and denote the set (system) of natural numbers and real numbers respectively

x: = means x is defined as

[ ], [ ], _t [ ] denote expectation, expectation w.r.t. and the expectation conditioned at time t

[ ] denotes the variance

given a security S, S _t (or with superscripts ) stand for the price at time t

the $ sign are often dropped, so we write x=10 instead of x=$10

no distinction is made between value and price

stock prices are assumed uniformly bounded away from 0

bounded means the absolute value is bounded by a finite number

S-continuous, S-bounded, are simply called continuous, bounded, when it is clear that they don t mean *continuous, *bounded,

C ⁿ means possessing continuous nth (and hence all lower) derivatives

( ) ⁺ is the maximum junction, i.e. (x) ⁺=max{ x, 0}

elements of are written as instead of a sequence