Irregularity of Brownian Motion Paths
Introduction to Mathematical Finance:
Part II: Continuous-Time Models
By definition, for (almost) all ω ∈ Ω, the trajectory Bt (ω), t ≥ 0 is
a continuous function of t.
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April-May 2012
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Differentiable?
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The Brownian Motion is nowhere Differentiable
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The trajectory of a standard Brownian motion B is nowhere
differentiable P.a.s.
P(ω : ∃t0 t 7→ Bt (ω) differentiable in t0 ) = 0.
∆B
Bt+∆t − Bt
dBt
= lim
= lim
.
∆t→0
∆t→0
dt
∆t
∆t
√
But since ∆B is of order ∆t, it then follows that
1
√
∆t
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Theorem (Paley-Wiener-Zygmund 1933)
Heuristic argument: Suppose we try to compute the derivative of
Bt in the usual sense, that is,
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Irregularity of Brownian Motion Paths
The self-similarity property implies that sample paths of a
Brownian motion are nowhere differentiable (technically, with
probability 1).
∆B
=O
∆t
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That is,
Bt − Bs
t −s
does not exist at any time t > 0 for almost all realizations of
the BM B process.
lim
s→t
,
so that dB/dt = ∞ as ∆t → 0.
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Irregularity of Brownian Motion Paths
other examples of non-differentiable function at any
point?:
Do you know other examples of non-differentiable function at any
point?
Example
Remark (implication)
Weierstrass function (1872).....
Hence, a definition of the form
Z t
Z t
Xs (ω)dBs (ω) :=
0
Xs (ω)
0
dBs (ω)
ds
ds
is impossible.
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Processes Related to Brownian Motion
Definition (Arithmetic Brownian Motion:)
Another reason for the central role played by BM is that it has
many “faces”. Indeed, BM is
I
a (strong) Markov process,
I
a diffusion,
I
a continuous martingale,
I
a process with independent and stationary increments,
I
a Gaussian process,
I
Brownian motion as the limit of symmetric random walks.
Let {Bt }t≥0 be a standard Brownian motion and let x0 ∈ R,
µ and σ are constants; the process
Xt = x0 + µt + σBt
t≥0
is called Arithmetic Brownian motion with drift (or growth rate) µ
and volatility σ.
It has stationary independent increments (with the normal
√
N(µ(t − s), σ(t − s)) distribution) and continuous paths starting
at X1 (0) = x0 .
Remark
Bachelier (1900) suggests an Arithmetic Brownian Motion process
as a model for stock price.
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Processes Related to Brownian Motion
Geometric Brownian motion:
Definition (Geometric Brownian Motion:)
Let {Bt }t≥0 be a standard Brownian motion and set
Xt := e x0 +µt+σBt
t≥0
This is called Geometric Brownian motion (occasionally,
exponential Brownian motion) with drift µ and volatility σ.
Remark
Maury Osborne’s seminal 1959 article, ”Brownian Motion in the
Stock Market.” Bachelier was a mathematician; Osborne is a
astrophysicist. The economic importance of geometric Brownian
motion was recognized by Paul A. Samuelson in his work from
1965, for which Samuelson received the Nobel Prize in Economics
in 1970.
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Geometric Brownian motion:
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Wald’s Martingale
Theorem (Wald’s Martingale)
Let λ ∈ R . The following process
{exp(λBt −
λ2
t)}t≥0
2
is a Martingale.
Proof. Clearly adapted!. Let Mt := exp(λBt −
check that E [|Mt |] < ∞.
λ2
t). Let us
2
2
λ2 exp(− u )
E [|Mt |] = E [Mt ] =
exp(λu − t) √ 2t du
2
2πt
−∞
Z ∞
=
2
Z ∞
exp(− (u−λt)
)
2t
√
−∞
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2πt
du = 1.
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Functions of Finite Variation on [0, ∞):
Wald’s Martingale
Let X a real-valued function defined on the time interval [0, ∞).
Let Π be the set of all partitions π of the interval [0, t] with
0 = t0 < t1 < · · · < tn = t. Consider the sum
Now it remains to show that, for any t , s > 0,
E [Mt+s |Ft ] = Mt .
Stπ := Sπ [X ; 0, t] =
Equivalently, (since Mt > 0)
n−1
X
|Xti+1 − Xti |
i=0
Mt+s
1
E [Mt+s |Ft ] = E [
|Ft ] = 1
Mt
Mt
?
Definition
The function X is of finite variation if, for every t,
Indeed,
St (X ) = sup Stπ < +∞ .
Mt+s
λ2
E[
|Ft ] = E [exp{λ(Bt+s − Bt ) − (t + s − t)}|Ft ]
Mt
2
2
λ
= E [exp{λBs − s}] = E [Ms ] = 1
2
π∈Π
The function t 7→ St (X ) is called the total variation of X and is
positive and increasing. The function X is of bounded variation if
lim St (X ) = sup St (X ) < +∞ .
