, 1 Introduction to Dynamic Asset Allocation
Definition 1.1. An incomplete market is a market where not all risks
can be hedged, i.e., a market where we cannot let risk disappear by holding
a well-chosen portfolio of liquidity traded assets.
2 Theory Warm-up: The discrete time case
Dynamic Asset Allocation can be referred to as optimal portfolio choice
as well as optimal investment. The problem is to find a long-term consump-
tion and investment strategy that maximizes expected utility, where we have
to take several trade-offs into consideration such as risk versus return, or
consuming now or later.
2.1 The Setting
We consider the following statements with respect to the discrete setting in
Dynamic Asset Allocation:
1. We have a time horizon [0, T ] partitioned into N time intervals [tn , tn+1 ]
of length ∆t, that means tn = n∆t and tN = N ∆t = T .
2. An agent can invest at each time point t ∈ T = {t0 , ..., tN −1 } into d + 1
assets with price processes Pti , i = 0, ..., d.
3. Asset 0 is locally risk-free, that means that at time t it is known that
0
Pt+∆t = (1 + rt ∆t)Pt0 . ’Locally’ means that the risk free interest rate
is only known for the upcoming period.
4. The remaining assets i > 0 are risky and their returns are denoted by
i
Pt+∆t − Pti
i
Rt+∆t = ,
Pti
moreover the column vector of the returns of the risky assets equals
1 d
′
Rt+∆t = Rt+∆t , . . . , Rt+∆t
3
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