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\noindent{\huge{\bf A report on the paper:}}

\bigskip

{\small

\noindent Griselda Deelstra and Fred Delbaen: ``Long-term returns
in stochastic interest rate models: different convergence
results''. Appl. Stochastic Models Data Anal. 13 (1998), no. 3-4,
401--407.

}

\noindent\dotfill

\bigskip


\noindent{\bf 1. Motivation}

Let $r_t$ denote the short-term interest rate at time $t\ge0$ of a
riskless asset. We assume that $(r_t)_{t\ge0}$ is a real-valued
stochastic process defined on some probability space. Accordingly,
the expected bond value and discount factor are given respectively
by
 \beqlb
\bfE\bigg[\exp\bigg\{\int_0^t r_sds\bigg\}\bigg]
 \quad\mbox{and}\quad
\bfE\bigg[\exp\bigg\{-\int_0^t r_sds\bigg\}\bigg].
 \eeqlb
In particular, if $r_t\equiv r_0$, the above values are given by
 \beqlb
\bfE[e^{tr_0}]
 \quad\mbox{and}\quad
\bfE[e^{-tr_0}].
 \eeqlb
In view of (2), we may rewrite (1) into
 \beqnn
\bfE\bigg[\exp\bigg\{t\bigg(\frac{1}{t}\int_0^t
r_sds\bigg)\bigg\}\bigg]
 \quad\mbox{and}\quad
\bfE\bigg[\exp\bigg\{-t\bigg(\frac{1}{t}\int_0^t
r_sds\bigg)\bigg\}\bigg]
 \eeqnn
and interpret
 \beqlb
\frac{1}{t}\int_0^t r_sds
 \eeqlb
as the averaging long-term interest rate or return.

In the classical Cox-Ingersoll-Ross model, the short-term interest
rate is defined by the stochastic differential equation
 \beqlb
dr_t = \kappa(\gamma-r_t)dt + \sigma\sqrt{r_t}dB_t,
 \eeqlb
where $(B_t)_{t\ge0}$ is a Brownian motion and $\kappa$, $\gamma$
and $\sigma$ are positive constants. In this case, many results
for the process $(r_t)_{t\ge0}$ is known; see, e.g., Pitman and
Yor (1982) and Revuz and Yor (1991). Many authors have also
considered various extensions of the CIR model; see, e.g., Brennan
and Schwartz (1982), Hull and White (1990), Longstaff and Schwartz
(1992) and Schaefer and Schwartz (1984).

\textsl{For all those models, a natural problem is to investigate
the asymptotic behavior of the random variable (3) as $t$ is
large. This problem is discussed in the present paper for an
extended CIR model.}


\bigskip

\noindent{\bf 2. Main results}

Let $\beta<0$ be a constant and $g(\cdot)$ a non-negative function
on $\dfR_+$ which vanishes at zero and there is a constant $b\ge0$
such that
 \beqnn
|g(x)-g(y)| \le b|x-y|^{1/2}, \qquad x,y\ge0.
 \eeqnn
Let $(\delta_t)_{t\ge0}$ be an adapted non-negative measurable
process such that
 \beqnn
\int_0^t \delta_sds < \infty, \qquad t\ge0.
 \eeqnn
The authors consider a process $(X_t)_{t\ge0}$ defined by
 \beqlb
dX_t = (2\beta X_t+\delta_t)dt + g(X_t)dB_t.
 \eeqlb

The following main results are given in this paper:


\medskip

\noindent{\bf Theorem 1~} {\sl Suppose that there is a random
variable $\bar \delta \ge0$ such that
 \beqnn
\frac{1}{t}\int_0^t \delta_sds \overset{\rm a.s.}\longrightarrow
\bar\delta \qquad (t\to\infty).
 \eeqnn
Then the following a.s.\ convergence holds:
 \beqlb
\frac{1}{t}\int_0^t X_sds \overset{\rm a.s.}\longrightarrow
-\frac{\bar\delta}{2\beta} \qquad (t\to\infty).
 \eeqlb
}

\medskip


A particular form of (5) is given by
 \beqlb
dX_t = (2\beta X_t+\delta_t)dt + 2\sqrt{X_t}dB_t.
 \eeqlb
For the process $(X_t)_{t\ge0}$ defined by (7), the authors give
the following

\medskip

\noindent{\bf Theorem 2~} {\sl Suppose that: (i) there is a
constant $\bar \delta>0$ such that
 \beqnn
\frac{1}{t}\int_0^t \delta_sds \overset{\rm a.s.}\longrightarrow
\bar\delta \qquad (t\to\infty);
 \eeqnn
(ii) there is a constant $k\ge0$ such that
 \beqnn
\sup_{t\ge 1}\frac{1}{t}\int_0^t \bfE[\delta_s^2]ds \le k;
 \eeqnn
(iii) for each $a\ge0$,
 \beqnn
\frac{1}{t}\int_{t-a}^t \bfE[\delta_s^2]ds \longrightarrow 0
\qquad (t\to\infty).
 \eeqnn
Then the following convergence in distribution holds:
 \beqlb
\bigg[\bigg(\frac{-2\beta^3}{\bar\delta n}\bigg)^{1/2} \int_0^{nt}
\bigg(X_s+\frac{\delta_s}{2\beta}\bigg)ds\bigg]_{t\ge0}
\overset{\rm L}\longrightarrow (W_t)_{t\ge0} \qquad (n\to\infty),
 \eeqlb
where $(W_t)_{t\ge0}$ is a Brownian motion.}

\medskip



The authors also apply the above theorems to the so-called CIR
two-factor model $(r_t,\gamma_t)$ defined by
 \beqlb
 dr_t &=& \kappa(\gamma_t+r_t)dt + \sigma\sqrt{r_t}dB_t,  \\
 d\gamma_t &=& \tilde\kappa(\tilde\gamma+\gamma_t)dt
 + \tilde\sigma\sqrt{\gamma_t}d\tilde B_t.
 \eeqlb
They obtain that
 \beqlb
\frac{1}{t}\int_0^t r_sds \overset{\rm a.s.}\longrightarrow
\bar\gamma \qquad (t\to\infty)
 \eeqlb
and
 \beqlb
(Y^{(n)}_t)_{t\ge0} :=
\bigg(\frac{\kappa}{\sigma\sqrt{\tilde\gamma n}} \int_0^{nt}
\big(r_s - \gamma_s\big)ds\bigg)_{t\ge0} \overset{\rm L}
\longrightarrow (W_t)_{t\ge0} \qquad (n\to\infty).
 \eeqlb
Based on those results, approximations of the long-term interest
rate are discussed.

\bigskip

\noindent{\bf 3. Proofs}

The proof of Theorem~1 relies on the theory of stochastic
differential equations and on Kroneck's lemma. To prove Theorem~2,
the authors first check the convergence of the finite-dimensional
distributions of the sequence $(Y^{(n)}_t)_{t\ge0}$. Then they
show that the sequence is tight by using Aldous' criterion.

The proofs of the results are actually not provided in this paper.
Instead, the authors refer the reader to Deelstra and Delbaen
(1995, two papers).



\bigskip

\noindent{\bf 4. Discussion}

A comparison of the results in this paper and those from some
other papers inspires a number of problems which might be of
interest to investigate. The problems are described as follows.

The CIR model is a special case of the continuous state branching
process with immigration introduced and studied in Kawazu and
Watanabe (1971). A more general class of interest rate models, the
so-called affine models, are introduced in Duffie et al (2003) and
studied in Dawson and Li (2004). A closely related classes of
measure-valued Markov processes are studied in Li (1996, 1998,
2002).

One might want to see the generalizations of the above two
theorems and their consequences for the models of Dawson and Li
(2004), Duffie et al (2003), Kawazu and Watanabe (1971) and Li
(1996, 1998, 2002) as well as those of Brennan and Schwartz
(1982), Hull and White (1990), Longstaff and Schwartz (1992) and
Schaefer and Schwartz (1984).




\bigskip\bigskip


\noindent{\bf References}

\small

\begin{enumerate}

\renewcommand{\labelenumi}{[\arabic{enumi}]}


\bibitem{DL04}
Dawson, D.A. and Li, Z.H. (2004): Skew convolution semigroups and
affine Markov processes. In preparation. [Preprint available at:
{\tt math.bnu.edu.cn/\~{}lizh}]

\bibitem{DFS03}
Duffie, D.; Filipovi\'c, D. and Schachermayer, W. (2003): Affine
processes and applications in finance. \textit{Ann. Appl. Probab.}
\textbf{13}, 984-1053.

\bibitem{KW71}
Kawazu, K. and Watanabe, S. (1971): Branching processes with
immigration and related limit theorems. \textit{Theory Probab.
Appl.} \textbf{16}, 36-54.

\bibitem{L96}
Li, Z.H. (1996): Immigration structures associated with
Dawson-Watanabe superprocesses. \textit{Stochastic Process. Appl.}
\textbf{62}, 73-86.

\bibitem{L98}
Li, Z.H. (1998): Immigration processes associated with branching
particle systems. \textit{Adv. Appl. Probab.} \textbf{30},
657-675.

\bibitem{L02}
Li, Z.H. (2002): Skew convolution semigroups and related
immigration processes. \textit{Theory Probab. Appl.} \textbf{46},
274-296.

[See also the references of  Deelstra and Delbaen (1998).]


\end{enumerate}




\bigskip\bigskip

\hfill (By Z.\ Li; October 18, 2004)








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