Forward–backward stochastic differential equations with monotone functionals and mean field games with common noise

We consider a system of forward–backward stochastic differential equations (FBSDEs) with monotone functionals. We show that such a system is well-posed by the method of continuation similarly to Peng and Wu (1999) for classical FBSDEs. As applications, we prove the well-posedness result for a mean field FBSDE with conditional law and show the existence of a decoupling function. Lastly, we show that mean field games with common noise are uniquely solvable under a linear-convex setting and weak-monotone cost functions and prove that the optimal control is in a feedback form depending only on the current state and conditional law.     

Solvability of forward–backward stochastic partial differential equations

In this paper we study the solvability of a class of fully-coupled forward–backward stochastic partial differential equations (FBSPDEs). These FBSPDEs cannot be put into the framework of stochastic evolution equations in general, and the usual decoupling methods for the Markovian forward–backward SDEs are difficult to apply. We prove the well-posedness of the FBSPDEs, under various conditions on the coefficients, by using either the method of contraction mapping or the method of continuation. These conditions, especially in the higher dimensional case, are novel in the literature.     

Existence and uniqueness of solutions to the backward 2D stochastic Navier–Stokes equations

The backward two-dimensional stochastic Navier–Stokes equations (BSNSEs, for short) with suitable perturbations are studied in this paper, over bounded domains for incompressible fluid flow. A priori estimates for adapted solutions of the BSNSEs are obtained which reveal a pathwise L^∞(H)$L^{\infty }\left(H\right)$ bound on the solutions. The existence and uniqueness of solutions are proved by using a monotonicity argument for bounded terminal data. The continuity of the adapted solutions with respect to the terminal data is also established.     

Time-consistent mean–variance hedging of longevity risk: Effect of cointegration

This paper investigates the time-consistent dynamic mean–variance hedging of longevity risk with a longevity security contingent on a mortality index or the national mortality. Using an HJB framework, we solve the hedging problem in which insurance liabilities follow a doubly stochastic Poisson process with an intensity rate that is correlated and cointegrated to the index mortality rate. The derived closed-form optimal hedging policy articulates the important role of cointegration in longevity hedging. We show numerically that a time-consistent hedging policy is a smoother function in time when compared with its time-inconsistent counterpart.     

Reflected forward–backward stochastic differential equations with continuous monotone coefficients

In this work, we prove that there exists at least one solution for the reflected forward–backward stochastic differential equations satisfying the obstacle constraint with continuous monotone coefficients. The distinct character of our result is that the coefficient of the forward SDEs contains the solution variable of the reflected BSDEs.     

On one-dimensional forward–backward diffusion equations with linear convection and reaction

We study the initial-Neumann boundary value problem for a class of one-dimensional forward–backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to contain two forward-diffusion phases. We prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many Lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases. Furthermore, we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant. The results are new even for the pure forward–backward diffusion problem without convection and reaction. Our primary approach relies on a combination of the convex integration and Baire's category methods for a related nonlocal differential inclusion.     

High-order compact solution of the one-dimensional heat and advection–diffusion equations

In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C¹$C^1$-spline collocation method for the resulting linear system of ordinary differential equations. The cubic C¹$C^1$-spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O(h⁴,k⁴)$O(h^4\text{,}{k}^4)$. Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C¹$C^1$-spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.     

Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations

In this paper we study stochastic optimal control problems with jumps with the help of the theory of Backward Stochastic Differential Equations (BSDEs) with jumps. We generalize the results of Peng [S. Peng, BSDE and stochastic optimizations, in: J. Yan, S. Peng, S. Fang, L. Wu, Topics in Stochastic Analysis, Science Press, Beijing, 1997 (Chapter 2) (in Chinese)] by considering cost functionals defined by controlled BSDEs with jumps. The application of BSDE methods, in particular, the use of the notion of stochastic backward semigroups introduced by Peng in the above-mentioned work allows a straightforward proof of a dynamic programming principle for value functions associated with stochastic optimal control problems with jumps. We prove that the value functions are the viscosity solutions of the associated generalized Hamilton–Jacobi–Bellman equations with integral-differential operators. For this proof, we adapt Peng’s BSDE approach, given in the above-mentioned reference, developed in the framework of stochastic control problems driven by Brownian motion to that of stochastic control problems driven by Brownian motion and Poisson random measure.     

