Fractional Poisson Fields and Martingales

Abstract

We present new properties for the Fractional Poisson process (FPP) and the Fractional Poisson field on the plane. A martingale characterization for FPPs is given. We extend this result to Fractional Poisson fields, obtaining some other characterizations. The fractional differential equations are studied. We consider a more general Mixed-Fractional Poisson process and show that this process is the stochastic solution of a system of fractional differential-difference equations. Finally, we give some simulations of the Fractional Poisson field on the plane.

Covariance Structure of Parameter-Changed Poisson Random Fields

In this Appendix, we prove a general result that can be used to compute the covariance structure of the parameter-changed Poisson random field:

$$\begin{aligned} Z\left( t_{1},t_{2}\right) =N(Y_{1}(t_{1}),Y_{2}(t_{2})),\ (t_{1},t_{2})\in {\mathbb {R}}_{+}^{2}, \end{aligned}$$

where $Y_{1}=\left\{ Y_{1}(t_{1}),t_{1}\ge 0\right\}$ and $Y_{2}=\left\{ Y_{2}(t_{2}),t_{2}\ge 0\right\}$ are independent non-negative non-decreasing stochastic processes, in general non-Markovian with non-stationary and non-independent increments, and $N= \{N(t_{1},t_{2}),(t_{1},t_{2})\in {\mathbb {R}}_{+}^{2}\}$ is a PRF with intensity $\lambda >0$. We also assume that $Y_{1}$ and $Y_{2}$ are independent of N.

For example, $Y_{1}$ and $Y_{2}$ might be inverse subordinators.

Theorem 11

Suppose that N is a PRF, $Y_{1}$ and $Y_{2}$ are two non-decreasing non-negative independent stochastic processes which are also independent of N. Then

• (1) if ${\mathrm {E}}Y_{1}(t_{1})=U_{1}(t_{1})$ and ${\mathrm {E}} Y_{2}(t_{2})=U_{2}(t_{2})$ exist, then ${\mathrm {E}}Z(t_{1},t_{2})$ exists and

  $$\begin{aligned} {\mathrm {E}}Z(t_{1},t_{2})={\mathrm {E}}N(1,1){\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}} Y_{2}(t_{2}); \end{aligned}$$

• (2) if $Y_{1}$ and $Y_{2}$ have second moments, so does Z and

  $$\begin{aligned} \mathrm {Var}Z(t_{1},t_{2})= & {} \left[ {\mathrm {E}}N(1,1)\right] ^{2}\left\{ {\mathrm {E}}Y_{1}^{2}(t_{1}){\mathrm {E}}Y_{2}^{2}(t_{2})-\left( {\mathrm {E}} Y_{1}(t_{1})\right) ^{2}\left( {\mathrm {E}}Y_{2}(t_{2})\right) ^{2}\right\} \\&+\mathrm {Var}N(1,1){\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}}Y_{2}(t_{2}) \end{aligned}$$

  and its covariance function

  $$\begin{aligned} {\mathrm {Cov}}(Z(t_{1},t_{2}),Z(s_{1},s_{2}))={\mathrm {Cov}}\left( N(Y_{1}(t_{1}),Y_{2}(t_{2})),N(Y_{1}(s_{1}),Y_{2}(s_{2}))\right) \end{aligned}$$

  for $s_{1}<t_{1},s_{2}<t_{2}$ is given by:

  $$\begin{aligned}&({\mathrm {E}}N(1,1))^{2} \Big \{ {\mathrm {Cov}}\left( Y_{1}(t_{1}),Y_{1}(s_{1})\right) {\mathrm {Cov}}\left( Y_{2}(t_{2}),Y_{2}(s_{2})\right) \nonumber \\&\quad +\,{\mathrm {E}}Y_{2}(t_{2}){\mathrm {E}}Y_{2}(s_{2}) {\mathrm {Cov}}\left( Y_{1}(t_{1}),Y_{1}(s_{1})\right) +{\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}}Y_{1}(s_{1}) {\mathrm {Cov}}\left( Y_{2}(t_{2}),Y_{2}(s_{2})\right) \Big \}\nonumber \\&\quad +\, \mathrm {Var}N(1,1){\mathrm {E}}Y_{1}(s_{1}){\mathrm {E}}Y_{2}(s_{2}) \end{aligned}$$

  (A.1)

  and for any $(s_{1},s_{2}),$ and $(t_{1},t_{2})$ from ${\mathbb {R}}_{+}^{2}$

  $$\begin{aligned}&({\mathrm {E}}N(1,1))^{2} \Big \{ {\mathrm {Cov}}\left( Y_{1}(t_{1}),Y_{1}(s_{1})\right) {\mathrm {Cov}}\left( Y_{2}(t_{2}),Y_{2}(s_{2})\right) \nonumber \\&\quad +\,{\mathrm {E}}Y_{2}(t_{2}){\mathrm {E}}Y_{2}(s_{2}) {\mathrm {Cov}}\left( Y_{1}(t_{1}),Y_{1}(s_{1})\right) +{\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}}Y_{1}(s_{1}) {\mathrm {Cov}}\left( Y_{2}(t_{2}),Y_{2}(s_{2})\right) \Big \} \nonumber \\&\quad +\,\mathrm {Var}N(1,1){\mathrm {E}}Y_{1}(\min (s_{1},t_{1})){\mathrm {E}}Y_{2}(\min (s_{2},t_{2})) \end{aligned}$$

  (A.2)

Remark 7

These formulae are valid for any Lévy random field $N=\{N(t_{1},t_{2})$,$(t_{1},t_{2})\in {\mathbb {R}}_{+}^{2}\}$, with finite expectation ${\mathrm {E}} N(1,1)$ and finite variance $\mathrm {Var}N(1,1),$ for PRF ${\mathrm {E}} N(1,1)=\lambda ;\ \mathrm {Var}N(1,1)=\lambda$ and to apply these formulae one needs to know

$$\begin{aligned} U_{1}(t_{1})={\mathrm {E}}Y_{1}(t),\ U_{2}(t_{2})={\mathrm {E}}Y_{2}(t),\ U_{1}^{(2)}(t_{1})={\mathrm {E}}Y_{1}^{2}(t),~U_{2}^{(2)}(t_{1})={\mathrm {E}} Y_{2}^{2}(t),\ \end{aligned}$$

and ${\mathrm {Cov}}\left( Y_{1}(t_{1}),Y_{1}(s_{1})\right) ,\; {\mathrm {Cov}} \left( Y_{2}(t_{2}),Y_{2}(s_{2})\right)$ which are available for many non-negative processes $Y_{1}(t)$ and $Y_{2}(t)$ induction inverse subordinators.

Remark 8

One can compute the following expression for the one-dimensional distribution of the parameter-changed PRF:

$$\begin{aligned} {\mathrm {P}}\left( N(Y_{1}(t_{1}), Y_{2}(t_{2}))=k\right)= & {} p_{k}(t_{1},t_{2}) \\= & {} \int _{0}^{\infty }\int _{0}^{\infty }\frac{e^{-\lambda x_{1}x_{2}}(\lambda x_{1}x_{2})^{k}}{k!}f_{1}(t_{1},x_{1})f_{2}(t_{2},x_{2})dx_{1}dx_{2},\\&\quad k=0,1,2,\ldots \end{aligned}$$

where

$$\begin{aligned} f_{i}(t_{i},x_{i}) = \frac{d}{d{x_i}}{\mathrm {P}}\left\{ Y_{i}(t_{i})\le x_{i}\right\} =\frac{d}{dx_{i}}G_{t_i}^{(i)}(x_{i}), \quad i=1,2. \end{aligned}$$

and its Laplace transform:

$$\begin{aligned} {\mathscr {L}} \left\{ p_{k}(t_{1},t_{2}); s_{1},s_{2} \right\}= & {} \int _{0}^{\infty }\int _{0}^{\infty }\frac{e^{-\lambda x_{1}x_{2}}(\lambda x_{1}x_{2})^{k}}{k!} {\mathscr {L}}\\&\left\{ f_{1}(t_{1},x_{1});s_{1}\right\} {\mathscr {L}} \left\{ f_{2}(t_{2},x_{2});s_{2}\right\} dx_{1}dx_{2}, \end{aligned}$$

where

$$\begin{aligned} {\mathscr {L}} \left\{ f_{i}(t_{i},x_{i});s_{i}\right\} =\int _{0}^{\infty }e^{-s_{i}t_{i}}f_{i}(t_{i},x_{i})dt_{i},\quad i=1,2. \end{aligned}$$

Proof of Theorem 11

We denote

$$\begin{aligned}&G_{t_{1}}^{(1)}(u_{1})={\mathrm {P}}\left\{ Y_{1}(t_{1})\le u_{1}\right\} ,&G_{t_{2}}^{(2)}(u_{2})={\mathrm {P}}\left\{ Y_{2}(t_{2})\le u_{2}\right\} . \end{aligned}$$

We know that for a PRF

$$\begin{aligned} {\mathrm {E}}{\varDelta } _{s_{1},s_{2}}N(t_{1},t_{2})= & {} {\mathrm {E}}N(1,1) \left( t_{1}-s_{1}\right) \left( t_{2}-s_{2}\right) =\mathrm {Var}{\varDelta } _{s_{1},s_{2}}N(t_{1},t_{2});\\ {\mathrm {E}}\left( {\varDelta } _{s_{1},s_{2}}N(t_{1},t_{2})\right) ^{2}= & {} {\mathrm {E}}N(1,1) \left( t_{1}-s_{1}\right) \left( t_{2}-s_{2}\right) +\left[ {\mathrm {E}}N(1,1)\left( t_{1}-s_{1}\right) \left( t_{2}-s_{2}\right) \right] ^{2}. \end{aligned}$$

To prove (1) we use simple conditioning arguments:

$$\begin{aligned} {\mathrm {E}}Z(t_{1},t_{2})=\int _{0}^{\infty }\int _{0}^{\infty }u\ v\ {\mathrm {E}} N(1,1)G_{t_{1}}^{(1)}(du)G_{t_{2}}^{(2)}(dv)={\mathrm {E}}N(1,1){\mathrm {E}} Y_{1}(t_{1}){\mathrm {E}}Y_{2}(t_{2}). \end{aligned}$$

Let us prove (2).

For the variance, we have

$$\begin{aligned} \mathrm {Var}Z(t_{1},t_{2})= & {} {\mathrm {E}}\left( N(Y_{1}(t_{1}),Y_{2}(t_{2})\right) ^{2}-\left( {\mathrm {E}} N(Y_{1}(t_{1}),Y_{2}(t_{2})\right) ^{2} \\= & {} \int _{0}^{\infty }\int _{0}^{\infty }\left( ({\mathrm {E}}N(u_{1},u_{2}))^{2}+ \mathrm {Var}N(u_{1},u_{2})\right) G_{t_{1}}^{(1)}(du_{1})G_{t_{2}}^{(2)}(du_{2}) \\&-\left( {\mathrm {E}}N(1,1){\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}} Y_{2}(t_{2})\right) ^{2} \\= & {} \int _{0}^{\infty }\int _{0}^{\infty }\left[ \left( {\mathrm {E}}N(1,1)\right) ^{2}u_{1}^{2}u_{2}^{2}+\mathrm {Var}N(1,1)u_{1}u_{2}\right] G_{t_{1}}^{(1)}(du_{1})G_{t_{2}}^{(2)}(du_{2}) \\&-\left( {\mathrm {E}}N(1,1){\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}} Y_{2}(t_{2})\right) ^{2} \\= & {} \left( {\mathrm {E}}N(1,1)\right) ^{2}{\mathrm {E}}Y_{1}^{2}(t_{1}){\mathrm {E}} Y_{2}^{2}(t_{2})+\mathrm {Var}N(1,1){\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}} Y_{2}(t_{2}) \\&-\left( {\mathrm {E}}N(1,1){\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}} Y_{2}(t_{2})\right) ^{2} \\= & {} \left( {\mathrm {E}}N(1,1)\right) ^{2}\left\{ {\mathrm {E}}Y_{1}^{2}(t_{1}) {\mathrm {E}}Y_{2}^{2}(t_{2})-({\mathrm {E}}Y_{1}(t_{1}))^{2}({\mathrm {E}} Y_{2}(t_{2}))^{2}\right\} \\&+\mathrm {Var}N(1,1){\mathrm {E}}Y_{1}(t_{1}){\mathrm {E}}Y_{2}(t_{2}). \end{aligned}$$

