Remark on writing down an SDE
If we are asked to write down an SDE for a process Xt , answer should be
presented in form of
dXt = µ(Xt )dt + σ(Xt )dBt .
The drift and diffusion coefficients should always be expressed in Xt .
Question from the week 10 class test: Suppose X satisfies
dXt = Xαt dt + dBt . Write down an SDE for Qt where Qt = Xt2 .
Applying Ito’s lemma will give
dQt = d(Xt2 ) = 2Xt dXt + (dXt )2
= (2α + 1)dt + 2Xt dBt .
But dQt = (2α + 1)dt + 2Xt dBt should not be presented as the final
solution since the coefficient of dBt is in X but not in Q. Instead we
should write X in terms of Q and the answer should be written as
p
dQt = (2α + 1)dt + 2 Qt dBt .
(Local) martingale property of a stochastic integral
Let X be a martingale, and C be an adapted process.
Pn
I In discrete time, we know that (C · X )n :=
k=1 Ck−1 (Xk − Xk−1 )
is a martingale.
Rt
I As a continuous time analogue, (C · X )t :=
C dXs is a local
0 s
martingale.
Rt
In the continuous time if we further know that 0 Cs dXs is a true (i.e not
just local) martingale, then by the martingale property of E(Mt ) = M0 ,
Rt
we can conclude E( 0 Cs dXs ) = 0.
Warning: In general a stochastic integral is not always a true martingale,
and hence it may not have zero expectation. In the ST909 module in
term 2, you will see some tools which help check whether a local
martingale is a true martingale.
Stochastic integral (against a Brownian motion) with
deterministic integrand
In the last part of Q4(B) of the Jan2015 exam paper, it asks about
identifying the distribution of Yt where
Z t
−κt
−κt
Yt = Y0 e
+ m(1 − e
)+
σe −κ(t−s) dBs .
0
The first two termsR are just constants. Thus it is all about asking what
t
the distribution of 0 σe −κ(t−s) dBs is. Indeed, this is having a normal
R t 2 −2κ(t−s)
distribution of N(0, 0 σ e
ds) (see the lemma below), and hence
Yt is distributed as
Z t
Yt ∼ N(Y0 e −κt + m(1 − e −κt ),
σ 2 e −2κ(t−s) ds))
0
σ2
−κt
−κt
−2κt
= N Y0 e
+ m(1 − e
), (1 − e
) .
2κ
Lemma
If C is a deterministic process, then
Rt
0
Cs dBs ∼ N(0,
Rt
0
Cs2 ds).
Sketch of proof for the lemma
Proof.
Rt
Let f (t) := 0 Cs2 ds, which is a deterministic and increasing function.
Let f −1 (t) be the (left-continuous) inverse function of f , which is
guaranteed to be well-defined since f is increasing.
R f −1 (t)
Cs dBs . We first claim that the process (It )t>0 is
Also define It := 0
a Brownian motion, since
1. It , as a stochastic integral, is a continuous local martingale;
2. The quadratic variation of I is given by
R f −1 (t) 2
[I ]t = 0
Cs ds = f (f −1 (t)) = t.
I is then a Brownian motion by Levy’s theorem. Then It ∼ N(0, t) and
R f −1 (t)
equivalently 0
Cs dBs ∼ N(0, t). On replacing t by f (t), we get
Z t
Z t
Cs dBs ∼ N(0, f (t)) = N(0,
Cs2 ds).
0
0