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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`

.

## 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**

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

#### The ‘setCarma’ Function

Defines a generic Continuous ARMA model.

**Continuous ARMA(3,1) process driven by a Brownian Motion**

\[ CARMA(3,1) \]

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

\[ CARMA(3,1) \]

#### The ‘setCogarch’ Function

Defines a generic Continuous GARCH model.

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

\[ COGARCH(1,1) \]

### 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 = 2011, time range [2012-05-18 ; 2020-05-15]
##
## Number of zoo time series: 1
## length time.min time.max delta
## FB.Close 2011 0 7.976 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`

.

## 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 0.2940395 0.003585685
## mu 0.1750000 0.080466180
##
## -2 log L: 24147.58
```

## 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.

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