Abstract
The main purpose of this chapter is to formulate the optimal control problem (to be studied in this book) for stochastic evolution equations in infinite dimensions, and review some related literature. To solve this problem, we introduce a vector-valued and an operator-valued backward stochastic evolution equation. Also, we give the definition of transposition solutions to these two equations.
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Notes
- 1.
Throughout this book, for any operator-valued process (resp. random variable) \(R\), we denote by \(R^*\) its pointwisely dual operator-valued process (resp. random variable). For example, if \(R\in L^{r_1}_{\mathbb {F}}(0,T; L^{r_2}(\varOmega ; {\fancyscript{L}}(H)))\), then \(R^*\in L^{r_1}_{\mathbb {F}}(0,T; L^{r_2}(\varOmega ; {\fancyscript{L}}(H)))\), and \(|R|_{L^{r_1}_{\mathbb {F}}(0,T; L^{r_2}(\varOmega ; {\fancyscript{L}}(H)))}=|R^*|_{L^{r_1}_{\mathbb {F}}(0,T; L^{r_2}(\varOmega ; {\fancyscript{L}}(H)))}\).
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Lü, Q., Zhang, X. (2014). Introduction. In: General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06632-5_1
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DOI: https://doi.org/10.1007/978-3-319-06632-5_1
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