---
title: "Constructs and Analysis of Bridges Stochastic Differential Equations"
author: 
- A.C. Guidoum^[Department of Probabilities & Statistics, Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail  (acguidoum@usthb.dz)] and K. Boukhetala^[Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail (kboukhetala@usthb.dz)]
date: "`r Sys.Date()`"
output: 
  knitr:::html_vignette:
    toc: yes
vignette: >
  %\VignetteIndexEntry{Constructs and Analysis of Bridges Stochastic Differential Equations}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, echo = F, message = F, results = 'hide',screenshot.force=FALSE}
library(Sim.DiffProc)
library(knitr)
knitr::opts_chunk$set(comment="",prompt=TRUE, fig.show='hold', warning=FALSE, message=FALSE)
options(prompt="R> ",scipen=16,digits=5,warning=FALSE, message=FALSE,
        width = 70)
```

# Bridges SDE's

Consider now a $d$-dimensional stochastic process $X_{t}$ defined on a probability space $(\Omega, \mathfrak{F},\mathbb{P})$. We say that the bridge associated to $X_{t}$ conditioned to the event $\{X_{T}= a\}$ is the process:
$$
\{\tilde{X}_{t}, t_{0} \leq t \leq T \}=\{X_{t}, t_{0} \leq t \leq T: X_{T}= a \}
$$
where $T$ is a deterministic fixed time and $a \in \mathbb{R}^d$ is fixed too.

# The `bridgesdekd()` functions

The (S3) generic function `bridgesdekd()` (where `k=1,2,3`) for simulation of 1,2 and 3-dim bridge stochastic differential equations,Ito or Stratonovich type, with different methods. The main arguments consist: 


- The `drift` and `diffusion` coefficients as R expressions that depend on the state variable `x` (`y` and `z`) and time variable `t`.
- `corr` the correlation structure of standard Wiener process ; must be a real symmetric positive-definite square matrix (2D and 3D, default: `corr=NULL`).
- The number of simulation steps `N`.
- The number of the solution trajectories to be simulated by `M` (default: `M=1`).
- Initial value `x0` at initial time `t0`.
- Terminal value `y` final time `T`
- The integration step size `Dt` (default: `Dt=(T-t0)/N`).
- The choice of process types by the argument `type="ito"` for Ito or `type="str"` for Stratonovich (by default `type="ito"`).
- The numerical method to be used by `method` (default `method="euler"`).

By Monte-Carlo simulations, the following statistical measures (`S3 method`) for class `bridgesdekd()` (where `k=1,2,3`) can be approximated for the process at any time $t \in [t_{0},T]$ (default: `at=(T-t0)/2`):

* The expected value $\text{E}(X_{t})$ at time $t$, using the command `mean`.
* The variance $\text{Var}(X_{t})$ at time $t$, using the command `moment` with `order=2` and `center=TRUE`.
* The median $\text{Med}(X_{t})$ at time $t$, using the command `Median`.
* The mode $\text{Mod}(X_{t})$ at time $t$, using the command `Mode`.
* The quartile of $X_{t}$ at time $t$, using the command `quantile`.
* The maximum and minimum of $X_{t}$ at time $t$, using the command  `min` and `max`.
* The skewness and the kurtosis of $X_{t}$ at time $t$, using the command `skewness` and `kurtosis`.
* The coefficient of variation (relative variability) of $X_{t}$ at time $t$, using the command `cv`.
* The central moments up to order $p$ of $X_{t}$ at time $t$, using the command `moment`.
* The result summaries of the results of Monte-Carlo simulation at time $t$, using the command `summary`.

We can just make use of the `rsdekd()` function (where `k=1,2,3`) to build our random number for class `bridgesdekd()` (where `k=1,2,3`) at any time $t \in [t_{0},T]$. the main arguments consist:

- `object` an object inheriting from class `bridgesdekd()` (where `k=1,2,3`).
- `at` time between $s=t0$ and $t=T$.

The function `dsde()` (where `k=1,2,3`) approximate transition density for class `bridgesdekd()` (where `k=1,2,3`), the main arguments consist: 

- `object` an object inheriting from class `bridgesdekd()` (where `k=1,2,3`).
- `at` time between $s=t0$ and $t=T$.
- `pdf` probability density function `Joint` or `Marginal`.

The following we explain how to use this functions.

# `bridgesde1d()`

Assume that we want to describe the following bridge sde in Ito form:
\begin{equation}\label{eq0166}
dX_t = \frac{1-X_t}{1-t} dt + X_t dW_{t},\quad X_{t_{0}}=3 \quad\text{and}\quad X_{T}=1
\end{equation}
We simulate a flow of $1000$ trajectories, with integration step size $\Delta t = 0.001$, and $x_0 = 3$ at time $t_0 = 0$, $y = 1$ at terminal time $T=1$.

