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# Long-term behavior of non-ferrous metal price models with jumps

- Jun Peng
^{1, 2, 3}Email author and - Jianbai Huang
^{1, 3}

**2014**:210

https://doi.org/10.1186/1687-1847-2014-210

© Peng and Huang; licensee Springer 2014

**Received: **20 January 2014

**Accepted: **17 July 2014

**Published: **4 August 2014

## Abstract

In this paper, we study the long-term behavior of a class of stochastic non-ferrous metal prices with jumps. Suppose that $X(t)$ is a stochastic model for some metal price with Poisson jumps. For a suitable $\mu \ge 1$, we prove that ${t}^{-\mu}{\int}_{0}^{t}X(s)\phantom{\rule{0.2em}{0ex}}ds$ converges almost surely as $t\to \mathrm{\infty}$. Finally, the model is applied to forecast the behavior of a two-factor affine model.

**MSC:**60H15, 86A05, 34D35.

## Keywords

- long-term behavior
- jump
- geometric Brownian motion
- convergence

## 1 Introduction

where $d{W}_{t}$ is the increment in a Gauss-Wiener process with drift *θ* and instantaneous standard deviation *σ*. A geometric Brownian motion or exponential Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion or a Wiener process. It is applicable to mathematical modeling of some phenomena in financial markets. GBM is used as a mathematical model in financial markets and in forecasting prices. The GBM formula means that ‘in any interval of time, prices will never be negative and can either go up or down randomly as a function of their volatility’.

One generalization of the GBM model is to use regime switching such as in [7, 8], to name a few. Hamilton [9] characterizes business cycles as periods of discrete regime shifts, *i.e.*, recessions are characterized belonging to one regime and expansions to another in a Markov-switching process. The main advantage of the Markov-switching space state model over the standard GARCH model is that in the case of the latter the unconditional variance is constant, while in the former the variance changes according to the state of the economy. However, the Markov-switching model has been criticized because it lacks transparency, is less robust and is difficult to apply [10].

where $X(t-):={lim}_{s\to t}X(s)$. The integral ${\int}_{U}g(X(t-),u)\tilde{N}(dt,du)$ depends on the Poisson measure and is regarded as a jump. Precise assumptions on the data of Equation (1) are given in Section 2.

The remainder of the paper is organized as follows. In the next section, we give the limit theorem and its proof for the stochastic metal price model with jumps. The limit theorem of a two-factor affine model is given in the last section.

## 2 The long-term behavior of stochastic metal price models with jumps

*d*-dimensional Brownian motion and

*N*a Poisson random measure on $(0,+\mathrm{\infty})\times (Z\setminus \{0\})$, where $Z\subset {R}^{d}$ is equipped with its Borel field ${\mathcal{B}}_{Z}$, with the Levy compensator $\tilde{N}(dt,dz)=dt\nu (dz)$,

*i.e.*, ${\{\tilde{N}((0,t]\times A)=(N-\tilde{N})((0,t]\times A)\}}_{t>0}$ is an ${\mathcal{F}}_{t}$ martingale for each $A\in {\mathcal{B}}_{Z}$. Hence $\nu (dz)$ is a Poisson

*σ*-finite measure satisfying ${\int}_{Z}\nu (dz)<\mathrm{\infty}$. We assume that there exist a sufficiently large constant $\gamma >0$ and a function $\rho :{R}^{k}\to {R}^{+}$ with ${\int}_{Z}{\rho}^{2}(z)\nu (dz)<\mathrm{\infty}$ such that

where $\u3008\cdot ,\cdot \u3009$ and $|\cdot |$ denote the Euclidean scalar product and the norm, respectively. Obviously, under the above assumptions, there exists a unique strong solution to (1) (see, *e.g.*, [14]).

