Stochastic Volterra Integral Equations Research Articles

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.

Dynamic Risk Measures for Anticipated Backward Doubly Stochastic Volterra Integral Equations.

Inspired by the consideration of some inside and future market information in financial market, a class of anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs) are introduced to induce dynamic risk measures for risk quantification. The theory, including the existence, uniqueness and a comparison theorem for ABDSVIEs, is provided. Finally, dynamic convex risk measures by ABDSVIEs are discussed.

A two-parameter Milstein method for stochastic Volterra integral equations

In this paper, a two-parameter Milstein method for stochastic Volterra integral equations is introduced. First, the method is proved to be strongly convergent with order one in Lp norm (p≥1). Then, we investigate the mean square stability of the exact and numerical solutions of a stochastic convolution test equation. Stability conditions are derived. Based on these conditions, analytical and numerical stability regions are plotted and compared with each other. The results show that additional implicitness offers benefits for numerical stability. Finally, some numerical experiments are carried out to confirm the theoretical results.

On a class of reflected backward stochastic Volterra integral equations and related time-inconsistent optimal stopping problems

We introduce a class of one-dimensional continuous reflected backward stochastic Volterra integral equations driven by Brownian motion, where the reflection keeps the solution above a given stochastic process (lower obstacle). We prove existence and uniqueness by a fixed point argument and derive a comparison result. Moreover, we show how the solution of our problem is related to a time-inconsistent optimal stopping problem and derive an optimal strategy.

Numerical methods for stochastic Volterra integral equations with weakly singular kernels

AbstractIn this paper we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations (SVIEs) with weakly singular kernels, with singularities $\alpha \in (0, 1)$ for the drift term and $\beta \in (0, 1/2)$ for the stochastic term. Subsequently, we propose a $\theta $-Euler–Maruyama scheme and a Milstein scheme to solve the equations numerically and obtain strong rates of convergence for both schemes in $L^{p}$ norm for any $p\geqslant 1$. For the $\theta $-Euler–Maruyama scheme the rate is $\min \big\{1-\alpha ,\frac{1}{2}-\beta \big\}~ $ and for the Milstein scheme is $\min \{1-\alpha ,1-2\beta \}$. These results on the rates of convergence are significantly different from those it is similar schemes for the SVIEs with regular kernels. The source of the difficulty is the lack of Itô formula for the equations. To get around this difficulty we use the Taylor formula subsequently carrying out a sophisticated analysis of the equation.

Fluctuation limits for mean-field interacting nonlinear Hawkes processes

We investigate the asymptotic behavior of networks of interacting non-linear Hawkes processes modeling a homogeneous population of neurons in the large population limit. In particular, we prove a functional central limit theorem for the mean spike-activity thereby characterizing the asymptotic fluctuations in terms of a stochastic Volterra integral equation. Our approach differs from previous approaches in making use of the associated resolvent in order to represent the fluctuations as Skorokhod continuous mappings of weakly converging martingales. Since the Lipschitz properties of the resolvent are explicit, our analysis in principle also allows to derive approximation errors in terms of driving martingales. We also discuss extensions of our results to multi-class systems.

Support characterization for regular path-dependent stochastic Volterra integral equations

We consider a stochastic Volterra integral equation with regular path-dependent coefficients and a Brownian motion as integrator in a multidimensional setting. Under an imposed absolute continuity condition, the unique solution is a semimartingale that admits almost surely Hölder continuous paths. Based on functional Itô calculus, we prove that the support of its law in Hölder norms can be described by a flow of mild solutions to ordinary integro-differential equations that are constructed by means of the vertical derivative of the diffusion coefficient.

A Modified Euler-Maruyama Scheme for Multi-Term Fractional Nonlinear Stochastic Differential Equations With Weakly Singular Kernels

A modified Euler-Maruyama (EM) scheme was constructed for a class of multi-term fractional nonlinear stochastic differential equations with weak singularity kernels, and the strong convergence of this modified EM scheme was proved. Specifically, according to the sufficient condition for stochastic integral decomposition, the multi-term fractional stochastic differential equation was equivalently transformed into the stochastic Volterra integral equation, and then the corresponding modified EM scheme and its strong convergence were derived and proved, respectively. The order of strong convergence is α_m-α_m-1, where α_i is the index of fractional derivative satisfying 0<α₁<…<α_m-1<α_m<1. Finally, numerical experiments verify the correctness of the theoretical results.

Infinite horizon backward stochastic Volterra integral equations and discounted control problems

Infinite horizon backward stochastic Volterra integral equations (BSVIEs for short) are investigated. We prove the existence and uniqueness of the adapted M-solution in a weightedL2-space. Furthermore, we extend some important known results for finite horizon BSVIEs to the infinite horizon setting. We provide a variation of constant formula for a class of infinite horizon linear BSVIEs and prove a duality principle between a linear (forward) stochastic Volterra integral equation (SVIE for short) and an infinite horizon linear BSVIE in a weightedL2-space. As an application, we investigate infinite horizon stochastic control problems for SVIEs with discounted cost functional. We establish both necessary and sufficient conditions for optimality by means of Pontryagin’s maximum principle, where the adjoint equation is described as an infinite horizon BSVIE. These results are applied to discounted control problems for fractional stochastic differential equations and stochastic integro-differential equations.

Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situations with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation is derived, through which the equilibrium value function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman–Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal valueZ(r,r) ofZ(⋅ , ⋅)) is naturally introduced and the well-posedness of such kind of equations is briefly presented.

Backward stochastic Volterra integral equations with jumps in a general filtration

In this paper, we study backward stochastic Volterra integral equations introduced in Lin [Stochastic Anal. Appl.20(2002) 165–183] and Yong [Stochastic Process. Appl.116(2006) 779–795] and extend the existence, uniqueness or comparison results for general filtration as in Papapantoleonet al.[Electron. J. Probab.23(2018) EJP240] (not only Brownian-Poisson setting). We also considerLp-data and explore the time regularity of the solution in the Itô setting, which is also new in this jump setting.

A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite family of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semi-linear partial differential equation of Hamilton–Jacobi–Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.

The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time

<p style="text-indent:20px;">Motivated by financial and empirical arguments and in order to introduce a more flexible methodology of pricing, we provide a new approach to asset pricing based on Backward Volterra equations. The approach relies on an arbitrage-free and incomplete market setting in continuous time by choosing non-unique pricing measures depending either on the time of evaluation or on the maturity of payoffs. We show that in the latter case the dynamics can be captured by a <em>time-delayed</em> backward stochastic Volterra integral equation here introduced which, to the best of our knowledge, has not yet been studied. We then prove an existence and uniqueness result for time-delayed backward stochastic Volterra integral equations. Finally, we present a Lucas-type consumption-based asset pricing model that justifies the emergence of stochastic discount factors matching the term structure of Sharpe ratios.

Collocation methods for nonlinear stochastic Volterra integral equations

Influenced by Xiao et al. (J Integral Equations Appl 30(1):197–218, 2018), collocation methods are developed to study strong convergence orders of numerical solutions for nonlinear stochastic Volterra integral equations under the Lipschitz condition in this paper. Some properties of exact solutions are discussed. These properties include the mean-square boundedness, the Holder condition, and conditional expectations. In addition, this paper considers the solvability, the mean-square boundedness, and strong convergence orders of numerical solutions. At last, we validate our conclusions by numerical experiments.

On the fuzzy stability results for fractional stochastic Volterra integral equation

&lt;p style='text-indent:20px;'&gt;By a fuzzy controller function, we stable a random operator associated with a type of fractional stochastic Volterra integral equations. Using the fixed point technique, we get an approximation for the mentioned random operator by a solution of the fractional stochastic Volterra integral equation.&lt;/p&gt;

Asymptotic separation for stochastic Volterra integral equations with doubly singular kernels

In this paper, we study the exact asymptotic separation rate of two distinct solutions of doubly singular stochastic Volterra integral equations (SVIEs) with two different initial values. The main tool is the Gronwall inequality with doubly singular kernel. For a special linear singular SVIEs, we obtain a bound for the leading coefficient of the asymptotic separation rate for the two distinct solutions which reveals that our asymptotic results are sharp.

Impulsive Stochastic Volterra Integral Equations Driven by Lévy Noise

The purpose of this paper is to establish the existence and uniqueness of solutions to impulsive stochastic Volterra integral equations driven by Levy process (ISVIEs, in short). By employing the averaging method, we propose a simplified form for ISVIEs. Moreover, we prove that the solutions of ISVIEs can be approximated by the solutions of the simplified one without impulses in the sense of mean square and in probability. Finally, two examples are presented to illustrate the theoretical results.

Extended backward stochastic Volterra integral equations and their applications to time-Inconsistent stochastic recursive control problems

<p style='text-indent:20px;'>In this paper, we study extended backward stochastic Volterra integral equations (EBSVIEs, for short). We establish the well-posedness under weaker assumptions than the literature, and prove a new kind of regularity property for the solutions. As an application, we investigate, in the open-loop framework, a time-inconsistent stochastic recursive control problem where the cost functional is defined by the solution to a backward stochastic Volterra integral equation (BSVIE, for short). We show that the corresponding adjoint equations become EBSVIEs, and provide a necessary and sufficient condition for an open-loop equilibrium control via variational methods.

Itô Differential Representation of Singular Stochastic Volterra Integral Equations

In this paper we obtain an Ito differential representation for a class of singular stochastic Volterra integral equations. As an application, we investigate the rate of convergence in the small time central limit theorem for the solution.

Improved ϑ-methods for stochastic Volterra integral equations

The paper introduces improved stochastic ϑ-methods for the numerical integration of stochastic Volterra integral equations. Such methods, compared to those introduced by the authors in Conte et al. (2018)[14], have better stability properties. This is here made possible by inheriting the stability properties of the corresponding methods for systems of stochastic differential equations. Such a superiority is confirmed by a comparison of the stability regions.