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t→∞
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Finite Variation Illustration
t≥0
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A continuous function, but not of bounded variation:
Define the function f : [0, 1] → R by
(
f (x ) =
S[t0 ,T ] (f ) = {f (t1 ) − f (t0 )} − {f (t2 ) − f (t1 )} + {f (T ) − f (t2 )}
Z t1
=
Z t2
f (t)dt +
t0
Z T
=
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0
t0
0
{−f (t)}dt +
t1
Z T
f 0 (t)dt
0
if x = 0
π
6 0
x cos( x ) if x =
This function is continuous, but is not of bounded variation
because it wobbles too much near x = 0.
t2
|f 0 (t)|dt
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A continuous function, but not of bounded variation:
Typical Examples of Processes of Finite Variation:
For each m ∈ {1, 2, · · · } consider the partition
1
1
, 2m−1
, · · · , 13 , 12 , 1}. The values of f at the points of
πm := {0, 2m
1
1
this partition are f (πm ) := {0, 2m
, − 2m−1
, · · · , − 13 , 12 , −1}. For
this partition,
n
X
|f (xi ) − f (xi−1 )| = |
i=1
1
1
1
− 0| + | −
−
| + ···
2m
2m − 1 2m
Another typical example
of a process of Finite Variation
R
are the integrals Xt = 0t Ys ds, where Y is a stochastic
process, provided that they are well defined i.e.
|Ys | ds < ∞ a.s.
∀t ≥ 0
In this case, t 7→ Xt is a.s. differentiable.
Important: You need to recognize functions/processes of
Finite Variation
P
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I
0
1
The series ∞
diverges. So given any M , there is a partition
k=2
Pn k
πm for which i=1 |f (xi ) − f (xi−1 )| > M.
R. Ghomrasni
X a real-valued monotone i.e. increasing or decreasing
function defined on the time interval [0, ∞) is of Finite
Variation.
Z t
1
+| − 1 − |
2
1
1
1
=2
+
+ ··· +
+1
2m 2m − 1
2
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Examples of Finite Variation Functions on [0, ∞):
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Mean Value Theorem:
Example
If f : [0, ∞) → R is C 1 , then for any partition
π := {0 = t0 < t1 < · · · < tn−1 < tn = t} of [0, t], we have by the
Mean Value Theorem
n
X
i=1
|f (ti )−f (ti−1 )| =
n
X
|f 0 (ti? )(ti −ti−1 )| ≤
i=1
n
X
M(ti −ti−1 ) = M t
i=1
Thus f is of Finite Variation and St (f ) ≤ M t < ∞ for any t ≥ 0.
We actually have
Z t
St (f ) =
|f 0 (s)| ds .
0
This means that, in general, the total variation of a C 1 fct will be
different from zero.
Monotone finite functions are of finite variation and conversely we
have the following Jordan theorem
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Jordan Decomposition Theorem:
Connection Between Finite Variation and Arc Length :
Theorem (Jordan Theorem)
A function f is of finite variation on [0, ∞) if and only if it can be
written as the difference of two increasing functions on [0, ∞).
Remark
This decomposition is not unique....
Proposition (Arc Length)
Example
For X = (Xt =
Rt
0
The arc-length of a continuous curve y = f (x ) is finite if and
only if f is of finite variation on [a, b].
Ys ds)t≥0 as in the previous slide, we have
Xt = X̄t − Xt
where
Z t
X̄t =
(Ys )+ ds,
0
Z t
Xt =
(Ys )− ds
0
and moreover,
Z t
St (X ) = X̄t + Xt =
|Ys | ds .
0
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The Importance of Processes of Finite Variation in Finance:
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Second Variation or Quadratic Variation:
We will see that a risk-free portfolio is such that the portfolio’s
wealth follows an adapted, continuous process of Finite
Variation.
A special case, which suffices for all practical purposes, is
Definition
For any partition π = {0 = t0 < t1 < t2 < · · · < tn = t} of
[0, t] ⊂ [0, ∞) and for any process {Xt } on [0, ∞), the π-quadratic
variation of the process {Xt } is defined to be the random variable
Z t
exp{
rs ds}
0
for some adapted process r (the local risk-free rate). why it is of
FV?
Qtπ (X ) :=
Proposition (Change of Variables)
n
X
(Xti − Xti−1 )2 .
i=1
Let A be an Finite Variation process with continuous paths, and
let f ∈ C 1 (i.e. its derivative f 0 exists and is continuous). Then
(f (At ))t≥0 is an Finite Variation process and
f (At ) − f (A0 ) =
Z t
f 0 (As ) dAs .
0
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Quadratic Variation Cont’d:
Important Remarks:
Definition
A real-valued process X is of finite quadratic variation if there
exists a unique finite process V = {Vt }t≥0 such that for every t
and every sequence {πn } of subdivisions of [0, t] such that
|πn | → 0 as n → ∞, Qtπn (X ) converges to Vt
Remark
Let X = (Xt )t≥0 be a real-valued continuous adapted process on a
filtered probability space (Ω, F, (Ft )t≥0 , P). If its quadratic
variation process hX i = (hX it )t≥0 exists, then:
lim Qtπn (X ) = Vt .
