# Overview of the yuima and yuimaGUI R packages

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The YUIMA Project is an open source academic project aimed at developing a complete environment for estimation and simulation of Stochastic Differential Equations and other Stochastic Processes via the R package called `yuima`

and its Graphical User Interface `yuimaGUI`

.

## Quickstart

# install the package install.packages('yuima') # load the package require(yuima)

## The YUIMA Object

The main object is the `yuima`

object which allows to describe the **model** in a mathematically sound way. Then the **data** and the **sampling** structure can be included as well for estimation and simulation purposes.

### Model

#### The ‘setModel’ function

The `setModel()`

function defines a stochastic differential equation with or without jumps of the following form:

\[ dX_t = a(t,X_t, \alpha)dt + b(t,X_t,\beta)dW_t^H + c(t,X_t,\gamma)dZ_t \] where

- \(a(t,X_t,\alpha)\) is the drift term. Described by the
`drift`

argument - \(b(t,X_t,\beta)\) is the diffusion term. Described by the
`diffusion`

argument - \(c(t,X_t,\gamma)\) is the jump term. Described by the
`jump.coeff`

argument - \(H\) is the Hurst coefficient. Described by the
`hurst`

argument - \(Z_t\) is the Levy noise. Described by the
`measure.type`

and`measure`

arguments

**Deterministic Model**

\[ dU_t = \sin(\alpha t) dt \]

setModel(drift = "sin(alpha*t)", # the drift term solve.variable = "u", # the solve variable time.variable = "t") # the time variable

**Geometric Brownian Motion**

\[ dX_t = \mu X_t \; dt + \sigma X_t \; dW_t \]

setModel(drift = "mu*x", # the drift term diffusion = "sigma*x", # the diffusion term solve.variable = "x") # the solve variable

**CKLS Model**

\[ dX_t = (\theta_1+\theta_2 X_t) \; dt + \theta_3 X_t^{\theta_4} \; dW_t \]

setModel(drift = "theta1+theta2*x", # the drift term diffusion = "theta3*x^theta4", # the diffusion term solve.variable = "x") # the solve variable

**2-Dimensional Diffusion with 3 Noises**

\[ \begin{cases} dX_t^1 = -3X_t^1 \; dt + dW_t^1 + X_t^2 dW_t^3 \\ dX_t^2 = -(X_t^1+2X_t^2) \; dt + X_t^1 dW_t^1 + 3 dW_t^2 \end{cases} \]

setModel(drift = c("-3*x1","-x1-2*x2"), # the drift vector diffusion = matrix(c("1","x1","0","3","x2","0"), 2, 3), # the diffusion matrix solve.variable = c("x1","x2")) # the solve variables

**Fractional Ornstein-Uhlenbeck**

\[ dX_t = -\theta X_t \; dt + \sigma \; dW_t^H \]

setModel(drift = "-theta*x", # the drift term diffusion="sigma", # the diffusion term hurst = NA, # the hurst coefficient solve.variable = "x") # the solve variable

**Jump Process with Compound Poisson Measure**

\[ dX_t = -\theta X_t dt + \sigma dW_t + dZ_t \]

setModel(drift = "-theta*x", # the drift term diffusion="sigma", # the diffusion term jump.coeff = "1", # the jump term measure.type = "CP", # the measure type measure = list( # the measure intensity = "lambda", # constant intensity df = "dnorm(z, mu_jump, sigma_jump)" # jump density function ), solve.variable = "x") # the solve variable

#### The ‘setPoisson’ Function

Defines a generic Compound Poisson model.

