Infinite horizon BSDEs in infinite dimensions with continuous driver and applications (Q862255)

The backward stochastic differential equation (BSDE): \[ dY_t= -BY_tdt+\lambda Y_t dt-\psi(t, X_t, Y_t, Z_t)\,dt+ Z_t dW_t,\quad t\in [0,\infty),\tag{1} \] is considered. In the above, \(\lambda\) is a real number, \(W\) is a cylindrical Wiener process in a Hilbert space \(\Xi\), \(({\mathcal J}_t)\) denotes its Brownian filtration, \(B\) is the infinitesimal generator of a strongly continuous dissipative compact semigroup \((e^{tB})\) in a Hilbert space \(K\), \(X\) is a given \(({\mathcal J}_t)\)-predictable process in the Hilbert space \(K\), \(\psi(t,x,y,z)\) is a deterministic function with values in \(K\). The unknown processes \(Y\) and \(Z\) take values in the Hilbert spaces \(K\) and \(L_2(\Xi,K)\), respectively. The authors prove existence of a solution to the BSDE (1) assuming that \(\psi\) is continuous and with linear growth with respect to \((y, z)\) and \(X = \{X^{t,x}_s,s\geq 0\}\) is the Ornstein-Uhlenbeck process defined by: \[ X^{t,x}_s= e^{(s-t)A}x+ \int^s_t e^{(s-r)A} G\,dW_r,\quad s\geq t,\qquad X^{t,x}_s= x,\;s\in [0,t]. \] The existence result is applied to prove existence of a Nash equilibrium in a stochastic game with infinitely many players.