Transition Probabilities. Consider the stochas-
tic process Y = {Yι, £ = 0,1,2, ...N} where
Y∣ is defined through equation (8). For a given
control sequence U( and equidistant discretisation
times, the iterative scheme (8) can be abbreviated
to
Y,t+1 =Yt+δft + bt^Wt (11)
where fι and ⅛ denote, respectively,
fl = f(Yι, uι, τι) bt = b(Ye, ue, τt).
The increments
XWt = W(τt+1) - W(τt), for £ = 0,1,2,... N
refer to the Wiener process W = {W(t), t ≥ 0}
and are known (cf [5]) to be independent Gaus-
sian random variables with mean and variance:
IF.(Δ∏G ) = 0 E∏ΔW,∏ = δ.
The iterative scheme (11) thus defined is the
simplest strong Taylor approximation of an Ito
diffusion process (1), see [5]. Now, suppose that
at some time ∏, Y, = Y, ∈ Xt.
Deterministic process. Assume, for the time being,
that there is no noise in the process (11) so, for a
given control value uι, the process moves to Y,ι+ι
which is defined by:
Yι+1=Yι + δft. (12)
If there is a pair of states of X,ι+ι adjacent to
Yι+ι then the transition probabilities are as-
signed using an inverse distance method. Let
V‰ < Yf+1 be the pair of states adjacent to
Yι+ι- Define
h . — V ® — V®
nl — Г ι+1 Г ι+1
and assign the following non-zero transition prob-
abilities
p(Yι, Y,t+1 ∖uι) =----——ɪ- (13)
V® —V. . .
p(Vf,Vθ1H)= f+1 i+1. (14)
nι
A Weak Taylor Approximation. If the Gaussian
noise is present in (11) a value of Y,t+ι is not
deterministic. For this situation, the strong Euler
scheme (11) will be replaced by a weak Euler
scheme (see [5])
Yι+1 = Yi + δfl + blXWl. (15)
The difference is in Δ∏G, which is a “convenient”
approximation of the random increments ΔII',
of the Wiener process that has similar moment
properties to those of XW∣. In the portfolio
model, we will use an easily generated two point
random variable taking values ±∖∕δ i.e.,
p (∆∏G = ±√J) = ∣. (16)
This approximation of the continuously distributed
perturbation ΔII', by a two-value noise is of
course arbitrary. However, it is sufficient for the
approximating solutions’ convergence. One can
obviously use other more realistic discrete repre-
sentations of ΔWb e.g., it can be modelled as a
three-point distributed random variable Ti with
p(Tι = ±Viδ)=- P(Ti = O) = -. (17)
No matter how simple or complex these approxi-
mations are, they should preserve the original dis-
tribution’s first and second moments and depend
on the partition interval’s length. The latter fea-
ture guarantees that, for all such approximations,
the smaller δ the less diffuse the states become, to
which the process transits.
For the noise representation (16), the definition
of the transition probabilities in the stochastic
case is only slightly different from (13), (14).
Let Yι+ι be determined through (12). The noise
discretisation method means that for δ > 0 the
process reaches, at £ + 1:
Yi+-l = Y.i+1 — bι Vδ with prob, ɪ (18)
= Y,ι+ι + bι Vδ with prob. ɪ. (19)
If there are two adjacent states to Yi+-l and V)(4~1
then apply the inverse distance method as in (13),
(14) but weight the two probabilities by ɪ. Thus,
for example, if V^1 X,t+ι but there exist
Y i^1 < Yt+1 in X,ι+ι adjacent to Yi+1 then the
transition probabilities are
ι Vr - V.®
p(Yι, Vz® ∖ul) = J e+1 e+1 (20)
2 III
,-, 1 Ÿ7® — v-
p(Yι, Yi θ ∖ui) = I e+1 t+1 (21)
2 hi
where hi = Yf J∣ — Y i^1 ■ If any of the states
V-θ, V+θ, etc. overlaps another, the respective
probabilities have to be summed up.
As evident, the above discretisation method is
very simple and intuitive. However, as noted, it