**Compound Poisson with constant intensity and Gaussian jumps**

\[ X_t = X_0+\sum_{i=0}^{N_t} Y_i \; : \;\;\; N_t \sim Poi\Bigl(\int_0^t \lambda(t)dt\Bigl) , \;\;\;\; Y_i \sim N(\mu_{jump}, \; \sigma_{jump}) \\ \lambda(t)=\lambda \]

setPoisson(intensity = "lambda", # the intensity function df = "dnorm(z, mean = mu_jump, sd = sigma_jump)", # the density function solve.variable = "x") # the solve variable

**Compound Poisson with exponentially decaying intensity and Student-t jumps**

\[ X_t = X_0+\sum_{i=0}^{N_t} Y_i \; : \;\;\; N_t \sim Poi\Bigl(\int_0^t \lambda(t)dt\Bigl) , \;\;\;\; Y_i \sim t( \nu_{jump}, \; \mu_{jump} ) \\ \lambda(t)=\alpha \; e^{-\beta t} \]

setPoisson(intensity = "alpha*exp(-beta*t)", # the intensity function df = "dt(z, df = nu_jump, ncp = mu_jump)", # the density function solve.variable = "x") # the solve variable

#### The ‘setCarma’ Function

Defines a generic Continuous ARMA model.

**Continuous ARMA(3,1) process driven by a Brownian Motion**
\[ CARMA(3,1) \]

setCarma(p = 3, # autoregressive coefficients q = 1) # moving average coefficients

**Continuous ARMA(3,1) process driven by a Compound Poisson with Gaussian jumps**
\[ CARMA(3,1) \]

setCarma(p = 3, # autoregressive coefficients q = 1, # moving average coefficients measure.type = "CP", # compound poisson measure = list( # cp measure intensity = "lambda", # intensity function df = "dnorm(z, 'mu', 'sigma')" # density function ))

#### The ‘setCogarch’ Function

Defines a generic Continuous GARCH model.

**Continuous COGARCH(1,1) process driven by a Compound Poisson with Gaussian jumps**
\[ COGARCH(1,1) \]

setCogarch(p = 1, # autoregressive coefficients q = 1, # moving average coefficients measure.type = "CP", # compound poisson measure = list( # cp measure intensity = "lambda", # intensity function df = "dnorm(z, 'mu', 'sigma')" # density function ))

### Data

The `setData()`

function prepares the data for model estimation. The `delta`

argument describes the time increment between observations. If we have monthly data and want to measure time in years, then `delta`

should be \(1/12\). If we have daily data and want to measure time in months, then `delta`

should be \(1/30\). If we have financial daily data and want to measure time in years, then `delta`

should be \(1/252\), since 252 is the average number of trading days in one year. In general, if we want to measure time in unit \(T\), `delta`

should be 1 over the average number of observations in a period \(T\). The unit of measure of time affects the estimated value of the model parameters.

The following example downloads and sets some financial data (see tutorial on Data Acquisition in R).

# Install the quantmod package if needed: # install.packages('quantmod') # load quantmod require(quantmod) # download Facebook quotes fb <- getSymbols(Symbols = 'FB', src = 'yahoo', auto.assign = FALSE) # setData with time in years -> delta = 1/252 # (there are 252 observations in 1 year) setData(fb$FB.Close, delta = 1/252, t0 = 0) ## ## ## Number of original time series: 1 ## length = 2016, time range [2012-05-18 ; 2020-05-22] ## ## Number of zoo time series: 1 ## length time.min time.max delta ## FB.Close 2016 0 7.996 0.003968254

### Sampling

The `setSampling()`

function describes the simulation grid. If `delta`

is not specified, it is calculated as `(Terminal-Initial)/n`

. If `delta`

is specified, the `Terminal`

is adjusted to be equal to `Initial+n*delta`

.

# define a regular grid using delta setSampling(Initial = 0, delta = 0.01, n = 1000) # define a regular grid using Terminal setSampling(Initial = 0, Terminal = 2, n = 1000)

## Simulation

Simulation of a generic model is perfomed with the `simulate()`

function.

