A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications

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.Comment: 22 pagea, 1 figur

Product-form solutions for integrated services packet networks and cloud computing systems

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.Comment: 26 pages, 3 figures, short conference version is reported at MICAI 200

Optimal Rate Scheduling via Utility-Maximization for J-User MIMO Markov Fading Wireless Channels with Cooperation

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.Comment: 53 pages, Originally submitted on June 17, 2010; Revised version submitted on December 24, 201

Mean-variance hedging based on an incomplete market with external risk factors of non-Gaussian OU processes

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.Comment: 36 page

Brownian approximations for queueing networks with finite buffers : modeling, heavy traffic analysis and numerical implementations

Ph.D.Jiangang Da