arXiv subjects
Publications and source records attributed to Wanyang Dai.
Studying SLE$_κ$ on $S^2$ provides a new and interesting perspective for the conformality of some 2-dimensional physical models. We prove the existence and some basic properties of the spherical Minkowski content of SLE$_κ$, which is finite and can be viewed as the natural parametrization of SLE$_κ$ on $S^2$.
We study n-qubit operation rules on (n+1)-sphere with the target to help developing a (photon or other technique) based programmable quantum computer. In the meanwhile, we derive the scaling limits (called reflecting Gaussian random fields on a (n+1)-sphere) for n-qubit quantum computer based queueing systems under two different heavy traffic regimes. The queueing systems are with multiple classes of users and batch quantum random walks over the $(n+1)$-sphere as arrival inputs. In the first regime, the qubit number $n$ is fixed and the scaling is in terms of both time and space. Under this regime, performance modeling during deriving the scaling limit in terms of balancing the arrival and service rates under first-in first-out and work conserving service policy is conducted. In the second regime, besides the time and space scaling parameters, the qubit number $n$ itself is also considered as a varying scaling parameter with the additional aim to find a suitable number of qubits for the design of a quantum computer. This regime is in contrast to the well-known Halfin-Whitt regime.
Internet of quantum blockchains (IoB) will be the future Internet. In this paper, we make two new contributions to IoB: developing a block based quantum channel networking technology to handle its security modeling in face of the quantum supremacy and establishing IoB based FinTech platform model with dynamic pricing for stable digital currency. The interaction between our new contributions is also addressed. In doing so, we establish a generalized IoB security model by quantum channel networking in terms of both time and space quantum entanglements with quantum key distribution (QKD). Our IoB can interact with general structured things (e.g., supply chain systems) having online trading and payment capability via stable digital currency and can handle vector-valued data streams requiring synchronized services. Thus, within our designed QKD, a generalized random number generator for private and public keys is proposed by a mixed zero-sum and non-zero-sum resource-competition pricing policy. The effectiveness of this policy is justified by diffusion modeling with approximation theory and numerical implementations.
We study four systems and their interactions. First, we formulate a unified system of coupled forward-backward stochastic partial differential equations (FB-SPDEs) with Levy jumps, whose drift, diffusion, and jump coefficients may involve partial differential operators. A solution to the FB-SPDEs is defined by a 4-tuple general dimensional random vector-field process evolving in time together with position parameters over a domain (e.g., a hyperbox or a manifold). Under an infinite sequence of generalized local linear growth and Lipschitz conditions, the well-posedness of an adapted 4-tuple strong solution is proved over a suitably constructed topological space. Second, we consider a unified system of FB-SDEs, a special form of the FB-SPDEs, however, with skew boundary reflections. Under randomized linear growth and Lipschitz conditions together with a general completely-S condition on reflections, we prove the well-posedness of an adapted 6-tuple weak solution with boundary regulators to the FB-SDEs by the Skorohod problem and an oscillation inequality. Particularly, if the spectral radii in some sense for reflection matrices are strictly less than the unity, an adapted 6-tuple strong solution is concerned. Third, we formulate a stochastic differential game (SDG) with general number of players based on the FB-SDEs. By a solution to the FB-SPDEs, we get a solution to the FB-SDEs under a given control rule and then obtain a Pareto optimal Nash equilibrium policy process to the SDG. Fourth, we study the applications of the FB-SPDEs/FB-SDEs in queueing systems and quantum statistics while we use them to motivate the SDG.
In this paper, we prove the global risk optimality of the hedging strategy of contingent claim, which is explicitly (or called semi-explicitly) constructed for an incomplete financial market with external risk factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) processes. Analytical and numerical examples are both presented to illustrate the effectiveness of our optimal strategy. Our study establishes the connection between our financial system and existing general semimartingale based discussions by justifying required conditions. More precisely, there are three steps involved. First, we firmly prove the no-arbitrage condition to be true for our financial market, which is used as an assumption in existing discussions. In doing so, we explicitly construct the square-integrable density process of the variance-optimal martingale measure (VOMM). Second, we derive a backward stochastic differential equation (BSDE) with jumps for the mean-value process of a given contingent claim. The unique existence of adapted strong solution to the BSDE is proved under suitable terminal conditions including both European call and put options as special cases. Third, by combining the solution of the BSDE and the VOMM, we reach the justification of the global risk optimality for our hedging strategy.
