## Infinite horizon forward-backward stochastic differential equations.(English)Zbl 0997.60062

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.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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### References:

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