Two-Dimensional (Bivariate) Jump-Diffusion Processes

Abstract

In this chapter we provide a generalization of jump-diffusion precesses (12.1) in two dimensions by considering a class of coupled systems that are described by a bivariate state vector $\mathbf{x }(t)$ contained in a two-dimensional state space $\{\mathbf{x}\}$. The evolution of the state vector $\mathbf{x}(t)$ is assumed to be governed by a deterministic part to which diffusion parts and jump contributions are added. Generalization of the results to higher dimensions is straightforward. We note that for N multivariate time series, by assuming the presence of two-body type of interactions between time series, the analysis will reduce to analysing ${N(N-1)}/{2}$ pairwise bivariate time series.

[Type][CrossLinking]The original version of this chapter was revised: Belated correction has been incorporated. The correction to this chapter is available at https://doi.org/10.1007/978-3-030-18472-8_24

Problems

13.1

Kramers–Moyal coefficients

Fill in the details in the derivation of Eq. (13.9).

13.2

Correlated Wiener processes

Assume that two Wiener processes are dependent as,

$$ \langle dW_1(t) dW_2(t) \rangle = \rho dt. $$

Show that two correlated Wiener processes $dW_1(t)$ and $dW_2(t)$ can be written in terms of two independent Wiener processes $dw_1$ and $dw_2$ as,

$$\begin{aligned} dW_1= & {} \frac{\sqrt{1+\rho }}{\sqrt{2}} dw_1 + \frac{\sqrt{1+\rho }}{\sqrt{2}} dw_2 \\\nonumber \\ dW_2= & {} \frac{\sqrt{1+\rho }}{\sqrt{2}} dw_1 - \frac{\sqrt{1+\rho }}{\sqrt{2}} dw_2, \end{aligned}$$

or

$$\begin{aligned} dW_1= & {} dw_1 \\\nonumber \\ dW_2= & {} \rho dw_1 - \sqrt{1-\rho ^2} dw_2. \end{aligned}$$

13.3

Positive correlated Poisson distributed jump processes

One simple way to correlate two independent Poisson-distributed jumps $J_1$ and $J_2$ is to use three Poisson-distributed jumps $n_1$, $n_2$ and $n_3$ (with jump rates $\lambda _1$, $\lambda _2$ and $\lambda _3$), so that $J_1$ and $J_2$ are defined as:

$$ J_1= n_1 + n_3, J_2= n_2 + n_3. $$

(a) With this definitions, using the statistical moment generating functions show that changing the construction of the Poisson process $J_i$ in this way does not change it’s distribution (the mean will be $\mu _i=\lambda _i + \lambda _3$).

(b) Show that the mean, covariance and correlation coefficients of $J_1$ and $J_2$ are given by,

$$ \langle dJ_i \rangle = (\lambda _i + \lambda _3) ~ dt, $$

$$ cov(J_1,J_2) = \langle J_1 J_2 \rangle - \langle J_1 \rangle \langle J_2 \rangle = var(n_3) = \lambda _3 $$

and

$$ corr[J_1,J_2] = \frac{\lambda _3}{\sqrt{(\lambda _1+\lambda _3)(\lambda _2+\lambda _3)}} $$

where $\lambda _i >0$ is the jump rate of process $n_i$. We note that using this method one will be able construct two positively correlated Poisson-distributed jumps $J_1$ and $J_2$.

13.4

Negative correlated Poisson distributed jump processes

One can use a more advanced approach for construction of negatively correlated Poisson processes (which is based on the idea of the backward simulation of the Poisson processes [3]) to generate two Poisson-distributed jump processes with negative correlations.

Follow steps presented in [3] and synthesis two Poisson jump processes with jump rates $\lambda _1 =0.1$, $\lambda _2=0.2$ and correlation coefficient $corr[J_1,J_2]=-0.5$.

13.5

Correlated Poisson distributed jump and Wiener processes

(a) Suppose that Poisson jump process has jump rate $\lambda$ and show that

$$ \langle W(t) J(t) \rangle = \rho (t) \sqrt{\lambda }~ t $$

where $\rho (t)$ is correlation coefficient of W(t) and J(t).

(b) Use backward simulation of the Poisson process with jump rate $\lambda =0.1$ and synthesis correlated jump and Wiener processes with $\rho (t)=\pm 0.2$ [3].