A maximum principle for a stochastic control problem with multiple random terminal times (Q2128538)

The authors consider the stochastic differential system \(dX^{i;0}(t)=\mu ^{i;0}(t,X^{i;0}(t),\alpha ^{i;0}(t))dt+\sigma ^{i;0}(t,X^{i;0}(t),\alpha ^{i;0}(t))dW^{i}(t)\), \(i=1,\ldots ,n\), where \(W^{i}(t)\) is a standard Brownian motion, \(\mu ^{i;0},\sigma ^{i;0}:[0,T]\times \mathbb{R}\times A\rightarrow \mathbb{R}\) are Lipschitz continuous coefficients with at most linear growth, \(\alpha ^{i}\) is a control which belongs to \(\mathcal{A} ^{i}=\{\alpha ^{i;0}\in L_{ad}^{2}([0,T];\mathbb{R}):\alpha ^{i;0}(t)\in A^{i}\) a.e. \(t\in \lbrack 0,T]\}\), \(A^{i}\) being a convex and closed subset of \(\mathbb{R}\), and \(L_{ad}^{2}([0,T];\mathbb{R})\) being the space of \(( \mathcal{F}_{t})_{t\in \lbrack 0,T]}\)-adapted processes such that \(\mathbb{E} \int_{0}^{T}\left\vert \alpha ^{i;0}(t)\right\vert ^{2}dt<1\). The authors write this system in a vectorial form as \(dX^{0}(t)=B^{0}(t,X^{0}(t),\alpha ^{0}(t))dt+\Sigma ^{0}(t,X^{0}(t),\alpha ^{0}(t))dW(t)\) and they add the initial condition \(X^{0}(t)=x_{0}^{0}\). They further introduce the cost functional \(J(x,\alpha )=\mathbb{E}\int_{0}^{\widehat{\tau } ^{1}}L^{0}(t,X^{0}(t),\alpha ^{0}(t))dt+G^{0}(\widehat{\tau }^{1},X^{0}( \widehat{\tau }^{1}))\) where \(L^{0}:[0,T]\times \mathbb{R}^{n}\times A^{0}\rightarrow \mathbb{R}\) and \(G^{0}:[0,T]\times \mathbb{R} ^{n}\rightarrow \mathbb{R}\) are measurable and continuous functions such that there exist two\ positive constants \(K,k\) and for any \(t\in \lbrack 0,T] \), \(x\in \mathbb{R}^{n}\) and \(a\in A^{0}\), it holds \(\left\vert L^{0}(t,x,a)\right\vert \leq K(1+\left\vert x\right\vert ^{k}+\left\vert a\right\vert ^{k})\), \(\left\vert G^{0}(t,x)\right\vert \leq K(1+\left\vert x\right\vert ^{k})\), \(\widehat{\tau }^{1}\) being a vector of stopping times. The authors prove a necessary and a sufficient maximum principle for such systems. Finally, considering the system \(dX(t)=B(t,X(t),\alpha (t))dt+\Sigma (t,X(t))dW(t)\) where both the drift and the volatility coefficients are supposed to be linear, with running and terminal cost functionals which satisfy suitable quadratic weighted averages of the distance from the stopping boundaries, the authors prove the existence of an optimal control to which they give an explicit formula.