To compute the covariance structure, first we consider the case when $s_{1}<t_{1},\ s_{2}<t_{2}$. Then

$$\begin{aligned}&{\mathrm {E}}N(s_{1},s_{2})N(t_{1},t_{2}) \\&={\mathrm {E}} \Big (N(s_{1},s_{2}) \Big \{N(t_{1},t_{2})-N(t_{1},s_{2})-N(s_{1},t_{2})+N(s_{1},s_{2})\\&\quad +N(t_{1},s_{2})+N(s_{1},t_{2})-N(s_{1},s_{2}) \Big \}\Big ) \\&={\mathrm {E}}{\varDelta } _{s_{1},s_{2}}N(t_{1},t_{2}){\mathrm {E}}N(s_{1},s_{2})+ {\mathrm {E}}N(t_{1},s_{2})N(s_{1},s_{2})\\&\quad +{\mathrm {E}} N(s_{1},t_{2})N(s_{1},s_{2})-{\mathrm {E}}N^{2}(s_{1},s_{2}). \end{aligned}$$

Using the facts that

$$\begin{aligned} \begin{aligned} {\mathrm {E}}{\varDelta } _{s_{1},s_{2}}N(t_{1},t_{2}){\mathrm {E}} N(s_{1},s_{2})&=(t_{1}-s_{1})(t_{2}-s_{2})\left[ {\mathrm {E}}N(1,1)\right] ^{2}s_{1}s_{2}, \\ {\mathrm {E}}N(t_{1},s_{2})N(s_{1},s_{2})&={\mathrm {E}} \{N(t_{1},s_{2})-N(s_{1},s_{2})+N(s_{1},s_{2})\}N(s_{1},s_{2}) \\&={\mathrm {E}}{\varDelta } _{s_{1},0}N(t_{1},s_{2}){\mathrm {E}}N(s_{1},s_{2})+{\mathrm {E}} N^{2}(s_{1},s_{2})\\&=\left[ {\mathrm {E}}N(1,1)\right] ^{2}(t_{1}-s_{1})s_{1}s_{2}^{2}+{\mathrm {E}}N^{2}(s_{1},s_{2}), \end{aligned} \end{aligned}$$

it is easy to obtain

$$\begin{aligned}&{\mathrm {E}}N(s_{1},s_{2})N(t_{1},t_{2})=\left[ {\mathrm {E}}N(1,1)\right] ^{2}t_{1}t_{2}s_{1}s_{2}+s_{1}s_{2}\mathrm {Var}N(1,1). \end{aligned}$$

Since the processes $N,Y_{1},Y_{2}$ are independent, a conditioning argument yields (A.1) and (A.2). In a similar way, one can consider the case $s_{1}>t_{1}, s_{2}<t_{2}$. $\square$

Proof of Proposition 2

It follows from Theorem 11 and Proposition 1. $\square$