Infinite horizon forward-backward stochastic differential equations
Let \(B\) be a standard \(d\)-dimensional Wiener process defined on a probability space \((\Omega ,\mathfrak F,P)\), let \((\mathfrak F_{t})\) be the (augmented) natural filtration of \(B\). An infinite horizon forward-backward stochastic differential equation \[ \begin{aligned} & dX(t) = b(t,X(t),Y(t),Z(t)) dt + \sigma (t,X(t), Y(t),Z(t)) dB(t), \tag{1} \\ -& dY(t) = f(t,X(t),Y(t),Z(t)) dt - Z(t) dB(t) \tag{2} \end{aligned} \] with an initial condition \( X(0) = x_0\) is studied. Considerable attention has been paid to the problem (1), (2) (with an additional condition \(Y(T) = \Phi (X(T))\)) on compact time intervals \([0,T]\); the paper under review, however, is devoted to the case \(T=+\infty \). The solution \((X,Y,Z)\) to (1), (2) should be an \((\mathfrak F_{t})\)-adapted process square integrable over \(\Omega \times \left [0,\infty \right [\). Let \(b,f: \Omega \times \mathbb R_{+}\times \mathbb R^{n}\times \mathbb R^{n}\times \mathbb R^{n\times d}\to \mathbb R^{n}\), \(\sigma : \Omega \times \mathbb R_{+}\times \mathbb R^{n}\times \mathbb R^{n}\times \mathbb R^{n\times d}\to \mathbb R^{n\times d}\) satisfy the following hypotheses: Set \(A(t,u) = (-f,b,\sigma)(t,(x,y,z))\) and assume that \(A(\cdot ,u)\) is an \((\mathfrak F_{t})\)-adapted process for each \(u\), \(A(\cdot ,0)\) is square integrable over \(\Omega \times \mathbb R_{+}\), \(A(t,\cdot)\) satisfies the Lipschitz condition uniformly in \(t\geq 0\), and the monotonicity hypothesis \(\langle A(t,u) - A(t,v), u-v\rangle \leq -\mu |u-v|^{2}\) for a \(\mu >0\) and all \(u,v\) holds. Then there exists a unique solution to (1), (2). Moreover, the authors prove a comparison theorem and investigate global asymptotical stability.
forward-backward stochastic differential equations
global exponential asymptotical stability
scientific article; zbMATH DE number 1792492
scientific article