Uncertainty and economic growth in a stochastic R&D model

The paper examines an R&D model with uncertainty from the population growth, which is a stochastic cooperative Lotka-Volterra system, and obtains a suciently condition for the existence of the globally positive solution. The long-run growth rate of the economic system is ultimately bounded in mean and fluctuation of its growth will not be faster than the polynomial growth. When uncertainty of the population growth, in comparison with its expectation, is suciently large, the growth rate of the technological progress andthe capital accumulation will converge to zero. Inversely, when uncertainty of the population growth is suciently small or its expected growth rate is suciently high, the economic growth rate will not decay faster than the polyno-mial speed. The paper explicitly computes the sample average of the growth rates of both the technology and the capital accumulation in time and compares them with their counterparts in the corresponding deterministic model

The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications

This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem andthe linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type

On solutions of a class of infinite horizon FBSDEs

In this paper, we study the solvability of a class of infinite horizon forward-backward stochastic differential equations (FBSDEs, for short). Under some mild assumptions on the coefficients in such FBSDEs, the existence and uniqueness result of adapted solutions is established. The method adopted here is based on constructing a contraction mapping related to the solution of the forward SDE in the FBSDEs.

Hilbert space-valued forward–backward stochastic differential equations with Poisson jumps and applications

AbstractIn this paper, we study a class of Hilbert space-valued forward–backward stochastic differential equations (FBSDEs) with bounded random terminal times; more precisely, the FBSDEs are driven by a cylindrical Brownian motion on a separable Hilbert space and a Poisson random measure. In the case where the coefficients are continuous but not Lipschitz continuous, we prove the existence and uniqueness of adapted solutions to such FBSDEs under assumptions of weak monotonicity and linear growth on the coefficients. Existence is shown by applying a finite-dimensional approximation technique and the weak convergence theory. We also use these results to solve some special types of optimal stochastic control problems

Optimal investment and consumption strategies for an investor with stochastic economic factor in a defaultable market

This paper considers the issue of optimal investment and consumption strategies for an investor with stochastic economic factor in a defaultable market. In our model, the price process is composed of a money market account and a default-free risky asset, assuming they rely on a stochastic economic factor described by a diffusion process. A defaultable perpetual bond is depicted by the reduced-form model, and both the default risk premium and the default intensity of it rely on the stochastic economic factor. Our goal is to maximize the infinite horizon expected discounted power utility of the consumption. Applying the dynamic programming principle, we derive the Hamilton--Jacobi--Bellman (HJB) equations and analyze them using the so-called sub-super solution method to prove the existence and uniqueness of their classical solutions. Next, we use a verification theorem to derive the explicit formula for optimal investment and consumption strategies. Finally, we provide a sensitivity analysis

Comments on "finite-time stability theorem of stochastic nonlinear systems" [Automatica 46 (2010) 2105-2108]

In this comment, we will point out some errors existing in Chen and Jiao (2010) from definitions to the proof of the main result, where the authors discussed the finite-time stability of stochastic nonlinear systems and proved a Lyapunov theorem on the finitetime stability

A finite-time stability theorem of stochastic nonlinear systems and stabilization designs

This paper focuses on the finite-time stability and stabilization designs of stochastic nonlinear systems. We first present and discuss a definition on the finite-time stability in probability of stochastic nonlinear systems, then we introduce a stochastic Lyapunov theorem on the finite-time stability, which has been established by Yin et al [8]. We also employ this theorem to design a continuous state feedback controller that makes a class of stochastic nonlinear systems to be stable in finite time. An example and a simulation are given to illustrate the theoretical analysis