Long-term behavior of non-ferrous metal price models with jumps
© Peng and Huang; licensee Springer 2014
Received: 20 January 2014
Accepted: 17 July 2014
Published: 4 August 2014
In this paper, we study the long-term behavior of a class of stochastic non-ferrous metal prices with jumps. Suppose that is a stochastic model for some metal price with Poisson jumps. For a suitable , we prove that converges almost surely as . Finally, the model is applied to forecast the behavior of a two-factor affine model.
MSC:60H15, 86A05, 34D35.
where 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  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 .
where . The integral 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
where and 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., ).
Lemma 1 If satisfies (1) and , then .
It is easy to observe that and if or otherwise ; if or otherwise .
where if or otherwise .
Hence as and the nonnegative property of the solution follows. □
where is a bounded stopping time.
where is arbitrary. Due to , we choose and ϵ such that . □
Now we turn to the proof of the convergence theorem.
Proof We use Kronecker’s lemma in .
Hence , and consequently, we only need to prove the existence a.e. of on , where .
3 The long-time behavior of affine models
a two-dimensional Brownian motion ;
, represent Poisson counting measures with characteristic measures and , respectively.
For model (2), we are interested in the almost sure convergence of the long-term behavior for some .
The process has a reversion level which is a stochastic process itself. From Dawson and Li , the equation system has a unique strong solution . Moreover, is an affine Markov process. Now, we give the main theorem of this section.
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.
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).
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