New Developments in Backward Stochastic Riccati Equations and Their Applications

The following backward stochastic Riccati differential equation (BSRDE in short) $$ \begin{gathered} d{K_t} = - \left[ {{{A'}_t}{K_t} + {K_t}{A_t} + \sum\limits_{i = 1}^d {{{C'}_{it}}{K_t}{C_{it}} + {Q_t} + \sum\limits_{i = 1}^d {({{C'}_{it}}{L_{it}} + {L_{it}}{C_{it}})} } } \right. \hfill \\ - \left( {{K_t}{B_t} + \sum\limits_{i = 1}^d {{{C'}_{it}}{K_t}{D_{it}} + \sum\limits_{i = 1}^d {{L_{it}}{D_{it}}} } } \right){\left( {{N_t} + \sum\limits_{i = 1}^d {{{D'}_{it}}{K_t}{D_{it}}} } \right)^{ - 1}} \hfill \\ \times \left. {{{\left( {{K_t}{B_t} + \sum\limits_{i = 1}^d {{{C'}_{it}}{K_t}{D_{it}} + \sum\limits_{i = 1}^d {{L_{it}}{D_{it}}} } } \right)}^\prime }} \right]dt + \sum\limits_{i = 1}^d {{L_{it}}d{w_{it}}} \hfill \\ 0 \leqslant t < T, \hfill \\ {K_T} = M. \hfill \\ \end{gathered} $$ is motivated, and is then studied. Various properties are presented. The existence and uniqueness of a global adapted solution to a BSRDE has been open for the case Di # 0 for more than two decades. Recently, we have made a breakthrough on this problem. On this topic, the literature is reviewed, and our recent results are surveyed. Finally, applications are outlined, both in finance and control, which contain another unique feature of our recent works — exposing the novel crucial role of stochastic maximum principle and the BMO-property of the solutions for BSRDEs in obtaining the necessary regularities, so as to overcome the new difficulty which arises from applications of BSRDEs and which roots in a general combination of both the random nature of the coefficients and the control-dependent nature of the system noise.