In this paper, we investigate stochastic partial differential equations driven by multi-parameter anisotropic fractional Levy noises, including the stochastic Poisson equation, the linear heat equation, and the quasi-linear heat equation. Well-posedness of these equations under the fractional noises will be addressed. The multi-parameter anisotropic fractional Levy noise is defined as the formal derivative of the anisotropic fractional Levy random field. In doing so, there are two folds involved. First, we consider the anisotropic fractional Levy random field as the generalized functional of the path of the pure jump Levy process. Second, we build} the Skorohod integration with respect to the multi-parameter anisotropic fractional Levy noise by white noise approach.
We iteratively derive the product-form solutions of stationary distributions of priority multiclass queueing networks with multi-sever stations. The networks are Markovian with exponential interarrival and service time distributions. These solutions can be used to conduct performance analysis or as comparison criteria for approximation and simulation studies of large scale networks with multi-processor shared-memory switches and cloud computing systems with parallel-server stations. Numerical comparisons with existing Brownian approximating model are provided to indicate the effectiveness of our algorithm.
In this paper, based on the white noise analysis of square integrable pure-jump Levy process given by [1], we define the formal derivative of fractional Levy process defined by the square integrable pure-jump Levy process as the fractional Levy noises by considering fractional Levy process as the generalized functional of Levy process, and then we define the Skorohod integral with respect to the fractional Levy process. Moreover, we propose a class of stochastic Volterra equations driven by fractional Levy noises and investigate the existence and uniqueness of their solutions; In addition, we propose a class of stochastic differential equations driven by fractional Levy noises and prove that under the Lipschtz and linear conditions there exists unique stochastic distribution-valued solution.
We study the adapted solution, numerical methods, and related convergence analysis for a unified backward stochastic partial differential equation (B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients may involve nonlinear and high-order partial differential operators. Under certain generalized Lipschitz and linear growth conditions, the existence and uniqueness of adapted solution to the B-SPDE are justified. The methods are based on completely discrete schemes in terms of both time and space. The analysis concerning error estimation or rate of convergence of the methods is conducted. The key of the analysis is to develop new theory for random field based Malliavin calculus to prove the existence and uniqueness of adapted solutions to the first-order and second-order Malliavin derivative based B-SPDEs under random environments.
We design a dynamic rate scheduling policy of Markov type via the solution (a social optimal Nash equilibrium point) to a utility-maximization problem over a randomly evolving capacity set for a class of generalized processor-sharing queues living in a random environment, whose job arrivals to each queue follow a doubly stochastic renewal process (DSRP). Both the random environment and the random arrival rate of each DSRP are driven by a finite state continuous time Markov chain (FS-CTMC). Whereas the scheduling policy optimizes in a greedy fashion with respect to each queue and environmental state and since the closed-form solution for the performance of such a queueing system under the policy is difficult to obtain, we establish a reflecting diffusion with regime-switching (RDRS) model for its measures of performance and justify its asymptotic optimality through deriving the stochastic fluid and diffusion limits for the corresponding system under heavy traffic and identifying a cost function related to the utility function, which is minimized through minimizing the workload process in the diffusion limit. More importantly, our queueing model includes both J-user multi-input multi-output (MIMO) multiple access channel (MAC) and broadcast channel (BC) with cooperation and admission control as special cases. In these wireless systems, data from the J users in the MAC or data to the J users in the BC is transmitted over a common channel that is fading according to the FS-CTMC. The J-user capacity region for the MAC or the BC is a set-valued stochastic process that switches with the FS-CTMC fading. In any particular channel state, we show that each of the J-user capacity regions is a convex set bounded by a number of linear or smooth curved facets. Therefore our queueing model can perfectly match the dynamics of these wireless systems.
In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions (FBMs) or certain type of long-range dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little's law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.
We formulate a new class of stochastic partial differential equations (SPDEs), named high-order vector backward SPDEs (B-SPDEs) with jumps, which allow the high-order integral-partial differential operators into both drift and diffusion coefficients. Under certain type of Lipschitz and linear growth conditions, we develop a method to prove the existence and uniqueness of adapted solution to these B-SPDEs with jumps. Comparing with the existing discussions on conventional backward stochastic (ordinary) differential equations (BSDEs), we need to handle the differentiability of adapted triplet solution to the B-SPDEs with jumps, which is a subtle part in justifying our main results due to the inconsistency of differential orders on two sides of the B-SPDEs and the partial differential operator appeared in the diffusion coefficient. In addition, we also address the issue about the B-SPDEs under certain Markovian random environment and employ a B-SPDE with strongly nonlinear partial differential operator in the drift coefficient to illustrate the usage of our main results in finance.