ABSTRACT
In this chapter we develop and analyze tools for efficient numerical solution of the dynamic programming equation (7.3.3) Y T ( h ) = f ( X T ( h ) ) , Y t i ( h ) = E Y t i + 1 ( h ) + h g ( t i , X t i ( h ) , Y t i + 1 ( h ) ) | X t i ( h ) , i = N - 1 , ⋯ , 0 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315368757/e88f714d-2ba9-44cc-aeea-7a975525e77e/content/eq1995.tif"/> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315368757/e88f714d-2ba9-44cc-aeea-7a975525e77e/content/eq557.tif"/> The discussion at the end of the previous chapter shows (see equation (7.4.2)) that the discrete time process ( Y t i ( h ) ) 0 ≤ i ≤ N https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315368757/e88f714d-2ba9-44cc-aeea-7a975525e77e/content/eq1996.tif"/> can also be represented as Y t i ( h ) = u ( h ) ( t i , X t i ( h ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315368757/e88f714d-2ba9-44cc-aeea-7a975525e77e/content/eq1997.tif"/> for a sequence of functions (u (h)(t i,.))0 ≤ i ≤ N , each of which is written as an expectation that depends on the others (see (7.4.4)): u ( h ) ( t i , x ) = E u ( h ) ( t i + 1 , X t i + 1 ( h ) , t i , x ) + h g t i , x , u ( h ) ( t i + 1 , X t i + 1 ( h ) , t i , x ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315368757/e88f714d-2ba9-44cc-aeea-7a975525e77e/content/eq1998.tif"/> https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315368757/e88f714d-2ba9-44cc-aeea-7a975525e77e/content/eq559.tif"/>
