1 Continuous Time Processes - IPM

The partial differential equation (1.3.4) tells us considerably more than
just the expectation of Xt. The basic theory of a single linear first order
partial differential equation is well understood. Recall that the solution to a
first order ordinary differential equation is uniquely determined by specifying
one initial condition. Roughly speaking, the solution to a linear first order
partial differential equation in two variables is uniquely determined by specifying
a function of one variable. Let us see how this works for our equation
(1.3.4). For a function g(s) of a real variable s we want to substitute for s
a function of t and ξ such that (1.3.4) is necessarily valid regardless of the
choice of g. If for s we substitute λt + φ(ξ), then by the chain rule
∂g(λt + φ(ξ))
∂t
= λg ′ (λt + φ(ξ)),
∂g(λt + φ(ξ))
∂ξ
= φ ′ (ξ)g ′ (λt + φ(ξ)).
where g ′ denote the derivative of the function g. Therefore if φ is such that
φ ′ (ξ) = 1
ξ2 , then, regardless of what function we take for g, equation (1.3.4)
−ξ
is satisfied by g(λt + φ(ξ)). There is an obvious choice for φ (by solving the
) namely the function
differential equation φ ′ (ξ) = 1
ξ 2 −ξ
φ(ξ) = log
1 − ξ
,
ξ
for 0 < ξ < 1. Now we incorporate the initial condition X◦ = 1 which in
terms of the generating function FX means FX(0, ξ) = ξ. In terms of g this
translates into
g(log
1 − ξ
) = ξ.
ξ
That is, g should be the inverse to the mapping ξ → log 1−ξ
. It is easy to
ξ
see that
g(s) = 1
1 + e s
is the required function. Thus we obtain the expression
FX(t, ξ) =
ξ
ξ + (1 − ξ)e λt
25
(1.3.6)