- Source: P-variation
In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number
p
≥
1
{\displaystyle p\geq 1}
. p-variation is a measure of the regularity or smoothness of a function. Specifically, if
f
:
I
→
(
M
,
d
)
{\displaystyle f:I\to (M,d)}
, where
(
M
,
d
)
{\displaystyle (M,d)}
is a metric space and I a totally ordered set, its p-variation is
‖
f
‖
p
-var
=
(
sup
D
∑
t
k
∈
D
d
(
f
(
t
k
)
,
f
(
t
k
−
1
)
)
p
)
1
/
p
{\displaystyle \|f\|_{p{\text{-var}}}=\left(\sup _{D}\sum _{t_{k}\in D}d(f(t_{k}),f(t_{k-1}))^{p}\right)^{1/p}}
where D ranges over all finite partitions of the interval I.
The p variation of a function decreases with p. If f has finite p-variation and g is an α-Hölder continuous function, then
g
∘
f
{\displaystyle g\circ f}
has finite
p
α
{\displaystyle {\frac {p}{\alpha }}}
-variation.
The case when p is one is called total variation, and functions with a finite 1-variation are called bounded variation functions.
Link with Hölder norm
One can interpret the p-variation as a parameter-independent version of the Hölder norm, which also extends to discontinuous functions.
If f is α–Hölder continuous (i.e. its α–Hölder norm is finite) then its
1
α
{\displaystyle {\frac {1}{\alpha }}}
-variation is finite. Specifically, on an interval [a,b],
‖
f
‖
1
α
-var
≤
‖
f
‖
α
(
b
−
a
)
α
{\displaystyle \|f\|_{{\frac {1}{\alpha }}{\text{-var}}}\leq \|f\|_{\alpha }(b-a)^{\alpha }}
.
If p is less than q then the space of functions of finite p-variation on a compact set is continuously embedded with norm 1 into those of finite q-variation. I.e.
‖
f
‖
q
-var
≤
‖
f
‖
p
-var
{\displaystyle \|f\|_{q{\text{-var}}}\leq \|f\|_{p{\text{-var}}}}
. However unlike the analogous situation with Hölder spaces the embedding is not compact. For example, consider the real functions on [0,1] given by
f
n
(
x
)
=
x
n
{\displaystyle f_{n}(x)=x^{n}}
. They are uniformly bounded in 1-variation and converge pointwise to a discontinuous function f but this not only is not a convergence in p-variation for any p but also is not uniform convergence.
Application to Riemann–Stieltjes integration
If f and g are functions from [a, b] to
R
{\displaystyle \mathbb {R} }
with no common discontinuities and with f having finite p-variation and g having finite q-variation, with
1
p
+
1
q
>
1
{\displaystyle {\frac {1}{p}}+{\frac {1}{q}}>1}
then the Riemann–Stieltjes Integral
∫
a
b
f
(
x
)
d
g
(
x
)
:=
lim
|
D
|
→
0
∑
t
k
∈
D
f
(
t
k
)
[
g
(
t
k
+
1
)
−
g
(
t
k
)
]
{\displaystyle \int _{a}^{b}f(x)\,dg(x):=\lim _{|D|\to 0}\sum _{t_{k}\in D}f(t_{k})[g(t_{k+1})-g({t_{k}})]}
is well-defined. This integral is known as the Young integral because it comes from Young (1936). The value of this definite integral is bounded by the Young-Loève estimate as follows
|
∫
a
b
f
(
x
)
d
g
(
x
)
−
f
(
ξ
)
[
g
(
b
)
−
g
(
a
)
]
|
≤
C
‖
f
‖
p
-var
‖
g
‖
q
-var
{\displaystyle \left|\int _{a}^{b}f(x)\,dg(x)-f(\xi )[g(b)-g(a)]\right|\leq C\,\|f\|_{p{\text{-var}}}\|\,g\|_{q{\text{-var}}}}
where C is a constant which only depends on p and q and ξ is any number between a and b.