```{r}
set.seed(1234, kind = "L'Ecuyer-CMRG")
f <- expression((1-x)/(1-t))
g <- expression(x)
mod <- bridgesde1d(drift=f,diffusion=g,x0=3,y=1,M=1000)
mod
```
In Figure 1, we present the flow of trajectories, the mean path (red lines) of solution of $X_{t}|X_{0}=3,X_{T}=1$:

```{r 01,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
plot(mod,ylab=expression(X[t]))
lines(time(mod),apply(mod$X,1,mean),col=2,lwd=2)
legend("topleft","mean path",inset = .01,col=2,lwd=2,cex=0.8,bty="n")
```

Hence we can just make use of the `rsde1d()` function to build our random number generator for $X_{t}|X_{0}=3,X_{T}=1$ at time $t=0.55$: 

```{r}
x <- rsde1d(object = mod, at = 0.55) 
head(x, n = 3)
```

The function `dsde1d()` can be used to show the kernel density estimation for $X_{t}|X_{0}=3,X_{T}=1$ at time $t=0.55$ (`hist=TRUE` based on `truehist()` function in [MASS](https://cran.r-project.org/package=MASS) package):

```{r 04,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE}
dens <- dsde1d(mod, at = 0.55)
plot(dens,hist=TRUE) ## histgramme
plot(dens,add=TRUE)  ## kernel density
```

```{r 33, echo=FALSE, fig.cap='Bridge sde 1D. Histgramme and kernel density estimation for $X_{t}|X_{0}=3,X_{T}=1$', fig.env='figure*',fig.width=7,fig.height=7}
knitr::include_graphics(c("Figures/fig03.png","Figures/fig1008.png"))
```

[Return to bridgesde1d()](#bridgesde1d)

# `bridgesde2d()`

Assume that we want to describe the following $2$-dimensional bridge SDE's in Stratonovich form:

\begin{equation}\label{eq:09}
\begin{cases}
dX_t = -(1+Y_{t}) X_{t} dt +  0.2 (1-Y_{t})\circ dB_{1,t},\quad X_{t_{0}}=1 \quad\text{and}\quad X_{T}=1\\
dY_t = -(1+X_{t}) Y_{t} dt +  0.1 (1-X_{t}) \circ dB_{2,t},\quad Y_{t_{0}}=-0.5 \quad\text{and}\quad Y_{T}=0.5
\end{cases}
\end{equation}
with $(B_{1,t},B_{2,t})$ are two correlated standard Wiener process:
$$
\Sigma=
\begin{pmatrix}
1 & 0.3\\
0.3 & 1 
\end{pmatrix}
$$

We simulate a flow of $1000$ trajectories, with integration step size $\Delta t = 0.01$:

```{r}
set.seed(1234, kind = "L'Ecuyer-CMRG")
fx <- expression(-(1+y)*x , -(1+x)*y)
gx <- expression(0.2*(1-y),0.1*(1-x))
Sigma <-matrix(c(1,0.3,0.3,1),nrow=2,ncol=2)
mod2 <- bridgesde2d(drift=fx,diffusion=gx,x0=c(1,-0.5),y=c(1,0.5),Dt=0.01,M=1000,type="str",corr=Sigma)
mod2
```
In Figure 2, we present the flow of trajectories of $X_{t}|X_{0}=1,X_{T}=1$ and $Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$:

```{r 06,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
plot(mod2,col=c('#FF00004B','#0000FF82'))
```
```{r 333, echo=FALSE, fig.cap='  Bridge sde 2D ', fig.env='figure*',fig.width=5,fig.height=5}
knitr::include_graphics("Figures/fig04.png")
```

Hence we can just make use of the `rsde2d()` function to build our random number generator for the couple $X_{t},Y_{t}|X_{0}=1,Y_{0}=-0.5,X_{T}=1,Y_{T}=0.5$ at time $t=5$: 

```{r}
x2 <- rsde2d(object = mod2, at = 5) 
head(x2, n = 3)
```

The marginal density of $X_{t}|X_{0}=1,X_{T}=1$ and $Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$ at time $t=5$ are reported using `dsde2d()` function. A `contour` plot of joint density obtained from a realization of the couple $X_{t},Y_{t}|X_{0}=1,Y_{0}=-0.5,X_{T}=1,Y_{T}=0.5$ at time $t=5$, see e.g. Figure 3:

```{r 09,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE}
## Marginal 
denM <- dsde2d(mod2,pdf="M",at = 5)
plot(denM, main="Marginal Density")
## Joint
denJ <- dsde2d(mod2, pdf="J", n=100,at = 5)
plot(denJ,display="contour",main="Bivariate Transition Density at time t=5")
```

```{r 103, echo=FALSE, fig.cap='The marginal and joint density of $X_{t}|X_{0}=1,X_{T}=1$ and $Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$ at time $t=5$', fig.env='figure*',fig.width=7,fig.height=7}
knitr::include_graphics(c("Figures/fig1009.png","Figures/fig1010.png"))
```

A $3$D plot of the transition density at $t=5$ obtained with: 

