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seCurity and PrivaCy in smart Grids
an action on an observed sequence of random variables to maxi-
mize expected rewards or minimize expected costs.
4
Specifically,
let (X
1
; X
2
; …) denote a random process whose joint distribution is
assumed known, and (y
0
; y
1
(x
1
); y
2
(x
1
; x
2
); …; y
∞
(x
1
; x
2
; …)) denote a
sequence of real-valued reward functions. We need to choose a stop-
ping time N that satisfies {N = n} ∈ F
n
where F
n
is the σ algebra
generated by (X
j
; j < n) to maximize or minimize the expected return
E[Y
N
]. It has been used effectively in statistics, economics, mathemat-
ics, finance, and networks. In our scenario, an action means an elec-
tricity user or appliance starts to run; the observation is the electricity
price, and the objective is to min ...