If f and g are continuous, the indefinite integral
F
(
w
)
=
∫
a
w
f
(
x
)
d
g
(
x
)
{\displaystyle F(w)=\int _{a}^{w}f(x)\,dg(x)}
is a continuous function with finite q-variation: If a ≤ s ≤ t ≤ b then
‖
F
‖
q
-var
;
[
s
,
t
]
{\displaystyle \|F\|_{q{\text{-var}};[s,t]}}
, its q-variation on [s,t], is bounded by
C
‖
g
‖
q
-var
;
[
s
,
t
]
(
‖
f
‖
p
-var
;
[
s
,
t
]
+
‖
f
‖
∞
;
[
s
,
t
]
)
≤
2
C
‖
g
‖
q
-var
;
[
s
,
t
]
(
‖
f
‖
p
-var
;
[
a
,
b
]
+
f
(
a
)
)
{\displaystyle C\|g\|_{q{\text{-var}};[s,t]}(\|f\|_{p{\text{-var}};[s,t]}+\|f\|_{\infty ;[s,t]})\leq 2C\|g\|_{q{\text{-var}};[s,t]}(\|f\|_{p{\text{-var}};[a,b]}+f(a))}
where C is a constant which only depends on p and q.
Differential equations driven by signals of finite p-variation, p < 2
A function from
R
d
{\displaystyle \mathbb {R} ^{d}}
to e × d real matrices is called an
R
e
{\displaystyle \mathbb {R} ^{e}}
-valued one-form on
R
d
{\displaystyle \mathbb {R} ^{d}}
.
If f is a Lipschitz continuous
R
e
{\displaystyle \mathbb {R} ^{e}}
-valued one-form on
R
d
{\displaystyle \mathbb {R} ^{d}}
, and X is a continuous function from the interval [a, b] to
R
d
{\displaystyle \mathbb {R} ^{d}}
with finite p-variation with p less than 2, then the integral of f on X,
∫
a
b
f
(
X
(
t
)
)
d
X
(
t
)
{\displaystyle \int _{a}^{b}f(X(t))\,dX(t)}
, can be calculated because each component of f(X(t)) will be a path of finite p-variation and the integral is a sum of finitely many Young integrals. It provides the solution to the equation
d
Y
=
f
(
X
)
d
X
{\displaystyle dY=f(X)\,dX}
driven by the path X.
More significantly, if f is a Lipschitz continuous
R
e
{\displaystyle \mathbb {R} ^{e}}
-valued one-form on
R
e
{\displaystyle \mathbb {R} ^{e}}
, and X is a continuous function from the interval [a, b] to
R
d
{\displaystyle \mathbb {R} ^{d}}
with finite p-variation with p less than 2, then Young integration is enough to establish the solution of the equation
d
Y
=
f
(
Y
)
d
X
{\displaystyle dY=f(Y)\,dX}
driven by the path X.
Differential equations driven by signals of finite p-variation, p ≥ 2
The theory of rough paths generalises the Young integral and Young differential equations and makes heavy use of the concept of p-variation.
For Brownian motion
p-variation should be contrasted with the quadratic variation which is used in stochastic analysis, which takes one stochastic process to another. In particular the definition of quadratic variation looks a bit like the definition of p-variation, when p has the value 2. Quadratic variation is defined as a limit as the partition gets finer, whereas p-variation is a supremum over all partitions. Thus the quadratic variation of a process could be smaller than its 2-variation. If Wt is a standard Brownian motion on [0, T], then with probability one its p-variation is infinite for
p
≤
2
{\displaystyle p\leq 2}
and finite otherwise. The quadratic variation of W is
[
W
]
T
=
T
{\displaystyle [W]_{T}=T}
.
Computation of p-variation for discrete time series
For a discrete time series of observations X0,...,XN it is straightforward to compute its p-variation with complexity of O(N2). Here is an example C++ code using dynamic programming:
There exist much more efficient, but also more complicated, algorithms for
R
{\displaystyle \mathbb {R} }
-valued processes
and for processes in arbitrary metric spaces.
References
Young, L.C. (1936), "An inequality of the Hölder type, connected with Stieltjes integration", Acta Mathematica, 67 (1): 251–282, doi:10.1007/bf02401743.
External links
Continuous Paths with bounded p-variation Fabrice Baudoin
On the Young integral, truncated variation and rough paths Rafał M. Łochowski
Kata Kunci Pencarian:
7.3
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