Mathematical and Statistical Methods for Actuarial Sciences and Finance

Optimal dynamic asset allocation in a non–Gaussian world 275

• uk∈Uk⊆Rmis the control input, representing the portfolio allocation at timek;
  and


• wk∈Rmis a random vector describing the asset classes’ returns at timek∈N.

Equation (1) describes the time evolution of the portfolio value:uTkwk+ 1 quantifies the
percentage return of the portfolio allocationukat timekin the time interval [k,k+ 1 )
due to market performanceswk+ 1.
Let(,F,P)be the probability space associated with the stochastic system in
(1). Portfolio valuexkat timek=0isassumedtobeknownandsettox 0 =1. The
mathematical model in (1) is characterised by no specific distribution on the asset
classes’ returns. We model asset classes’ returns by means of Mixtures of Multivari-
ate Gaussian Models (MMGMs), which provide accurate modelling of non–Gaussian
distributions while being computationally simple to be implemented for practical is-
sues^1. We recall that a random vectorYis said to be distributed according to a MMGM
if its probability density functionpYcan be expressed as the convex combination of
probability density functionspYiof some multivariate Gaussian random variablesYi,
i.e.,

pY(y)=

∑N

i= 1

λipYi(y), λi∈[0,1],

∑N

i= 1

λi= 1.

Some further constraints are usually imposed on coefficientsλiso that the resulting
random variableYis well behaved, by requiring, for example, semi-definiteness of the
covariance matrix and unimodality in the marginal distribution. The interested reader
can refer to [8] for a comprehensive exposition of the main properties of MMGMs.
The class of control inputs that we consider in this work is the one of Markov
policies [6]. Given a finite time horizonN∈N, a Markov policy is defined by the
sequence
π={u 0 ,u 1 ,...,uN− 1 }

of measurable mapsuk:X→Uk. Denote byUkthe set of measurable mapsuk:

X →Ukand by (^) Nthe collection of Markov policies. For further purposes let
πk={uk,uk+ 1 ,...,uN− 1 }.
Let us consider a finite time horizonNwhich represents the lifetime of the consid-
ered investment. Our approach in the portfolio construction deals with how to select
a Markov policyπin order to fulfill some specifications on the portfolio valuexkat
timesk= 1 ,...,N. The specifications are defined by means of a sequence of target
sets{ 1 , 2 ,...,N}withi⊆X. The investor wishes to have a portfolio valuexk
at timekthat is ink. Typical target setskare of the formk=[xk,+∞[and
aim to achieve a performance that is downside bounded by xk∈R. This formulation
of specifications allows the investor to have a portfolio evolution control during its
lifetime, since target setskdepend on timek.
The portfolio construction problem is then formalized as follows:
(^1) We stress that MMGM modelling is only one of the possible choices: formal results below
hold without any assumptions on the return stochastic dynamics.