# 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 OrnsteinUhlenbeck 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 OrnsteinUhlenbeck 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)

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: