[Yuima-commits] r234 - pkg/yuimadocs/inst/doc/JSS
noreply at r-forge.r-project.org
noreply at r-forge.r-project.org
Sat Mar 9 23:06:29 CET 2013
Author: iacus
Date: 2013-03-09 23:06:29 +0100 (Sat, 09 Mar 2013)
New Revision: 234
Modified:
pkg/yuimadocs/inst/doc/JSS/article-new.Rnw
pkg/yuimadocs/inst/doc/JSS/bibliography.bib
Log:
update
Modified: pkg/yuimadocs/inst/doc/JSS/article-new.Rnw
===================================================================
--- pkg/yuimadocs/inst/doc/JSS/article-new.Rnw 2013-03-05 23:19:43 UTC (rev 233)
+++ pkg/yuimadocs/inst/doc/JSS/article-new.Rnw 2013-03-09 22:06:29 UTC (rev 234)
@@ -72,14 +72,14 @@
Masaaki Fukasawa\\Osaka University\And
Hideitsu Hino\\Waseda University\And
Stefano M. Iacus\\University of Milan \AND
- Kengo Kamatani\\The University of Tokyo \And
- Yuta Koike\\The University of Tokyo\And
+ Kengo Kamatani\\University of Tokyo \And
+ Yuta Koike\\University of Tokyo\And
Hiroki Masuda\\Kyushu University \And
- Ryosuke Nomura\\The University of Tokyo\AND
+ Ryosuke Nomura\\University of Tokyo\AND
Teppei Ogihara\\Osaka University \And
Yasutaka Shimuzu\\Osaka University \And
Masayuki Uchida\\Osaka University \And
- Nakahiro Yoshida\\The University of Tokyo
+ Nakahiro Yoshida\\University of Tokyo
}
\title{The YUIMA Project: a Computational Framework for Simulation and Inference of Stochastic Differential Equations}
@@ -237,6 +237,9 @@
the occasional user should consider to install the stable version first.
The package can be installed from R-Forge using
\code{install.packages("yuima",repos="http://R-Forge.R-project.org")}.
+If, the R-Forge system does not provide binary builds of the \pkg{yuima} package, the
+user can also try
+\code{install.packages("yuima",repos="http://R-Forge.R-project.org", type="source")}.
\subsection{The Main Object and Classes}
@@ -399,7 +402,7 @@
@
\subsection{Multidimensional Processes}\label{sec:multi}
-Next is an example of a system of two stochastic differential equations for the couple $(X_{t,1}, X_{t,2})$ driven by three independent Brownian motions $(W_{t,1}, W_{t,2}, W_{t,3})$
+Next is an example of a system of two stochastic differential equations for the couple $(X_{1,t}, X_{2,t})$ driven by three independent Brownian motions $(W_{1,t}, W_{2,t}, W_{3,t})$
$$
\begin{aligned}
\de X_{1,t} &= -3 X_{1,t} \de t + \de W_{1,t} + X_{2,t} \de W_{3,t}\\
@@ -411,7 +414,7 @@
\left(\begin{array}{c}\de X_{1,t} \\\de X_{2,t}\end{array}\right)=
\left(\begin{array}{c} -3 X_{1,t} \\ -X_{1,t} - 2X_{2,t}\end{array}\right)\de t +
\left(\begin{array}{ccc}1 & 0 & X_{2,t} \\X_{1,t} & 3 & 0\end{array}\right)
-\left(\begin{array}{c}\de W_{t,1} \\ \de W_{t,2} \\ \de W_{t,3}\end{array}\right)
+\left(\begin{array}{c}\de W_{1,t} \\ \de W_{2,t} \\ \de W_{3,t}\end{array}\right)
$$
For this system it now necessary to instruct \pkg{yuima} about the state variable on bot the left-hand side of the equation and the right-hand side of the equation, i.e. the \code{solve.variable}.
