Variation (math)

This article covers the variation of a function. For the variation in measure theory, see variation (measure theory) , total variation norm and vectorial measure . For variations in combinatorics, see Variation (combinatorics) .

In mathematics , especially the calculus of variations and the theory of stochastic processes , the variation (also called total variation ) of a function is a measure of the local oscillation behavior of the function. In the case of stochastic processes, the variation is of particular importance, since it divides the class of time-continuous processes into two fundamentally different subclasses: those with finite and those with infinite variation.

definition

Let be a function on the real interval . The variation of is defined by ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle [a,b]}$ ${\displaystyle |f|_{[a,b]}}$ ${\displaystyle f}$

${\displaystyle |f|_{[a,b]}:=\sup \left\{\left.\sum _{k=0}^{n-1}\left|f\left(t_{k+1}^{(n)}\right)-f\left(t_{k}^{(n)}\right)\right|\ \right|n\in \mathbb {N} ,a\leq t_{0}^{(n)}<t_{1}^{(n)}\dotsb <t_{n}^{(n)}\leq b\right\}}$,

thus by the smallest upper bound ( supremum ), which majorizes all sums that result from any fine subdivision of the interval . (If no real number can be found that dominates all sums, the supremum is set to plus infinity .) ${\displaystyle a\leq t_{0}^{(n)}<t_{1}^{(n)}\dotsb <t_{n}^{(n)}\leq b}$ ${\displaystyle [a,b]}$

The following theorem applies to piecewise monotonic , continuous functions:

If in the intervals with increasing or decreasing monotonically, the equation applies to the variation of ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle [t_{0},t_{1}],[t_{1},t_{2}],\ldots ,[t_{n-1},t_{n}]}$ ${\displaystyle a=t_{0},b=t_{n}}$ ${\displaystyle f}$

${\displaystyle |f|_{[a,b]}=\sum _{k=0}^{n-1}|f(t_{k+1})-f(t_{k})|}$.

The above definition of the variation can be transferred to functions that are defined at unrestricted intervals and to those that assume values ​​in the complex numbers or in normalized vector spaces .

Example of a continuous function with infinite variation

We want to show that for the function that is continuous on the unit interval ${\displaystyle [0,1]}$

${\displaystyle f(t)={\begin{cases}0&{\mbox{falls }}t=0,\\t\cos {\frac {\pi }{2t}}&{\mbox{falls }}t\in (0,1],\end{cases}}}$

${\displaystyle |f|_{[0,1]}=\infty }$applies. For each be ${\displaystyle n\in \mathbb {N} }$

${\displaystyle t_{k}^{(n)}={\begin{cases}0&{\mbox{falls }}k=0,\\{\frac {1}{n+1-k}}&{\mbox{falls }}k\in \{1,\dots ,n\}.\end{cases}}}$

Then

${\displaystyle \sum _{k=0}^{2n-1}\left|f(t_{k+1}^{(2n)})-f(t_{k}^{(2n)})\right|=\dotsc =\sum _{l=1}^{n}{\frac {1}{l}}}$

which because of the divergence of the harmonic series for tends to infinity. ${\displaystyle n\to \infty }$

Application in the calculus of variations

In the calculus of variations one often encounters optimization problems of the following kind:

${\displaystyle \min _{f\in {\mathcal {C}}}|f|_{[a,b]}}$,

where is a given set of functions, roughly all twice continuously differentiable functions with additional properties such as ${\displaystyle {\mathcal {C}}}$

${\displaystyle f(a)=0,\ f(b)=1,\ f\left({\frac {2a+b}{3}}\right)=-f\left({\frac {a+2b}{3}}\right).}$

Similar problems lead to the definition of the splines , for example .

Another reason for the spread of the variation in optimization problems is the following statement: If the function describes the course of an object in a one-dimensional space over time, then it indicates the distance covered in the period . ${\displaystyle f}$ ${\displaystyle |f|_{[a,b]}}$ ${\displaystyle [a,b]}$

Application in stochastics

See also : stochastics

In the theory of stochastic processes , the concept of variation plays a special role: An important characterization of processes (in addition to the division into classes such as Markov , Lévy or Gaussian processes ) consists in their property that they are almost certainly finite or infinite over finite intervals To show variation:

• Example of a process of almost certainly finite variation: for a Poisson process with intensity , because of the monotony ; ${\displaystyle (N_{t}),\;t\geq 0}$ ${\displaystyle \lambda }$ ${\displaystyle |N|_{[0,t]}\sim \mathrm {Poi} (\lambda t)}$
• Example of a process of almost certainly infinite variation: the Wiener process, on the other hand, almost certainly has infinite variation on each interval . ${\displaystyle [0,t],\;t>0}$

This property has fatal consequences for the application of the Wiener process in physics to explain Brownian molecular motion : A particle whose motion follows a Wiener process would cover an infinite distance in every time interval - in stark contradiction to the laws of physics. Such a particle would not have a defined instantaneous speed (especially not even a direction of movement ) and certainly not a defined acceleration , so that it makes no sense to talk about forces acting on the particle (cf. second Newton's law ).

Quadratic variation

See also : Quadratic Variation Process

Another interesting property of the Wiener process is also related to its variation: If one replaces in the above definition

${\displaystyle |f(t_{i+1}^{(n)})-f(t_{i}^{(n)})|}$through , ${\displaystyle (f(t_{i+1}^{(n)})-f(t_{i}^{(n)}))^{2}}$

so one arrives at the concept of the quadratic variation of a stochastic process on the interval (for ): ${\displaystyle [X,X]_{t}}$ ${\displaystyle X}$ ${\displaystyle [0,t]}$ ${\displaystyle t\geq 0}$

${\displaystyle [X,X]_{t}:=\sup \left\{\left.\sum _{k=0}^{n-1}\left(X\left(t_{k+1}^{(n)}\right)-X\left(t_{k}^{(n)}\right)\right)^{2}\ \right|n\in \mathbb {N} ,0\leq t_{0}^{(n)}<t_{1}^{(n)}\dotsb <t_{n}^{(n)}\leq t\right\}\;.}$

An important result, which is reflected, for example, in Itō's lemma , is the following: If a (standard) Wiener process is almost certainly true for its quadratic variation ${\displaystyle W}$

${\displaystyle \!\,[W,W]_{t}=t}$.

In general, a distinction is made between two forms of quadratic variation / covariation.

1. Let it be a - martingale . Then the clearly determined growing process from the Doob-Meyer decomposition of , with martingale and predictable growing process, is called the predictable quadratic variation or (angle) bracket of . Spelling or short . ${\displaystyle (X_{t},{\mathcal {F}}_{t})_{t\geq 0}}$ ${\displaystyle L^{2}}$ ${\displaystyle (A_{t})_{t\geq 0}}$ ${\displaystyle X^{2}}$ ${\displaystyle X_{t}^{2}=X_{0}+M_{t}+A_{t}}$ ${\displaystyle (M_{t})_{t\geq 0}}$ ${\displaystyle (A_{t})_{t\geq 0}}$ ${\displaystyle (X_{t})_{t\geq 0}}$ ${\displaystyle \langle X,X\rangle _{t}}$ ${\displaystyle \langle X\rangle _{t}}$

The predictable quadratic covariation for two martingales and is defined as: ${\displaystyle L^{2}}$ ${\displaystyle (X_{t},{\mathcal {F}}_{t})_{t\geq 0}}$ ${\displaystyle (Y_{t},{\mathcal {F}}_{t})_{t\geq 0}}$

${\displaystyle \langle X,Y\rangle _{t}={\frac {1}{4}}\left(\langle X+Y,X+Y\rangle _{t}-\langle X-Y,X-Y\rangle _{t}\right)}$.

2. The quadratic covariation of two semimartingals and / or the quadratic variation of , if , is the following process: ${\displaystyle (X_{t})_{t\geq 0}}$ ${\displaystyle (Y_{t})_{t\geq 0}}$ ${\displaystyle (X_{t})_{t\geq 0}}$ ${\displaystyle Y=X}$

${\displaystyle [X,Y]_{t}=X_{t}Y_{t}-X_{0}Y_{0}-\int _{0}^{t}(X_{s-}){\rm {d}}Y_{s}-\int _{0}^{t}(Y_{s-}){\rm {d}}X_{s}}$.

Relationship between the two definitions:

Let and two semimartingales. Then applies to everyone ${\displaystyle (X_{t})_{t\geq 0}}$ ${\displaystyle (Y_{t})_{t\geq 0}}$ ${\displaystyle t\geq 0}$

${\displaystyle [X,Y]_{t}=\langle X^{c},Y^{c}\rangle _{t}+\sum _{0<s\leq t}\Delta X_{s}\Delta Y_{s}}$,

where and denote the continuous martingale parts. ${\displaystyle X^{c}}$ ${\displaystyle Y^{c}}$

See also

• First variation
• Limited variation

literature

• Philip Protter: Stochastic Integration and Differential Equations . 2nd Edition. Springer, Berlin 2003, ISBN 978-3-540-00313-7 .
• Jean Jacod and Albert N. Shiryaev: Limit Theorems for Stochastic Processes . Springer, 1987, ISBN 3-540-17882-1 .