Open-loop equilibrium reinsurance-investment strategy under mean–variance criterion with stochastic volatility

This paper investigates the open-loop equilibrium reinsurance-investment (RI) strategy under general stochastic volatility (SV) models. We resolve difficulties arising from the unbounded volatility process and the non-negativity constraint on the reinsurance strategy. The resolution enables us to derive the existence and uniqueness result for the time-consistent mean variance RI policy under both situations of constant and state-dependent risk aversions. We apply the general framework to popular SV models including the Heston, the 3/2 and the Hull–White models. Closed-form solutions are obtained for the aforementioned models under constant risk aversion, and the non-leveraged models under state-dependent risk aversion.     

Global existence of a weak solution to 3d stochastic Navier–Stokes equations in an exterior domain

In this paper we consider the existence of a weak solution to a 3d stochastic Navier–Stokes equation perturbed by a noise g(X(t))dW, where W(t) is a cylindrical Wiener process, in an exterior domain D: $$dX(t) = [-AX(t) + B\left( X(t)\right)]dt + g(X(t))dW(t),$$ where \({A = -P_{2}\Delta}\) is the Stokes operator and g satisfies some conditions.     

Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations

BIT Numerical Mathematics - In this paper we derive error estimates of the backward Euler–Maruyama method applied to multi-valued stochastic differential equations. An important example of...     

Local solvability and solution blow-up of one-dimensional equations of the Yajima–Oikawa–Satsuma type

We consider one-dimensional equations of the type of the Yajima–Oikawa–Satsuma ion acoustic wave equation and prove the local solvability. Using the test function method, we obtain sufficient conditions for solution blow-up and estimate the blow-up time.     

Multi-soliton solution,rational solution of the Boussinesq–Burgers equations

In this paper we consider the Boussinesq–Burgers equations and establish the transformation which turns the Boussinesq–Burgers equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and abundant new exact solutions, including the multi-solitary wave solution and the rational series solutions. Besides the new trigonometric function periodic solutions are obtained by using the generalized tan h method.     

On the existence of solution to one–dimensional forward–backward sdes

In this paper, we study the existence of the solution to one-dimensional forward–backward stochastic differential equations with neither the smooth condition nor the monotonicity condition for the coefficients. Under the nondegeneracy condition for the forward equation, we prove the existence of the solution to one-dimensional forward–backward stochastic differential equations. And we apply this result to establish the existence of the viscosity solution to a certain one-dimensional quasilinear parabolic partial differential equation     

Global classical solution of the Cauchy problem to 1D compressible Navier–Stokes equations with large initial data

In this paper, we prove that the 1D Cauchy problem of the compressible Navier–Stokes equations admits a unique global classical solution (ρ,u)$(\rho ,\mathrm{u})$ if the viscosity μ(ρ)=1+ρ^β$\mu (\rho )=1+{\rho }^{\beta }$ with β?0$\beta ?0$. The initial data can be arbitrarily large and may contain vacuum. Some new weighted estimates of the density and velocity are obtained when deriving higher order estimates of the solution.     

Optimal mean–variance asset-liability management with stochastic interest rates and inflation risks

This paper considers an optimal asset-liability management problem with stochastic interest rates and inflation risks under the mean–variance framework. It is assumed that there are \(n+1\) assets available in the financial market, including a risk-free asset, a default-free zero-coupon bond, an inflation-indexed bond and \(n-2\) risky assets (stocks). Moreover, the liability of the investor is assumed to follow a geometric Brownian motion process. By using the stochastic dynamic programming principle and Hamilton–Jacobi–Bellman equation approach, we derive the efficient investment strategy and efficient frontier explicitly. Finally, we provide numerical examples to illustrate the effects of model parameters on the efficient investment strategy and efficient frontier.     

On the well-posedness of coupled forward–backward stochastic differential equations driven by Teugels martingales

We deal with a class of fully coupled forward–backward stochastic differential equations (FBSDEs), driven by Teugels martingales associated with a general Lévy process. Under some assumptions on the derivatives of the coefficients, we prove the existence and uniqueness of a global solution on an arbitrarily large time interval. Moreover, we establish stability and comparison theorems for the solutions of such equations. Note that the present work extends known results proved for FBSDEs driven by a Brownian motion, by using martingale techniques related to jump processes, to overcome the lack of continuity.     

Optimal mean–variance investment and reinsurance problem for an insurer with stochastic volatility

This paper considers an optimal investment and reinsurance problem for an insurer under the mean–variance criterion. The stochastic volatility of the stock price is modeled by a Cox-Ingersoll-Ross (CIR) process. By applying a backward stochastic differential equation (BSDE) approach, we obtain a BSDE related to the underlying investment and reinsurance problem. Then solving the BSDE leads to closed-form expressions for both the efficient frontier and the efficient strategy. In the end, numerical examples are presented to analyze the economic behavior of the efficient frontier.