Estimating doubly stochastic Poisson process with affine intensities by Kalman filter

Abstract

This paper proposes a Kalman filter formulation for parameter estimation of doubly stochastic Poisson processes (DSPP) with stochastic affine intensities. To achieve this aim, an analytical expression for the probability distribution functions of the corresponding DSPP for any intensity from the class of affine diffusions is obtained. More detailed results are provided for one- and two-factor Feller and Ornstein–Uhlenbeck diffusions. A Monte Carlo study indicates that the proposed method is a reliable procedure for moderate sample sizes. An empirical analysis of one- and two-factor Feller and Ornstein–Uhlenbeck models is carried out using high frequency transaction data.

Appendices

Appendix 1: Proof of Result 2

To convey the spirit of proofs used for this kind of results, we give here a short demonstration. Substituting the form of processes $\mathbf X(t)$, ${t \ge 0}$, and $\lambda (t)$, ${t \ge 0}$, as in (3) and (2), we obtain:

$$\begin{aligned} \begin{array}{l} \displaystyle 0=-(\rho _0+\varvec{\rho } \cdot \mathbf {X}(t))F(\mathbf x,\,t) +\frac{\partial F(\mathbf x,\,t)}{\partial t}\\ \displaystyle \qquad +\frac{\partial F(\mathbf x\,t)}{\partial \mathbf x}(\mathbf {K_0}+ \mathbf {K_1}\cdot \mathbf {X}(t)) + \frac{1}{2}\sum _{i,j}\frac{\partial ^2F(\mathbf x,\,t)}{\partial x_i\partial x_j}(a_{ij}+b_{ij}\cdot \mathbf {X}(t)). \end{array} \end{aligned}$$

(54)

Inserting $F(\mathbf x,t)=e^{\alpha (t)+\beta (t)\cdot \mathbf x}$ into the PDE above and grouping the terms in $\mathbf x$:

$$\begin{aligned} u(\cdot )\mathbf x+v(\cdot )=0 \end{aligned}$$

Where

$$\begin{aligned} u(\cdot )&= -\beta ^\shortmid (t)+\rho _1-\mathbf {K_1}^\top \beta (t)-\frac{1}{2}\beta (t)^\top \mathbf {b} \beta (t) \end{aligned}$$

(55)

$$\begin{aligned} v(\cdot )&=\alpha ^\shortmid (t)+\rho _0-\mathbf {K_0}\beta (t)-\frac{1}{2}\beta (t)^\top \mathbf {a} \beta (t) \end{aligned}$$

(56)

Use the separation of variable technique to obtain that $\alpha$ and $\beta$ satisfy a Ricatti equation with boundary condition $\alpha (0)=0$ and $\beta (0)=0$. $\square$

Appendix 2: Proof of Corollary 1 and 2

To obtain in a closed-form the PDF of a DSPP with a given affine intensity, we need from Theorem 1 to solve:

$$\begin{aligned} \left\{ \begin{array}{ccccccc} 0&{}= &{}-\beta ^\shortmid (t)&{} + &{}\rho _1-\mathbf {K_1}^\top \beta (t)&{}-&{}\frac{1}{2}\beta (t)^\top \mathbf {b} \beta (t)\\ 0&{}= &{}\alpha ^\shortmid (t)&{} + &{} \rho _0-\mathbf {K_0}\beta (t)&{}-&{}\frac{1}{2}\beta (t)^\top \mathbf {a} \beta (t) \end{array} \right. \end{aligned}$$

(57)

The exact solution for each case can be obtained after replacing the appropriate parametrization:

• 1. Feller intensity $\mathbf {K_0} = \kappa \theta$, $\mathbf {K_1} = -\kappa$, $\mathbf {a} = 0$ and $\mathbf {b} = \sigma ^2$

• 2. O–U intensity $\mathbf {K_0} = \kappa \theta$, $\mathbf {K_1} = -\kappa$, $\mathbf {a} = \sigma ^2$ and $\mathbf {b} = 0$

on (57) and solving a Ricatti equation for $\alpha$ and $\beta$ with boundary condition $\alpha (0)=0$ e $\beta (0)=0$.

As the multidimensional case is merely a sum of decoupled one-dimensional solutions, its derivation is identical to described above.$\square$

Appendix 3: Conditions to existence of Laplace transform for $\int \limits _t^T\lambda (t){\mathrm{d}}t$

Without going into detail of the Albanese–Lawi result, we describe below the class of (scalar) processes introduced in Albanese and Lawi (2004). It consists of diffusions $X(t)$, $t \ge 0$, solving the following SDE:

$$\begin{aligned} \mathrm{d} X(t) =2\frac{h'(X(t))}{h(X(t))}\frac{A(X(t))^2}{R(X(t))} \mathrm{d}t +\frac{\sqrt{2}A(X(t))}{\sqrt{R(X(t))}}\mathrm{d}W(t). \end{aligned}$$

(58)

Here $A(x)$, $R(x)$ and $h(x)$ are second-order polynomials and in addition:

• 1. $A(x)$ belongs to the set $\{1,x,x(1-x),x^2+1\}$ and $R(X(t))\ge 0$;

• 2. the function $h(x)$ is a linear combination of hypergeometric functions of a confluent type $_1F_1$ if $A(x) \in \{1,x\}$ and of a Gaussian type $_2F_1$ if $A(x)\in \{x(1-x),x^2+1\}$.

A hypergeometric function in its general form may be written as

$$\begin{aligned} {}_pF_q(\alpha _1,\ldots ,\alpha _p;\gamma _1,\ldots ,\gamma _q;z). \end{aligned}$$

For $p \le q+1, \gamma _j \in \mathbb {C}\setminus \mathbb {Z}_+$ it can be represented by using Taylor’s expansion around $z=0$:

$$\begin{aligned} {}_pF_q(\alpha _1,\ldots ,\alpha _p;\gamma _1,\ldots ,\gamma _q;z) =\sum _0^\infty \frac{(\alpha _1)_n \cdots (\alpha _p)_n}{(\gamma _1)_n \cdots (\gamma _q)_n} \frac{z^n}{n!}\,. \end{aligned}$$

As an example of application of the Albanese–Lawi, let the intensity process $\lambda (t)$ follows a one-dimensional Feller diffusion. To this end, assume that the polynomials $A(x)$, $h(x)$ and $R(x)$ are defined as:

$$\begin{aligned} A(x)= x, \quad R(x) = \frac{2x}{\sigma ^2}, \quad h(x)=x^{a/\sigma ^2}e^{-\frac{b}{\sigma ^2}x} \end{aligned}$$

(59)

Substituting the polynomial into (58) and performing the change of parameters $a=\kappa \theta /\sigma ^2$ and $\kappa /\sigma ^2$ we conclude our example. $\square$