TransformedProcess

TransformedProcess[expr,xproc,t]

represents the transformed process of expr where the variable x follows the random process proc and t denotes the time.

TransformedProcess[expr,{x1proc1,x2proc2,…},t]

represents a transformed process where x1, x2, … are independent and follow the processes proc1, proc2, ….

Details and Options

  • TransformedProcess is typically used to represent functional transformations of one or a finite number of time slices.
  • xproc can be entered as x dist proc or x \[Distributed]proc.
  • The processes proci must all be discrete-time or continuous-time processes.
  • The expression expr must be a function of xi[fij[t]] for a finite number of different fij.
  • Assumptions on parameters can be specified using the option Assumptions->assum.
  • TransformedProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.

Examples

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Basic Examples  (1)

Define a transformed Wiener process:

Simulate the process:

Mean and variance functions:

Covariance function:

Scope  (8)

Single Process Single Time  (4)

Square of a Poisson process:

Simulate the process:

Covariance and correlation functions for the process:

Cube of an AR process:

Simulate the process:

Mean and variance functions:

Monomial transformation of a Wiener process:

Mean function:

Evaluate the mean function for specific values of k:

Quadratic transformation of an Ornstein–Uhlenbeck process:

Simulate the transformed process:

Mean and variance functions are constant:

Verify the results by simulating a slice of the process:

Multiple Process Single Time  (2)

Sum of a Wiener process and a geometric Brownian motion process:

Skewness and kurtosis functions:

Create a jump-diffusion process as the sum of a Poisson process and a Wiener process:

Simulate the process:

Visualize the paths:

Compute slice properties for the process:

Single Process Multiple Times  (1)

Difference of two time slices for a Wiener process:

Probability density function:

The time slices of this process follow a normal distribution:

Multiple Processes Multiple Times  (1)

Sum of two different time slices of a Wiener process and an Ornstein–Uhlenbeck process:

Compute probabilities and expectations:

Options  (1)

Assumptions  (1)

Specify assumptions for a transformation parameter:

Mean for the slice distribution:

Compare with the mean for general values of the parameter:

Applications  (14)

Periodic Signal Corrupted with Noise  (1)

Add white noise to a periodic signal:

Stochastic Exponential Function  (1)

Define the stochastic exponential function:

The corresponding differential equation is u[t]u[t]b[t]:

Bessel Process  (3)

Define a one-dimensional Bessel process:

The mean and variance functions:

Correlation function:

The slice distributions:

Compute the probability of an event:

Define a two-dimensional Bessel process:

Mean and variance functions:

Define a squared Bessel process in 2D:

Mean and variance functions:

Brownian Bridge  (1)

Define a Brownian bridge process:

Simulate the process:

The mean of the process:

Compare the variance with that of BrownianBridgeProcess:

Moving Average Process  (1)

Define a moving average process:

Simulate the process:

Mean, variance, and kurtosis for the process:

Compare with the property values for the corresponding MAProcess:

Telegraph Process  (1)

Define the telegraph process in terms of PoissonProcess:

Simulate the process:

Probability density function for a time slice of the process:

Compare with the PDF for TelegraphProcess:

Gaussian Processes  (1)

Linear transformations of Gaussian processes are Gaussian:

Verify that single slices of both processes are Gaussian:

TemporalData  (1)

Square of a TemporalData object:

Simulate the process:

Mean and variance for a time slice of the process:

Compare with the values obtained from simulation:

Insurance Process  (1)

Simulate the surplus process for insurance, given that the insurer's initial surplus is 70 and the total annual premium is 61.2, if the number of claims follows a Poisson process with mean 60 and the losses are distributed exponentially with mean 1:

Delayed Process  (1)

Define a delayed process:

Simulate the original and delayed processes:

Standardized Process  (1)

Standardize a random process to one with zero mean and unit variance:

Merton's Jump-Diffusion Model  (1)

Define Merton's jump-diffusion model for option pricing:

Simulate the process:

Slice properties for the process:

Properties & Relations  (2)

SliceDistribution relates TransformedProcess to TransformedDistribution:

The resulting distributions are equal:

Transformed Wiener processes are related to ItoProcess:

Mean, variance, etc. functions agree:

Neat Examples  (2)

Generate Brownian motion on the unit circle:

A family of transformed Wiener processes:

Wolfram Research (2014), TransformedProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/TransformedProcess.html.

Text

Wolfram Research (2014), TransformedProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/TransformedProcess.html.

CMS

Wolfram Language. 2014. "TransformedProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TransformedProcess.html.

APA

Wolfram Language. (2014). TransformedProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TransformedProcess.html

BibTeX

@misc{reference.wolfram_2024_transformedprocess, author="Wolfram Research", title="{TransformedProcess}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/TransformedProcess.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_transformedprocess, organization={Wolfram Research}, title={TransformedProcess}, year={2014}, url={https://reference.wolfram.com/language/ref/TransformedProcess.html}, note=[Accessed: 22-November-2024 ]}