Strange right-arrow symbol ([itex]\mapsto[/itex]) in stochastic calculus.

In summary, the strange right-arrow symbol ([itex]\mapsto[/itex]) is used to represent mappings or transformations between stochastic processes in stochastic calculus, allowing for the analysis and modeling of complex systems. It is also used in other areas of mathematics and computer science to represent mappings and functions. In differential geometry and topology, it is used with other symbols to denote specific types of mappings.
  • #1
operationsres
103
0
What does the right arrow mean in this context:

"
...Then the process [itex]t \mapsto \int_{0}^t \phi_s dM_s[/itex] are well-defined continuous local martingales, whose quadratic variations are given by ...

"

Is this supposed to mean "the process [itex]X[/itex] that is the mapping [itex]X: t \mapsto \int_{0}^t \phi_s dM_s[/itex]"
 
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  • #2
Yes, that indicates a "mapping". t is mapped to [itex]\int_0^t \phi_s dM_s[/itex].
 

1. What does the strange right-arrow symbol ([itex]\mapsto[/itex]) represent in stochastic calculus?

The strange right-arrow symbol is used to denote a mapping or transformation from one stochastic process to another. This symbol is often used in the context of stochastic calculus to represent the mapping of a Wiener process (or Brownian motion) to a stochastic differential equation.

2. How is the strange right-arrow symbol ([itex]\mapsto[/itex]) used in stochastic calculus?

In stochastic calculus, the strange right-arrow symbol is used to define a stochastic process in terms of its underlying Wiener process. It represents the relationship between the two processes and allows for the analysis and modeling of complex stochastic systems.

3. What is the significance of the strange right-arrow symbol ([itex]\mapsto[/itex]) in stochastic calculus?

The strange right-arrow symbol is significant in that it allows for the representation and analysis of stochastic processes in terms of their underlying Wiener processes. This allows for the application of stochastic calculus techniques, such as Ito's lemma, to solve stochastic differential equations and make predictions about future behavior of the process.

4. Are there any other uses for the strange right-arrow symbol ([itex]\mapsto[/itex]) outside of stochastic calculus?

Yes, the strange right-arrow symbol is also used in other areas of mathematics, such as set theory and functional analysis, to denote mappings between sets or spaces. It is also commonly used in computer science to represent functions or transformations between data structures.

5. Is the strange right-arrow symbol ([itex]\mapsto[/itex]) used in other branches of calculus?

Yes, the strange right-arrow symbol is also used in differential geometry and differential topology to represent mappings between manifolds. In these fields, it is often used in conjunction with other symbols, such as the Greek letter phi (φ), to denote specific types of mappings.

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