Volume 26, Issue 3

In this paper, we give an approximation result for the adapted solution of the one-dimensional backward stochastic differential equation driven by a one-dimensional Brownian motion (BSDE for short). To prove our main result, we linearize the generator of the BSDE around a deterministic nominal reference trajectory by using a Taylor series expansion. We then find an approximate linear model of the BSDE. A test of our method is given with a numerical scheme driven by the Monte Carlo simulation. We believe that our result is new and valid for the multidimensional case.

In this paper, we deal with an anticipated backward stochastic differential equation driven by a fractional Brownian motion with Hurst parameter H∈(1/2,1){H\in(1/2,1)}. We essentially establish existence and uniqueness of a solution in the case of stochastic Lipschitz coefficients and prove a comparison theorem in a specific case.

We extend our earlier mean field approximation of the Bolker–Pacala model of population dynamics by dividing the population into N classes, using a mean field approximation for each class but also allowing migration between classes as well as possibly suppressive influence of the population of one class over another class. For N≥2{N\geq 2}, we obtain one symmetric nontrivial equilibrium for the system and give global limit theorems. For N=2{N=2}, we calculate all equilibrium solutions, which, under additional conditions, include multiple nontrivial equilibria. Lastly, we prove geometric ergodicity regardless of the number of classes when there is no population suppression across the classes.

We study backward doubly stochastic differential equations when the coefficients are continuous with stochastic linear growth. Via an approximation and comparison theorem, the existence of minimal and maximal solutions are obtained.

The analysis of spatial models has received much attention in the last three decades. It involves methods which take into account the data location for exploring and modelling spatial data. Spatial modelling has its applications in many fields like geology, geography, agriculture, meteorology, economics etc. In this paper, the unilateral second-order spatial autoregressive model, denoted as SAR⁡(2,1){\operatorname{SAR}(2,1)} model, is introduced. Then the necessary conditions for casual solutions of this model will be given. Since each casual model is a stationary model, these conditions will be stationary regions for the parameter space of the SAR⁡(2,1){\operatorname{SAR}(2,1)} model. Under the stationary conditions, we can estimate the model parameters.