```{r 11,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE}
plot(denJ,main="Bivariate Transition Density at time t=5")
```
```{r 33303, echo=FALSE, fig.cap='$3$D plot of the transition density of $X_{t}|X_{0}=1,X_{T}=1$ and $Y_{t}|Y_{0}=-0.5,Y_{T}=0.5$ at time $t=5$  ', fig.env='figure*',fig.width=5,fig.height=5}
knitr::include_graphics("Figures/fig1011.png")
```

We approximate the bivariate transition density over the set transition horizons $t\in [1,9]$ with $\Delta t = 0.005$ using the code:

```{r ,eval=FALSE, include=TRUE}
for (i in seq(1,9,by=0.005)){ 
plot(dsde2d(mod2, at = i,n=100),display="contour",main=paste0('Transition Density \n t = ',i))
}
```

[Return to bridgesde2d()](#bridgesde2d)

# `bridgesde3d()`

Assume that we want to describe the following bridges SDE's (3D) in Ito form:

\begin{equation}
\begin{cases}
dX_t = -4 (1+X_{t}) Y_{t} dt + 0.2 dW_{1,t},\quad X_{t_{0}}=0 \quad\text{and}\quad X_{T}=0\\
dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t},\quad Y_{t_{0}}=-1 \quad\text{and}\quad Y_{T}=-2\\
dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t},\quad Z_{t_{0}}=0.5 \quad\text{and}\quad Z_{T}=0.5
\end{cases}
\end{equation}

We simulate a flow of $1000$ trajectories, with integration step size $\Delta t = 0.001$.

```{r}
set.seed(1234, kind = "L'Ecuyer-CMRG")
fx <- expression(-4*(1+x)*y, 4*(1-y)*x, 4*(1-z)*y)
gx <- rep(expression(0.2),3)
mod3 <- bridgesde3d(x0=c(0,-1,0.5),y=c(0,-2,0.5),drift=fx,diffusion=gx,M=1000)
mod3
```

For plotting (back in time) using the command `plot`, and `plot3D` in space the results of the simulation are shown in Figure 5:

```{r 12,fig.env='figure*', fig.cap=' ',eval=FALSE, include=TRUE}
plot(mod3) ## in time
plot3D(mod3,display = "persp",main="3D Bridge SDE's") ## in space 
```

```{r 3333, echo=FALSE, fig.cap=' Bridge sde 3D ', fig.env='figure*',fig.width=7,fig.height=7}
knitr::include_graphics(c("Figures/fig05.png","Figures/fig06.png"))
```


Hence we can just make use of the `rsde3d()` function to build our random number generator for the triplet $X_{t},Y_{t},Z_{t}|X_{0}=0,Y_{0}=-1,Z_{0}=0.5,X_{T}=0,Y_{T}=-2,Z_{T}=0.5$ at time $t=0.75$: 

```{r}
x3 <- rsde3d(object = mod3, at = 0.75) 
head(x3, n = 3)
```
the density of $X_{t}|X_{0}=0,X_{T}=0$, $Y_{t}|Y_{0}=-1,Y_{T}=-2$ and $Z_{t}|Z_{0}=0.5,Z_{T}=0.5$ process at time $t=0.75$ are reported using `dsde3d()` function. For an approximate joint density for triplet $X_{t},Y_{t},Z_{t}|X_{0}=0,Y_{0}=-1,Z_{0}=0.5,X_{T}=0,Y_{T}=-2,Z_{T}=0.5$ at time $t=0.75$ (for more details, see package  [sm](https://cran.r-project.org/package=sm) or  [ks](https://cran.r-project.org/package=ks).)


```{r 15,fig.env='figure*', fig.cap='  ',eval=FALSE, include=TRUE}
## Marginal
denM <- dsde3d(mod3,pdf="M",at =0.75)
plot(denM, main="Marginal Density")
## Joint
denJ <- dsde3d(mod3,pdf="J",at=0.75)
plot(denJ,display="rgl")
```

 
[Return to bridgesde3d()](#bridgesde3d)

# Further reading

1. [`snssdekd()` & `dsdekd()` & `rsdekd()`- Monte-Carlo Simulation and Analysis of Stochastic Differential Equations](snssde.html).
2. [`bridgesdekd()` & `dsdekd()` & `rsdekd()` - Constructs and Analysis of Bridges Stochastic Differential Equations](bridgesde.html).
3. [`fptsdekd()` & `dfptsdekd()` - Monte-Carlo Simulation and Kernel Density Estimation of First passage time](fptsde.html).
4. [`MCM.sde()` & `MEM.sde()` - Parallel Monte-Carlo and Moment Equations for SDEs](mcmsde.html).
5. [`TEX.sde()` - Converting Sim.DiffProc Objects to LaTeX](sdetotex.html).
6. [`fitsde()` - Parametric Estimation of 1-D Stochastic Differential Equation](fitsde.html).



# References

1. Bladt, M. and Sorensen, M. (2007). Simple simulation of diffusion bridges with application to likelihood inference for diffusions. Working Paper, University of Copenhagen.

2. Guidoum AC, Boukhetala K (2020). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.8, URL https://cran.r-project.org/package=Sim.DiffProc.
