Quadratic variation process

A (quadratic) variation process is a special stochastic process in probability theory , a branch of mathematics . It is obtained from a further process (a martingale or a local martingale ) and in the case of discrete index sets, for example, allows equivalent formulations of the martingale convergence theorem . In the case of continuous time, the paths of the quadratic variation process almost certainly correspond to the quadratic variation of the paths of the underlying process.

In stochastic analysis , quadratic variation processes appear as integrators in the Ito integral .

Definition for a discrete index set

Let filtration be given and be a square integrable martingale . ${\displaystyle \mathbb {F} =({\mathcal {F}}_{t})}$ ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$

Then that predictable process is called by which the stochastic process ${\displaystyle \langle X\rangle =(\langle X\rangle _{n})_{n\in \mathbb {N} }}$

${\displaystyle Y=(X_{n}^{2}-\langle X\rangle _{n})_{n\in \mathbb {N} }}$

becomes a martingale, the quadratic variation process of . It is clearly determined. ${\displaystyle X}$

presentation

From the Doob decomposition follows directly

${\displaystyle \langle X\rangle _{n}=\sum _{i=1}^{n}\left(\operatorname {E} \left(X_{i}^{2}|{\mathcal {F}}_{i-1}\right)-X_{i-1}^{2}\right)}$,

from which the representation

${\displaystyle \langle X\rangle _{n}=\sum _{i=1}^{n}\operatorname {E} \left((X_{i}-X_{i-1})^{2}|{\mathcal {F}}_{i-1}\right)}$.

can be derived.

example

A sequence of independently identically distributed random variables with and is given . ${\displaystyle (Z_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \operatorname {E} (Z_{0})=0}$ ${\displaystyle \operatorname {Var} (Z_{0})<\infty }$

Then

${\displaystyle X_{n}:=\sum _{i=1}^{n}Z_{i}}$

a martingale with regard to canonical filtration and integrable with the square.

By means of the second of the two above representations and and follows ${\displaystyle X_{i}-X_{i-1}=Z_{i}}$ ${\displaystyle {\mathcal {F}}_{i}=\sigma (Z_{1},\dots ,Z_{i})}$

${\displaystyle \langle X\rangle _{n}=\sum _{i=1}^{n}\operatorname {E} \left((Z_{i})^{2}|Z_{1},\dots ,Z_{i-1}\right)=\sum _{i=1}^{n}\operatorname {E} \left(Z_{i}^{2}\right)=n\cdot \operatorname {Var} (Z_{0})}$,

according to the calculation rules for conditional expected values, since they are independent according to the assumption. In this case the quadratic variation process is purely deterministic. Generally this is not the case. ${\displaystyle Z_{i}}$

properties

From the second of the above two representations one obtains directly by forming the expected value

${\displaystyle \operatorname {Var} (X_{n}-X_{0})=\operatorname {E} (\langle X\rangle _{n})}$

But since, according to the martingale convergence theorem, it holds that a martingale converges almost certainly and in the square mean if and only if it is bounded in the square mean, the statement follows

It is if and only if the root mean square converges. ${\displaystyle \sup _{n\in \mathbb {N} }\operatorname {E} (\langle X\rangle _{n})<\infty }$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

A little weaker still applies

Is almost certain, so almost certainly converges . ${\displaystyle \sup _{n\in \mathbb {N} }\langle X\rangle _{n}<\infty }$ ${\displaystyle X}$

In addition, the quadratic variation process of a stopped process is the stopped quadratic variation process, so the commutation relation applies

${\displaystyle \langle X^{\tau }\rangle =\langle X\rangle ^{\tau }}$

for stop times . ${\displaystyle \tau }$

Definition with constant index set

A continuous local martingale is given . Then is the steady, monotonic and adapted process with , with the process ${\displaystyle M=(M_{t})_{t\geq 0}}$ ${\displaystyle \langle M\rangle =(\langle M_{t}\rangle )_{t\geq 0}}$ ${\displaystyle M_{0}=0}$

${\displaystyle Y=(M_{t}^{2}-\langle M_{t}\rangle )_{t\geq 0}}$

becomes a continuous local martingale, the quadratic variation process of . It is clearly determined. ${\displaystyle M}$

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 210.
2. ↑ Kusolitsch: Measure and probability theory. 2014, p. 275.
3. ↑ Klenke: Probability Theory. 2013, p. 227.
4. ↑ Klenke: Probability Theory. 2013, p. 513.