**Example** Solve an Ordinary Differential Equation

# model: ordinary differential equation model <- setModel(drift = 'sin(t)*t', solve.variable = 'x', time.variable = 't') # simulation scheme sampling <- setSampling(Initial = 0, Terminal = 10, n = 1000) # yuima object yuima <- setYuima(model = model, sampling = sampling) # simulation sim <- simulate(yuima) # plot plot(sim)

**Example** Simulate one trajectory of a jump diffusion model

# model: jump diffusion model <- setModel(drift = "-theta*x", diffusion="sigma", jump.coeff = "1", measure.type = "CP", measure = list( intensity = "lambda", df = "dnorm(z, mu_jump, sigma_jump)" ), solve.variable = "x") # simulation scheme sampling <- setSampling(Initial = 0, Terminal = 1, n = 1000) # yuima object yuima <- setYuima(model = model, sampling = sampling) # simulation sim <- simulate(yuima, # the yuima object xinit = 1, # the initial value true.parameter = list( # specify the parameters: theta = 1, # value for the 'theta' parameter sigma = 1, # value for the 'sigma' parameter lambda = 10, # value for the 'lambda' parameter mu_jump = 0, # value for the 'mu_jump' parameter sigma_jump = 2 # value for the 'sigma_jump' parameter )) # plot plot(sim)

## Estimation

The `qmle()`

function calculates the quasi-likelihood and estimate of the parameters of the stochastic differential equation by the maximum likelihood method or least squares estimator of the drift parameter.

**Example** Simulate a Geometric Brownian Motion and estimate its parameters

# model: geometric brownian motion model <- setModel(drift = 'mu*x', diffusion = 'sigma*x', solve.variable = 'x') # simulation scheme sampling <- setSampling(Initial = 0, Terminal = 1, n = 1000) # yuima object yuima <- setYuima(model = model, sampling = sampling) # simulation sim <- simulate(yuima, true.parameter = list(mu = 1.3, sigma = 0.25), xinit = 100) # estimation estimation <- qmle(sim, # the yuima object start = list(mu = 0, sigma = 1), # starting values for optimization lower = list(sigma = 0)) # lower bounds # estimates and standard errors summary(estimation) ## Quasi-Maximum likelihood estimation ## ## Call: ## qmle(yuima = sim, start = list(mu = 0, sigma = 1), lower = list(sigma = 0)) ## ## Coefficients: ## Estimate Std. Error ## sigma 0.2482216 0.005630911 ## mu 1.1125000 0.248221634 ## ## -2 log L: 3150.704

**Example** Estimate the yearly volatility (\(\sigma\) in the Geometric Brownian Motion) of Google stock quotes

# Install the quantmod package if needed: # install.packages('quantmod') # load quantmod require(quantmod) # download Google quotes goog <- getSymbols(Symbols = 'GOOG', src = 'yahoo', auto.assign = FALSE) # setData with time in years -> delta = 1/252 # (there are 252 observations in 1 year) data <- setData(goog$GOOG.Close, delta = 1/252, t0 = 0) # model: geometric brownian motion model <- setModel(drift = 'mu*x', diffusion = 'sigma*x', solve.variable = 'x') # yuima object yuima <- setYuima(model = model, data = data) # estimation estimation <- qmle(yuima, # the yuima object start = list(mu = 0, sigma = 0.5), # starting values for optimization lower = list(sigma = 0)) # lower bounds # estimates and standard errors summary(estimation) ## Quasi-Maximum likelihood estimation ## ## Call: ## qmle(yuima = yuima, start = list(mu = 0, sigma = 0.5), lower = list(sigma = 0)) ## ## Coefficients: ## Estimate Std. Error ## sigma 0.2939155 0.003581592 ## mu 0.1781250 0.080372571 ## ## -2 log L: 24191.29

## yuimaGUI

The yuimaGUI package provides a user-friendly interface for yuima. It simplifies tasks such as estimation and simulation of stochastic processes, including additional tools related to quantitative finance such as data retrieval of stock prices and economic indicators, time series clustering, change point analysis, lead-lag estimation.

The yuimaGUI is available online for free, but it is strongly recommended to install the application via the R package on your local machine for better performance and less downtime.

# install the package install.packages('yuimaGUI') # load the package require(yuimaGUI) # run the interface yuimaGUI()

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