Mathematical and Statistical Methods for Actuarial Sciences and Finance

276 Gianni Pola

Problem 1.(Optimal Dynamic Asset Allocation (ODAA))Given a finite time horizon
N∈Nand a sequence of target sets

{ 1 , 2 ,...,N}, (2)

wherekare Borel subsets ofX, find the optimal Markov policyπthat maximizes
the joint probability quantity

P({ω∈ :x 0 ∈ 0 ,x 1 ∈ 1 ,...,xN∈N}). (3)

The ODAA problem can be solved by using a dynamic programming approach [6] and
in particular by resorting to recent results on stochastic reachability (see e.g., [18]).
Since the solution of Problem 1 can be obtained by a direct application of the
results in the work of [1, 18], in the following we only report the basic facts which
lead to the synthesis of the optimal portfolio allocation. Givenx∈Xandu∈Rm,
denote bypf(x,u,wk)the probability density function of random variable:

f(x,u,wk+ 1 )=x( 1 +uTwk+ 1 ), (4)

associated with the dynamics of the system in (1). Given the sequence of target sets
in (2) and a Markov policyπ, we introduce the following cost functionV,which
associates a real numberV(k,x,πk)∈[0,1] to a triple(k,x,πk)by:

V(k,x,πk)=

⎧

⎪⎨

⎪⎩

Ik(x), ifk=N,

∫

k+ 1

V(k+ 1 ,z,πk+^1 )pf(z)dz, ifk=N− 1 ,N− 2 ,..., 0 ,

(5)
whereIN(x)is the indicator function of the Borel setN(i.e.IN(x)=1ifx∈N
andIN(x)=0, otherwise) andpfstands forpf(x,uk,wk+ 1 ). Results in [18] show
that cost functionVis related to the probability quantity in (3) as follows:

P({ω∈ :x 0 ∈ 0 ,x 1 ∈ 1 ,...,xN∈N})=V( 0 ,x 0 ,π).

Hence the ODAA problem can be reformulated, as follows:

Problem 2.(Optimal Dynamic Asset Allocation) Given a finite time horizonN∈N
and the sequence of target sets in (2), compute:

π∗=arg sup

π∈ (^) N
V( 0 ,x 0 ,π).
The above formulation of the ODAA problem is an intermediate step towards the
solution of the optimal control problem under study which can now be reported
hereafter.