ANTICIPATED BACKWARD STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH JUMPS AND APPLICATIONS TO DYNAMIC RISK MEASURES

In this paper, we focus on anticipated backward stochastic Volterra integral equations(ABSVIEs) with jumps. We solve the problem of the well-posedness of so-called M-solutions to this class of equation, and analytically derive a comparison theorem for them and for the continuous equilibrium consumption process. These continuous equilibrium consumption processes can be described by the solutions to this class of ABSVIE with jumps.Motivated by this, a class of dynamic risk measures induced by ABSVIEs with jumps are discussed.

Backward stochastic Volterra integral equations——a brief survey

In this paper, we present a brief survey on the updated theory of backward stochas-tic Volterra integral equations （BSVIEs, for short）. BSVIEs are a natural generalization of backward stochastic diff erential equations （BSDEs, for short）. Some interesting motivations of studying BSVIEs are recalled. With proper solution concepts, it is possible to establish the corresponding well-posedness for BSVIEs. We also survey various comparison theorems for solutions to BSVIEs.

SINGULAR CONTROL OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

This paper deals with optimal combined singular and regular controls for stochastic Volterra integral equations,where the solution X^(u,ξ)(t)=X(t)is given X(t)=φ(t)+∫_(0)^(t) b(t,s,X(s),u(s))ds+∫_(0)^(t)σ(t,s,X(s),u(s))dB(s)+∫_(0)^(t)h(t,s)dξ(s).by Here d B(s)denotes the Brownian motion It?type differential,ξdenotes the singular control(singular in time t with respect to Lebesgue measure)and u denotes the regular control(absolutely continuous with respect to Lebesgue measure).Such systems may for example be used to model harvesting of populations with memory,where X(t)represents the population density at time t,and the singular control processξrepresents the harvesting effort rate.The total income from the harvesting is represented by J(u, ξ) = E[∫_(0)^(t) f_(0)(t,X(t), u(t))dt + ∫_(0)^(t)f_(1)(t,X(t))dξ(t) + g(X(T))] for the given functions f0,f1 and g,where T>0 is a constant denoting the terminal time of the harvesting.Note that it is important to allow the controls to be singular,because in some cases the optimal controls are of this type.Using Hida-Malliavin calculus,we prove sufficient conditions and necessary conditions of optimality of controls.As a consequence,we obtain a new type of backward stochastic Volterra integral equations with singular drift.Finally,to illustrate our results,we apply them to discuss optimal harvesting problems with possibly density dependent prices.

L^p Solutions of Backward Stochastic Volterra Integral Equations

This paper is devoted to the unique solvability of backward stochastic Volterra integral equations （BSVIEs, for short）, in terms of both M-solution and the adapted solutions. We prove the existence and uniqueness of M-solutions of BSVIEs in Lp （1 〈 p 〈 2）, which extends the existing results on M-solutions. The unique solvability of adapted solutions of BSVIEs in Lp （p 〉 1） is also considered, which also generalizes the results in the existing literature.

STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR A CLASS OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS

In this paper,we consider the Euler-Maruyama method for a class of stochastic Volterra integral equations(SVIEs).It is known that the strong convergence order of the EulerMaruyama method is 12.However,the strong superconvergence order 1 can be obtained for a class of SVIEs if the kernelsσi(t,t)=0 for i=1 and 2;otherwise,the strong convergence order is 12.Moreover,the theoretical results are illustrated by some numerical examples.

STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR NONLINEAR STOCHASTIC VOLTERRA INTEGRAL EQUATIONS WITH TIME-DEPENDENT DELAY

We consider a nonlinear stochastic Volterra integral equation with time-dependent delay and the corresponding Euler-Maruyama method in this paper.Strong convergence rate(at fixed point)of the corresponding Euler-Maruyama method is obtained when coefficients f and g both satisfy local Lipschitz and linear growth conditions.An example is provided to interpret our conclusions.Our result generalizes and improves the conclusion in[J.Gao,H.Liang,S.Ma,Strong convergence of the semi-implicit Euler method for nonlinear stochastic Volterra integral equations with constant delay,Appl.Math.Comput.,348(2019)385-398.]

Path-dependent backward stochastic Volterra integral equations with jumps,differentiability and duality principle

We study the existence and uniqueness of a solution to path-dependent backward stochastic Volterra integral equations(BSVIEs)with jumps,where path-dependence means the dependence of the free term and generator of a path of a c`adl`ag process.Furthermore,we prove path-differentiability of such a solution and establish the duality principle between a linear path-dependent forward stochastic Volterra integral equation(FSVIE)with jumps and a linear path-dependent BSVIE with jumps.As a result of the duality principle we get a comparison theorem and derive a class of dynamic coherent risk measures based on path-dependent BSVIEs with jumps.

Backward stochastic differential equations and backward stochastic Volterra integral equations with anticipating generators

For a backward stochastic differential equation(BSDE,for short),when the generator is not progressively measurable,it might not admit adapted solutions,shown by an example.However,for backward stochastic Volterra integral equations(BSVIEs,for short),the generators are allowed to be anticipating.This gives,among other things,an essential difference between BSDEs and BSVIEs.Under some proper conditions,the well-posedness of such BSVIEs is established.Further,the results are extended to path-dependent BSVIEs,in which the generators can depend on the future paths of unknown processes.An additional finding is that for path-dependent BSVIEs,in general,the situation of anticipating generators is not avoidable,and the adaptedness condition similar to that imposed for anticipated BSDEs by Peng−Yang[22]is not necessary.

Linear quadratic stochastic integral games and related topics

This paper studies linear quadratic games problem for stochastic Volterra integral equations(SVIEs in short) where necessary and sufficient conditions for the existence of saddle points are derived in two different ways.As a consequence,the open problems raised by Chen and Yong(2007) are solved.To characterize the saddle points more clearly,coupled forward-backward stochastic Volterra integral equations and stochastic Fredholm-Volterra integral equations are introduced.Compared with deterministic game problems,some new terms arising from the procedure of deriving the later equations reflect well the essential nature of stochastic systems.Moreover,our representations and arguments are even new in the classical SDEs case.