In this paper, the authors are interested in the second-order equation \[ \begin{gathered} {\partial w\over\partial t}+ H(x, t, Dw, D^2w)+ G(x,t,Dw, D^2 w)= 0\quad\text{in }\mathbb{R}^N\times (0,T),\\ w(x,0)= \psi(x)\quad\text{in }\mathbb{R}^N.\end{gathered}\tag{1} \] The Hamiltonians \(H, G: \mathbb{R}^N\times [0,T]\times \mathbb{R}^N\times S_N(\mathbb{R})\to \mathbb{R}\) are continuous in all their variables and have the form \[ H(x, t,p,X)= \text{inf}_{\alpha\in A} \{\langle b(x, t,\alpha), p\rangle+ l(x,t,\alpha)- \text{Tr}[\sigma(x, t,\alpha)\sigma^T(x, t,\alpha)X]\} \] and \[ G(x, t,p,X)= \sup_{\beta\in B} \{-\langle g(x,t,\beta), p\rangle- f(x,t,\beta)- \text{Tr}[C(x, t,\beta) c^T(x, t,\beta)X]\}. \] This kind of equation is of particular interest for applications since it relies on differential game theory (Isaacs equations) or on deterministic and stochastic control problems when either \(H\equiv 0\) or \(G\equiv 0\) (Hamilton-Jacobi-Bellman equations). It is allowed that one of the control sets \(A\) or \(B\) be unbounded and the solutions to (1) may have quadratic growth. The model case is the linear quadratic problem which consists in minimizing the quadratic cost \[ V(x,t)= \inf_{\alpha_s\in{\mathcal A}_t} E\left\{\int^T_t [\langle X_s, Q(s) X_s\rangle+ R|\alpha_s|^2]\,ds+ \langle X_T, SX_T\rangle\right\}.\tag{2} \] The Hamilton-Jacobi equation associated to this problem is \[ \begin{gathered} -{\partial w\over\partial t}- \langle A(t) x,Dw\rangle- \langle x,Q(t)x\rangle+ {1\over 4R}|B(t)^T dW|^2- \text{Tr}[a(x, t)D^2 w]= 0,\\ w(x, T)= \langle x, Sx\rangle,\end{gathered}\tag{3} \] where \(a(x, t)= (C(t)x+ D(t))(C(t)x+ D(t))^T/2\). Note that this equation is of type (1) (with \(G\equiv 0\)).
The results obtained in this paper are beyond the classical comparison results for viscosity solutions because of the growth of both the solutions and the Hamiltonians. The use of an optimality principle to establish the connection between the control problem (2) and the equation (3) is more delicate than usual. Thus the authors follow another strategy which consists in comparing directly the value function with the unique solution of the Hamilton-Jacobi equation as long as this latter exists.
Finally, we point out that one of the main fields of application of these types of equations and problems is mathematical finance.