a different form, due to the fact that convection terms must be singular to the “linear part” which defines the heat semigroup. In order to study these parabolic type equations, a new kind of Sobolev inequalities for the Dirichlet form on the gasket will be established. These Sobolev inequalities, which are interesting on their own and in contrast to the case of Euclidean spaces, involve two norms with respect to two mutually singular measures. By examining properties of singular convolutions of the associated heat semigroup, we derive the space-time regularity of solutions to these parabolic equations under a few technical conditions. The Burgers equations on the Sierpinski gasket are also studied, for which a maximum principle for solutions is derived using techniques from backward stochastic differential equations, and the existence, uniqueness, and regularity of its solutions are obtained.

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Backward problems for stochastic differential equations on the Sierpinski gasket

Cited by 4 publications

References 36 publications

A Stochastic Pontryagin Maximum Principle on the Sierpinski Gasket

A Stochastic Pontryagin Maximum Principle on the Sierpinski Gasket

The Solution of Backward Heat Conduction Problem with Piecewise Linear Heat Transfer Coefficient

Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket

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