arXiv subjects
Publications and source records attributed to Kai Du.
The Dirichlet problem for a class of stochastic partial differential equations is studied in Sobolev spaces. The existence and uniqueness result is proved under certain compatibility conditions that ensure the finiteness of $L^{p}(Ω\times(0,T),W^{2,p}(G))$-norms of solutions. The Hölder continuity of solutions and their derivatives is also obtained by embedding.
In this paper, we establish a sharp $C^{2+α}$-theory for stochastic partial differential equations of parabolic type in the whole space.
This paper investigates analytic properties of American option prices under the finite moment log-stable (FMLS) model. Under this model the price of American options is characterised by the free boundary problem of a fractional partial differential equation (FPDE) system. Using the technique of approximation we prove that the American put price under the FMLS model is convex with respect the underlying price, and specify the impact of the tail index on option prices.
A linear quadratic optimal stochastic control problem with random coefficients and indefinite state/control weight costs is usually linked to an indefinite stochastic Riccati equation (SRE) which is a matrix-valued quadratic backward stochastic differential equation along with an algebraic constraint involving the unknown. Either the optimal control problem or the SRE is solvable only if the given data satisfy a certain structure condition that has yet to be precisely defined. In this paper, by introducing a notion of subsolution for the SRE, we derive several novel sufficient conditions for the existence and uniqueness of the solution to the SRE and for the solvability of the associated optimal stochastic control problem.
Considering stochastic partial differential equations of parabolic type with random coefficients in vector-valued Hölder spaces, we obtain a sharp Schauder estimate. As an application, the existence and uniqueness of solution to the Cauchy problem is also proved.
This note concerns a class of matrix Riccati equations associated with stochastic linear-quadratic optimal control problems with indefinite state and control weighting costs. A novel sufficient condition of solvability of such equations is derived, based on a monotonicity property of a newly defined set. Such a set is used to describe a family of solvable equations.
A general maximum principle is proved for optimal controls of abstract semilinear stochastic evolution equations. The control variable, as well as linear unbounded operators, acts in both drift and diffusion terms, and the control set need not be convex.
The general maximum principle is proved for an infinite dimensional controlled stochastic evolution system. The control is allowed to take values in a nonconvex set and enter into both drift and diffusion terms. The operator-valued backward stochastic differential equation, which characterizes the second-order adjoint process, is understood via the concept of "generalized solution" proposed by Guatteri and Tessitore [SICON 44 (2006)].
The goal of this paper is to prove a result conjectured in Föllmer and Schachermayer [FS07], even in slightly more general form. Suppose that S is a continuous semimartingale and satisfies a large deviations estimate; this is a particular growth condition on the mean-variance tradeoff process of S. We show that S then allows asymptotic exponential arbitrage with exponentially decaying failure probability, which is a strong and quantitative form of long-term arbitrage. In contrast to Föllmer and Schachermayer [FS07], our result does not assume that S is a diffusion, nor does it need any ergodicity assumption.
This paper is concerned with the existence and uniqueness of weak solutions to the Cauchy-Dirichlet problem of backward stochastic partial differential equations (BSPDEs) with nonhomogeneous terms of quadratic growth in both the gradient of the first unknown and the second unknown. As an example, we consider a non-Markovian stochastic optimal control problem with cost functional formulated by a quadratic BSDE, where the corresponding value function satisfies the above quadratic BSPDE.
A general stochastic maximum principle is proved for optimal controls of semilinear stochastic evolution equations. Stochastic evolution operators, and the control with values in a general set enter into both drift and diffusion terms.
Reconnection of the self-generated magnetic fields in laser-plasma interaction was first investigated experimentally by Nilson {\it et al.} [Phys. Rev. Lett. 97, 255001 (2006)] by shining two laser pulses a distance apart on a solid target layer. An elongated current sheet (CS) was observed in the plasma between the two laser spots. In order to more closely model magnetotail reconnection, here two side-by-side thin target layers, instead of a single one, are used. It is found that at one end of the elongated CS a fan-like electron outflow region including three well-collimated electron jets appears. The ($>1$ MeV) tail of the jet energy distribution exhibits a power-law scaling. The enhanced electron acceleration is attributed to the intense inductive electric field in the narrow electron dominated reconnection region, as well as additional acceleration as they are trapped inside the rapidly moving plasmoid formed in and ejected from the CS. The ejection also induces a secondary CS.
In this paper, we consider the Cauchy problem of semi-linear degenerate backward stochastic partial differential equations (BSPDEs in short) under general settings without technical assumptions on the coefficients. For the solution of semi-linear degenerate BSPDE, we first give a proof for its existence and uniqueness, as well as regularity. Then the connection between semi-linear degenerate BSPDEs and forward backward stochastic differential equations (FBSDEs in short) is established, which can be regarded as an extension of Feynman-Kac formula to non-Markov frame.
In this paper, we consider the backward Cauchy problem of linear degenerate stochastic partial differential equations. We obtain the existence and uniqueness results in Sobolev space $L^p(Ω; C([0,T];W^{m,p}))$ with both $m\geq 1$ and $p\geq 2$ being arbitrary, without imposing the symmetry condition for the coefficient $σ$ of the gradient of the second unknown---which was introduced by Ma and Yong [Prob. Theor. Relat. Fields 113 (1999)] in the case of $p=2$. To illustrate the application, we give a maximum principle for optimal control of degenerate stochastic partial differential equations.
This paper is concerned with the strong solution to the Cauchy-Dirichlet problem for backward stochastic partial differential equations of parabolic type. Existence and uniqueness theorems are obtained, due to an application of the continuation method under fairly weak conditions on variable coefficients and $C^2$ domains. The problem is also considered in weighted Sobolev spaces which allow the derivatives of the solutions to blow up near the boundary. As applications, a comparison theorem is obtained and the semi-linear equation is discussed in the $C^2$ domain.
This paper is concerned with semi-linear backward stochastic partial differential equations (BSPDEs for short) of super-parabolic type. An $L^p$-theory is given for the Cauchy problem of BSPDEs, separately for the case of $p\in (1,2]$ and for the case of $p\in (2, \infty)$. A comparison theorem is also addressed.
Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in the whole Euclidean space. Improved existence and uniqueness results are given in the Sobolev space $H^n$ ($=W^n_2$) under weaker assumptions than those used by X. Zhou [Journal of Functional Analysis 103, 275--293 (1992)]. As an application, a comparison theorem is obtained.
Backward stochastic partial differential equations of parabolic type with variable coefficients are considered in smooth domains. Existence and uniqueness results are given in weighted Sobolev spaces allowing the derivatives of the solutions to blow up near the boundary.