Ito integral of an arbitrary f(Wiener process)

In summary, the Ito integral is a mathematical tool used in stochastic calculus to integrate a function with respect to a Wiener process. A Wiener process is a type of stochastic process that models the random movement of particles in a fluid. The Ito integral is used to measure the area under the curve of a function as the Wiener process changes over time, allowing for the analysis of stochastic processes and calculation of expected values and variances. It differs from the Stratonovich integral in its interpretation, and has applications in finance, economics, engineering, and physics.
  • #1
saminator910
96
1
How would you take the Ito integral of an arbitrary [itex]f(W_T)[/itex] where [itex]W_T[/itex] is a standard Wiener process

[itex]X_T=\int^t_0 f(W_s)dW_s[/itex]

would you somehow use Ito's lemma? I have attempted, but it doesn't seem to make sense...
[itex]dX_T=f(W_T)dW_T[/itex], There doesn't seem to be a [itex]f(x)[/itex] that makes sense for this in Ito's lemma.

[itex]df = \frac{\partial f}{\partial x}dX_T + \frac{\partial^2 f}{\partial x^2}dX_T^2[/itex]
 
Physics news on Phys.org
  • #2
bump...
 

FAQ: Ito integral of an arbitrary f(Wiener process)

What is an Ito integral?

The Ito integral is a mathematical tool used in stochastic calculus to integrate a function with respect to a Wiener process. It is defined as the limit of a sum of products of the function evaluated at the discrete time points and the increments of the Wiener process.

What is a Wiener process?

A Wiener process is a type of stochastic process, also known as a Brownian motion, that models the random movement of particles in a fluid. It is characterized by its continuous and unpredictable trajectory, making it a useful tool in the study of random phenomena.

What is the relationship between the Ito integral and the Wiener process?

The Ito integral is used to integrate a function with respect to a Wiener process, meaning it measures the area under the curve of the function as the Wiener process changes over time. This allows for the analysis of stochastic processes and the calculation of expected values and variances.

What is the difference between Ito and Stratonovich integrals?

Both Ito and Stratonovich integrals are used to integrate functions with respect to a Wiener process. The main difference between the two lies in the interpretation of the integrals, with Ito integrals measuring the area under the curve and Stratonovich integrals measuring the area above the curve.

What are some applications of the Ito integral?

The Ito integral has many practical applications, especially in the fields of finance and economics. It is used to model and analyze various stochastic processes, such as stock prices, interest rates, and exchange rates. It is also used in engineering and physics to model and study random phenomena.

Similar threads

The chain rule for the Stratonovich integral
Replies
1
Views
552
I Definition of a diffusion coefficient
Replies
1
Views
1K
I The Euler-Lagrange equation and the Beltrami identity
Replies
1
Views
2K
Ito's lemma/taylor series vs. differential of a function
Replies
7
Views
3K
Existence/uniqueness of solution and Ito's formula
Replies
1
Views
1K
What is the solution to this ODE (and SDE)?
Replies
2
Views
2K
How do I solve this stochastic differential equation?
Replies
1
Views
1K
MHB Evaluate the surface integral $\iint\limits_{\sum}f\cdot d\sigma$
Replies
1
Views
2K
I Integration with different infinitesimal intervals
Replies
31
Views
2K
How Does Ito's Lemma Simplify the Stochastic Integral of W^n?
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top