Discrete stochastic integral

The discrete stochastic integral is a possibility in probability theory to link two stochastic processes in discrete time in order to create another stochastic process from them. If in particular one of the two processes is a martingale , one also speaks of the martingale transformation

definition

Let there be a filtration and a real process that is - adapted . Also , be another real process that is - predictable . Then it's called for through ${\displaystyle \mathbb {F} =({\mathcal {F}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {F} }$ ${\displaystyle (H_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mathbb {F} }$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle (H\cdot X)_{n}:=\sum _{m=1}^{n}H_{m}(X_{m}-X_{m-1})}$

defined stochastic process the discrete stochastic integral of respect . Is a martingale, the martingale transform of is called . ${\displaystyle (H\cdot X)}$ ${\displaystyle H}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle (H\cdot X)}$ ${\displaystyle X}$

Example: stopped process

Given a real stochastic process with generated filtration and a stop time with respect . Then the process is also predictable. The discrete stochastic integral is then ${\displaystyle X}$ ${\displaystyle \mathbb {F} }$ ${\displaystyle \tau }$ ${\displaystyle \mathbb {F} }$ ${\displaystyle H_{n}:=\mathbf {1} _{\{\tau \geq n\}}}$ ${\displaystyle \mathbb {F} }$

${\displaystyle (H\cdot X)_{n}=X_{0}+\sum _{m=1}^{n}\mathbf {1} _{\tau \geq m}(X_{m}-X_{m-1})={\begin{cases}X_{n}&{\text{ falls }}n<\tau \\X_{\tau }&{\text{ falls }}n\geq \tau \end{cases}}}$.

That is then exactly the stopped process regarding . ${\displaystyle (X_{\min(n,\tau )})_{n}}$ ${\displaystyle \tau }$

properties

Be an adapted, real process with . Then: ${\displaystyle X}$ ${\displaystyle \operatorname {E} (|X_{0}|)<\infty }$

• ${\displaystyle X}$is a (sub-) supermartingale if and only if a (sub-) supermartingale is for every predictable one that is locally restricted, that is, for all . ${\displaystyle H\cdot X}$ ${\displaystyle H\geq 0}$ ${\displaystyle 0\leq H_{n}<\infty }$ ${\displaystyle n}$
• ${\displaystyle X}$is a martingale if and only if a martingale is for every predictable one that is locally restricted, that is, for all . ${\displaystyle H\cdot X}$ ${\displaystyle H}$ ${\displaystyle |H_{n}|<\infty }$ ${\displaystyle n}$

This statement is also known as the martingale transform theorem.

Inferences

The following conclusion can be drawn from the above statement about the stability of martingales under the discrete stochastic integral: If you as a player take part in a fair game over several rounds with a game strategy that consists of placing a bet in the round , then there is none among these strategies that would be more beneficial to the player than others. The fair game corresponds to a martingale, the profit after the nth round is then the martingale transform of and . Since this is always a martingale, the game cannot be changed by a game strategy in such a way that it would be advantageous for the player, which would correspond to a submartingale. ${\displaystyle X}$ ${\displaystyle H}$ ${\displaystyle n}$ ${\displaystyle H_{n}}$ ${\displaystyle H}$ ${\displaystyle X}$

The Optional Stopping Theorem provides comparable statements about a possible improvement in overall profit through termination strategies .

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 .
• Christian Hesse: Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .