Digital Signal Processing Reference
In-Depth Information
60
50
0.0003
40
0.0002
30
20
0.0001
10
0
0
300
1
300
1
0.75
0.75
200
200
0.5
0.5
100
100
0.25
0.25
0
0
0
0
3
1
0.1
2.5
0.01
0.001
2
0.0001
1.5
1e-05
1e-06
1
1e-07
1e-08
0.5
1e-09
0.1
0
300
1e-10
1
1
0.075
0.75
0.75
200
0.05
0.5
0.5
100
0.025
0.25
0.25
0
0
0
0
Fig. 15.2
More examples of risk premia. Conventions follow those of Fig.
15.1
against the mean-variance model's in which we let
I
. The results are presented
in Figs.
15.1
and
15.2
. Notice that the mean-variance premium, which equals
a
Σ
=
/(
2
d
)
,
displays the simplest behavior (a linear plot, see upper-left in Fig.
15.1
).
15.4 On-line Learning in the Mean-Divergence Model
As previously studied by [
14
,
26
] in the mean-variance model, our objective is now
to track “efficient” portfolios at the market level, where a portfolio is all the more effi-
cient as its associated risk premium (
15.28
) is reduced. Let us denote these portfolios
reference
portfolios, and the sequence of their allocation matrices as:
O
0
,
O
1
,...
.
The natural market allocation may also shift over time, and we denote
Θ
0
,
Θ
1
,...
the sequence of natural parameter matrices of the market. Naturally, we could sup-
pose that
O
t
=
Θ
t
,
∀
t
, which would amount to tracking directly the natural market
allocation, but this setting would be too restrictive because it may be easier to track
some
O
t
close to
Θ
t
does not have (
e.g.
spar-
sity). Finally, we measure risk premia for references with the same risk aversion
parameter
a
as for the investor's.
Θ
t
but having specific properties that
