Stochastic Processes with Jumps and Non-vanishing Higher-Order Kramers–Moyal Coefficients

Abstract

In this chapter we study stochastic processes in the presence of jump discontinuity, and discuss the meaning of non-vanishing higher-order Kramers–Moyal coefficients. We describe in details the stochastic properties of Poisson jump processes. We derive the statistical moments of the Poisson process and the Kramers–Moyal coefficients for pure Poisson jump events. Growing evidence shows that continuous stochastic modeling (white noise-driven Langevin equation) of time series of complex systems should account for the presence of discontinuous jump components [1,2,3,4,5,6]. Such time series have some distinct important characteristics, such as heavy tails and occasionally sudden large jumps. Nonparametric (data-based) modeling of time series with jumps provides an attractive way of conducting research and gaining intuition of such processes. The focus in this chapter is introducing stochastic tools for investigation of time series with discontinuous jump components. We will start with the meaning of non-vanishing higher order KM coefficients and its impact on the continuity condition. Similarly to the role of the Wiener process in the Langevin modeling, the Poisson jump process plays an essential role in jump-diffusion modeling. Therefore we present stochastic properties of the Poisson jump process, such as its statistical moments, waiting time distribution, etc.

Problems

11.1

Exponential distribution of time between jumps for Poisson process

Fill in the details in the derivation of Eq. (11.13).

11.2

Mean inter-jump times

Show that the mean inter-jump times $\tau _k = T_{k+1} - T_k$ for Poisson jump process with constant jump rate $\lambda$ is given by,

$$ \langle \tau _k \rangle = \frac{1}{\lambda } \quad . $$

11.3

Correlation coefficient of Poisson process

Suppose $\{J(t), t\ge 0\}$ is a Poisson process with constant jump rate $\lambda$, then show that the correlation coefficient of J(t) and $J(t+s)$ ($s\ge 0$) is given by,

$$ \rho (t,s) = \frac{Cov[J(t), J(t+s)]}{ \sqrt{var [J(t)] \times var[J(t+s)] }} = \left\{ \frac{t}{t+s} \right\} ^{1/2}. $$

11.4

Statistical moments of Poisson jump process

Show that

$$ \langle (\varDelta J ) ^m \rangle = \lambda \varDelta t ~( 1 + \mathcal {O}(\lambda \varDelta t)) $$

for $\lambda \varDelta t \ll 1$, by induction for $m \ge 1$.

11.5

Poisson zero-one jump law

For Poisson jump process with jump rate $\lambda$, show that

$$\begin{aligned} Prob[\varDelta J(t) = 0]= & {} 1 - \lambda \varDelta t + \mathcal {O}(\lambda \varDelta t)^2 \\\nonumber \\ Prob[\varDelta J(t) = 1]= & {} \lambda \varDelta t + \mathcal {O}(\lambda \varDelta t)^2 \\\nonumber \\ Prob[\varDelta J(t) > 1]= & {} \mathcal {O}(\lambda \varDelta t)^2 \\\nonumber \\ Prob[(\varDelta J (t))^m= & {} \varDelta J (t) ] = 1- \frac{1}{2} (\lambda \varDelta t)^2 + \mathcal {O}(\lambda \varDelta t)^3 \end{aligned}$$

for $m \ge 2$. For $\lambda \varDelta t \ll 1$ we find $Prob[\varDelta J(t) > 1] \ll Prob[\varDelta J(t) = 1]$. This asymptotic relationship is specific characteristic of Poisson jump process.

11.6

Integral including Poisson jump process

Show that

$$ I[J](t) = \int _ 0 ^t J(s) dJ(s) {\mathop {=}\limits ^{\text {ms}}} I_{ms} [J](t) \equiv \frac{1}{2} J(t) ( J(t) -1) $$

where the mean square limit integral is,

$$ I_{ms} [J](t) {\mathop {=}\limits ^{\text {ms}}} {ms-lim}_{n \rightarrow \infty } \sum _{i=0} ^n J(t_i) \varDelta J(t_i). $$

11.7

Wiener and Poisson jump processes

Show that

$$\begin{aligned} dJ~dt&{\mathop {=}\limits ^{\text {dt}}}&0, \\\nonumber \\ dJ~dW&{\mathop {=}\limits ^{\text {dt}}}&0, \\\nonumber \\ (dJ)^m&{\mathop {=}\limits ^{\text {dt}}}&dJ {\mathop {=}\limits ^{\text {dt}}} \lambda dt \\\nonumber \\ \end{aligned}$$

for $m \ge 1$, where W(t) and J(t) are independent Wiener and Poisson jump processes, respectively. The symbol ${\mathop {=}\limits ^{\text {dt}}}$ means that the equality with an accuracy $\mathcal {O}(dt)$.

11.8

Poisson jump process with time dependent jump rate $\lambda (t)$

Consider a Poisson jump process with time dependent jump rate $\lambda (t)$ and suppose that the first jump occurs at time $t_0=0$.

(a) Show that the waiting time distribution $f(t|t_0=0)$ to have an another jump at time t, is given by,

$$ f(t|t_0=0) = \lambda (t) \exp \left\{ -\int _0 ^t \lambda (t') dt' \right\} . $$

(b) Show that for known $f(t|t_0=0)$, the time dependent jump rate $\lambda (t)$ can be found as

$$ \lambda (t) = \frac{f(t|t_0=0)}{1-\int _0 ^t f(t'|t_0=0) dt' } \quad . $$