When the quadratic variation of {Xt } exists and X is
continuous, it is denoted by hX it = hX , X it . (read
Angle-bracket process)
I
hX i0 = 0 a.s. (by convention),
I
hX i = (hX it )t≥0 is also Ft -adapted,
I
hX i = (hX it )t≥0 is non-negative: hX i ≥ 0,
I
hX i = (hX it )t≥0 is continuous and non-decreasing: (therefore
has finite variation.)
Remark
for 0 ≤ s ≤ t
In other words, we need the limit of ni=1 (Xti − Xti−1 )2 exists and
is independent of the choice of the sequence of partitions
{πn } of [0, t].
we have 0 ≤ hX is ≤ hX it < ∞
P
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Let X = (Xt )t≥0 a real-valued, continuous process. If X is of
finite variation, then its quadratic variation is identically zero.
continuous with finite variation
(Xti − Xti−1 )2 ≤ sup |Xti − Xti−1 | ·
i=1
Let X = (Xt )t≥0 a real-valued, continuous process. If X is of
finite variation, then its quadratic variation is identically zero.
=⇒ hX i ≡ 0.
X
n
X
X
Remark
In particular, for a smooth (C 1 ) function/process X , we have
hX i ≡ 0. That is why you never met the notion of Quadratic
variation in real analysis...
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=⇒ hX i ≡ 0.
The above proposition implies that continuous processes
X = (Xt )t≥0 with positive quadratic variation are of infinite
variation.
i=1
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continuous with finite variation
Remark
|Xti − Xti−1 |
St (X ) < ∞ implies that the 2nd term is bounded. By (uniform)
continuity of X the 1st term converges to zero with
|ti − ti−1 | → 0 as n → ∞.
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Proposition
Proof. One has, for any t ≥ 0,
n
X
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Important Consequence:
Proposition
X
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continuous with positive quadratic variation i.e.
∃ t0 > 0 s.t
hX it0 > 0
=⇒ St (X ) = ∞
∀t ≥ t0 .
The concepts of Finite Variation and of the Quadratic Variation
are not very friendly.
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Quadratic Variation of BM:
QV of BM: Different Paths, Same Deterministic Limit
Let us fix the horizon T , and set for N ≥ 1, ti = i T
N . We prove
that, for a.e ω
lim
N→∞
N
X
2
Bti (ω) − Bti−1 (ω)
=T
i=1
Not Intuitive Result...
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Quadratic Variation of BM:
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Infinite Variation of BM:
Theorem (Infinite Variation of Brownian sample paths)
Theorem
The Brownian paths are a.s. of infinite variation on any
interval, no matter how small the interval is. In other words, for
almost all standard Brownian sample paths,
Let B = (Bt , Ft )t≥0 a standard Brownian motion on a filtered
probability space (Ω, F, (Ft )t≥0 , P). The standard Brownian
motion B = (Bt , Ft )t≥0 is of finite quadratic variation and
hBit = hB, Bit = t
for all
t≥0
St (B(ω)) = sup
a.s.
π
We remark hBit = t = var(Bt ) but {hBit }t≥0 depends on the
whole path of the BM until t. In other words, the quadratic
variation result of BM is a much powerful result than the
observation that the variance of Bt equals t.
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|Bti (ω) − Bti−1 (ω)| = ∞ a.s.
i=1
where the supremum is taken over all possible partitions
π = {0 = t0 < · · · < tn = t} of [0, t]. In particular, if one measure
the path of a BM on any finite interval, it length would be
infinity!.
To Remember: The Quadratic Variation of the Brownian
motion is Finite and Deterministic (independent of ω)!
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n
X
Remark
It is this fact that is central to the difficulties in defining an
integral w.r.t Brownian motion (and martingales in general).
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BM Quadratic Variation with Refining Sequences
Theorem
(n)
(n)
Important Remark:
Remark
(n)
If πn = {0 = t0 < t1 < · · · < tmn = t} is a sequence of
refining partitions of [0, t] such that |πn | → 0 as n → ∞, then
mn
X
lim
n→∞
(B − B
tkn
n
tk−1
2
) =t
Unlike the first variation Vt (X ) of a continuous process X , for
the quadratic variation one has
hX it = lim
a.s.
X
|π| t ∈π
i
k=1
|Xti+1 − Xti |2 6= sup
X
π t ∈π
i
|Xti+1 − Xti |2
where the supremum is taken over all partitions
π = {0 = t0 < t1 < · · · < tn = t} of [0, t].
Remark
Refining partition means (i.e. πn ⊂ πn+1 , (in English “keeps the
original time-points and creates additional ones”)) sequence,
As we know, for the Brownian motion one has hBit = t for
almost all paths. However, the right-hand side equals +∞, for
almost all paths.
This property justifies the notation (dBt )2 = dt
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p = 2 is a Critical Value:
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Comments:
Remark
If we define the variation of order p of the sample path
t 7→ Bt (ω) along the dyadic partition
n
Vp(n) (t) :=
2 X
k
k − 1 p
t
−
B
t
B
,
n
n
k=1
2
2
p>0
we have, with probability one
lim V (n) (t)
n→∞ p
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=



∞
t


0
for
for
for
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0<p<2
p=2
p>2
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