<<echo=TRUE, print=FALSE,results=hide>>=
@@ -597,8 +600,8 @@
Assume that we want to simulate this model
$$
\begin{aligned}
-\de X^1_t &= -\theta X^1_t \de t + \de W^1_t + X^2_t \de W^3_t\\
-\de X^2_t &= -(X^1_t + \gamma X^2_t) \de t + X^1_t \de W^1_t + \delta \de W^2_t
+\de X_{1,t} &= -\theta X_{1,t} \de t + \de W_{1,t} + X_{2,t} \de W_{3,t}\\
+\de X_{2,t} &= -(X_{1,t} + \gamma X_{2,t}) \de t + X_{1,t} \de W_{1,t} + \delta \de W_{2,t}
\end{aligned}
$$
Now we prepare the model using the \code{setModel} constructor function
@@ -793,9 +796,11 @@
calculation of the expectation, while the asymptotic expansion of $F^{(\ve)}$ gives very fast
and accurate approximation by analytic formulation.
The \pkg{yuima} package provides functions to construct the functional $F^{(\ve)}$, and automatic asymptotic expansion based on the Malliavin calculus. starting from a \pkg{yuima} object.
-This asymptotic expansion approach to option pricing was proposed in \citep{Yoshida92b}
-and many related papers are available today.
+This asymptotic expansion approach to option pricing was proposed in early 1990's
+( \citep{Yoshida92b}, \citep{Takahashi99}, \citep{Kunitomo01}), and
+many related papers are available today.
+
Though our method can be applied to the nonlinear system (\ref{asy01})-(\ref{yuima:func1}),
just for an example, we shall consider
the Asian call option of the geometric Brownian motion of equation \eqref{eqyuima:gbm1} with $\mu=1$ and $x_0=1$,
@@ -853,7 +858,7 @@
asy2 <- asymp$d0 + e * asymp$d1+ e^2* asymp$d2 # 2nd order asymp. exp. of asian call price
asy2
@
-give the first and second order asymptotic expansions, respectively.
+give the first and second order asymptotic expansions, respectively.
We remark that the expansion of $\dE[g(\tilde{F}^{(\ep)})G^{(\ep)}]$ is also possible by the same method
for a functional $G^{(\ep)}$ having a stochastic expansion like (\ref{130206-8}).
@@ -870,7 +875,7 @@
@
and the relative difference between the two approximations is \Sexpr{round((asy2-levy)/asy2*100,1)}\%.
-\subsection{Asymptotic Expansion for General Stochastic Processes}
+\subsubsection{Asymptotic Expansion for General Stochastic Processes}
Of course, \pkg{yuima} approach is more general in that the above Levy approximation only holds when the process $X_t$ is the geometric Brownian motion. We now give an example when the underlying process $X_t$ is the following CIR model
$$
\de X_t = 0.9 X_t \de t + \ep \sqrt{X_t} \de W_t, X0 = 1
Modified: pkg/yuimadocs/inst/doc/JSS/bibliography.bib
===================================================================
--- pkg/yuimadocs/inst/doc/JSS/bibliography.bib 2013-03-05 23:19:43 UTC (rev 233)
+++ pkg/yuimadocs/inst/doc/JSS/bibliography.bib 2013-03-09 22:06:29 UTC (rev 234)
@@ -10,6 +10,28 @@
+
+ at article{Kunitomo01,
+ title = {The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims},
+ author = {Kunitomo, N. and Takahashi, A.},
+ journal = {Mathematical Finance},
+ year = {2001},
+ number={11},
+ issue={1},
+ pages={117--151}
+}
+
+ at article{Takahashi99,
+ title = {An asymptotic expansion approach to pricing financial contingent claims},
+ author = {Takahashi, A.},
+ journal = {Asia-Pacific Financial Markets},
+ year = {1999},
+ number={6},
+ issue={2},
+ pages={115--151}
+}
+
+
@article{Levy92,
title = {Pricing European Average Rate Currency Options},
author = {Levy, E.},
@@ -272,19 +294,18 @@
-
@Manual{ERRE,
title = {R: A Language and Environment for Statistical Computing},
author = {{R Core Team}},
organization = {R Foundation for Statistical Computing},
address = {Vienna, Austria},
- year = {2012},
+ year = {2013},
note = {{ISBN} 3-900051-07-0},
url = {http://www.R-project.org/},
}
+
-
@article{Yos05a,
author={Yoshida, N.},
title={General M-estimation for Stochastic Differential
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