**Lemma 1** *If* $X(t)$ *satisfies* (1) *and* $P(X(0)\ge 0)=1$, *then* $P(X(t)\ge 0\mathit{\text{for all}}t\ge 0)=1$.

*Proof*Let ${a}_{0}=1$ and ${a}_{k}=exp(-k(k+1)/2)$ for $k\ge 1$, so that ${\int}_{{a}_{k}}^{{a}_{k-1}}\frac{du}{u}=k$. For each $k\ge 1$, there clearly exists a continuous function ${\psi}_{k}(u)$ with support in $({a}_{k},{a}_{k-1})$ such that

It is easy to observe that $\phi \in {C}^{2}(R,R)$ and $-1\le {\phi}_{k}(x)\le 0$ if $x<-{a}_{k}$ or otherwise ${\phi}_{k}^{\prime}(x)=0$; ${\phi}^{\u2033}(x)\le \frac{2}{k{x}^{2}}$ if $-{a}_{k-1}<x<-{a}_{k}$ or otherwise ${\phi}^{\u2033}(x)=0$.

where ${x}^{-}=-x$ if $x<0$ or otherwise ${x}^{-}=0$.

Hence $E{X}^{-}(t)=0$ as $k\to \mathrm{\infty}$ and the nonnegative property of the solution ${\{X(t)\}}_{t\ge 0}$ follows. □

**Lemma 2**

*Let*$\beta <0$

*and assume that*$2\beta +{K}^{2}<0$.

*Then there exist*$k>0$

*and*$C>0$

*such that*

*where* $\tau >0$ *is a bounded stopping time*.

*Proof*By Ito’s formula, we have

*τ*and taking expectations on both sides, we have

where $\u03f5>0$ is arbitrary. Due to $2\beta +{K}^{2}<0$, we choose $K>0$ and *ϵ* such that $(2-k)\beta +2\u03f5+{K}^{2}=0$. □

Now we turn to the proof of the convergence theorem.

**Theorem 1**

*Let*$X(t)$

*be a solution to*(1)

*and assume that there is*$\mu \ge 1$

*and a nonnegative random variable*$\overline{\delta}$

*such that*

*Then the following convergence holds*:

*Proof* We use Kronecker’s lemma in [12].

*K*depending on

*ω*. A straightforward calculation shows that

Hence $\{{T}_{n}=\mathrm{\infty}\}\uparrow \mathrm{\Omega}$, and consequently, we only need to prove the existence a.e. of ${\int}_{0}^{\mathrm{\infty}}\frac{g{X}_{u}^{{T}_{n}}}{u+1}\phantom{\rule{0.2em}{0ex}}d{B}_{u}$ on $\{T=\mathrm{\infty}\}$, where $g(x)=x$.

*t*. For the second term, we apply Fubini’s theorem to find a bound which does not depend on

*t*,

□

## 3 The long-time behavior of affine models

- (i)
a two-dimensional Brownian motion $W(\cdot )=({W}_{1}(\cdot ),{W}_{2}(\cdot ))$;

- (ii)
${N}_{1}(dt,du)$, ${N}_{2}(dt,du)$ represent Poisson counting measures with characteristic measures ${\nu}_{1}(\cdot )$ and ${\nu}_{2}(\cdot )$, respectively.

For model (2), we are interested in the almost sure convergence of the long-term behavior ${t}^{-\mu}{\int}_{0}^{t}Y(s)\phantom{\rule{0.2em}{0ex}}ds$ for some $\mu \ge 1$.

The process $Y(t)$ has a reversion level $X(t)$ which is a stochastic process itself. From Dawson and Li [15], the equation system has a unique strong solution $(X(\cdot ),Y(\cdot ))$. Moreover, $(X(\cdot ),Y(\cdot ))$ is an affine Markov process. Now, we give the main theorem of this section.

**Theorem 2**

*Assume that*$(X(\cdot ),Y(\cdot ))$

*is a solution to the equation system*(2).

*Then we have*

*Proof* It can be obtained by similar arguments as Theorem 1. □

Another application of Theorem 1 is that if the average of the drift converges almost surely to a constant, then the long-term trend of the model will revert to a line almost surely.

**Corollary 3**

*Let*$\delta :\omega \times {R}_{+}\to {R}_{+}$

*and there exist constants*$\mu \ge 1$

*and*$\overline{\delta}\ge 0$

*such that*

*Then the following convergence holds for equation*(1):

## Declarations

### Acknowledgements

This work was supported by the Postdoctoral Foundation of Central South University, Major Program of the National Social Science Foundation of China (13&ZD024), the Hunan Provincial Natural Science Foundation of China (14JJ3019) and the National Natural Science Foundation of China (Grant No. 11101433).

## Authors’ Affiliations

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## Copyright

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