Stochastic optimal control and BSDEs with logarithmic growth (Q452075)

The authors investigate a backward stochastic differential equation (BSDE) with logarithmic growth in \(z\). More precisely, they consider a standard BSDE NEWLINE\[NEWLINEY_t= \xi+ \int^T_t \varphi(s, Y_s, Z_s)\,ds- \int^T_t Z_s dB_s,\tag{1}NEWLINE\]NEWLINE where \((B_s)\) is a standard Brownian motion, but where the growth of \(\varphi\) in \(z\) is of order \((|z|\sqrt{|\ln|z||})\) (and \(\xi\in L^p\), for some \(p>2\)).NEWLINENEWLINE The main result of the paper is that equation (1) has a unique solution \((Y,Z)\).NEWLINENEWLINE The proof uses a localization method which was developed by the first author [C. R. Acad. Sci., Paris, Sér. I, Math. 333, No. 5, 481--486 (2001; Zbl 1010.60052) and Electron. Commun. Probab. 7, Paper No. 17, 169--179 (2002; Zbl 1008.60075)].NEWLINENEWLINE This main result is then applied to a stochastic control problem and a zero-sum stochastic differential game: in both cases, it is shown that the value function is the solution of a BSDE where, even starting with standard regularity assumptions on the coefficients, the driver (which is the Hamiltonian of the control or game problem) satisfies the above logarithmic growth assumption.