Analysis of Nonlinear Stochastic Systems with Jumps Generated by Erlang Flow of Events ()
1. Introduction
In this paper we consider the stochastic systems with jumps generated by Erlang flow of events that lead to discontinuities of sample paths. These systems are called the jump-diffusion systems or the stochastic systems with random quantization period. Jumps may have different characteristics that describe intervals between them and their amplitudes [1,2].
Stochastic systems with jumps are used in various applications such as complex technical systems (control of moving objects, jam-resistant radars, radioisotope measuring systems, electrical circuits with impulse sources), financial mathematics (description of stock price movements and valuation of stock options), mathematical biology and medicine (biomass control and drug delivery model) [2,3].
The goal of this paper is to develop the spectral method [4-6] for a problem of the probabilistic analysis for jump-diffusion systems. The spectral method formalism has been used previously to the stochastic systems with jumps generated by Poisson flow of events. Here we consider more complex problem which assumes that we have Erlang flow of jumps. This allows to investigate the stochastic systems with jumps in sample paths at the random time moments. Intervals between these moments can be described by not only the exponential distribution, but Erlang distribution [7].
2. Problem Statement
We assume that the system behavior is described by a jump-diffusion process. This process can be represented as a solution of the stochastic differential equation [1]:
(1)
where 
is a state vector, 
,
;
is an s-dimensional standard Wiener process independent of
.
The component 
describes “extreme events” attended by jumps in sample paths of the process 
(e.g., a technical failure or a stock market crash). We assume that
Here 
is the 
-th order Erlang process (Erlang process 
is a “censored” Poisson process in which 
consecutive points are removed from a Poisson process 
with the transition intensity 
and then one point left unchanged [7]), 
are independent random variables from 
whose distribution is given by the probability density function
, i.e., state vector gets random increment at time moments 
associated with Erlang flow of events [1]:
The process 
may be represented as
where the value 
is used to “censor” 
Poisson flow events in succession and to select each event which is a multiple of N (
is the periodic function:
):
time moments 
conform to the events in Poisson flow:
.
Introduce a stochastic process 
with a finite state set
. These states are replaced sequentially starting from 1 with the transition intensity
:
when the state with number 
passes into the state with number 1, the state vector 
gets a random increment which leads to a jump in sample paths of the process 
(see Figure 1).
The introduction of the process 
allows to represent the probability density function 
of the state vector 
as follows:
where functions 
satisfy generalized Fokker-Planck equations [2,5,8]:
(2)
Figure 1. Sample paths of processes 
and
.
(3)
Here
(4)
The initial state 
is determined by a given probability density function
. The initial state of the process 
is fixed:
. So,
(5)
The last term on the right side of Equation (2) can be written in the operator form:
(6)
for all admissible functions
; 
is a linear operator which is a composition of the multiplication operator and the Fredholm operator with kernel 
.
The analysis problem of the stochastic systems with jumps described by Equation (1) is to find the probability density function 
of the state vector
.
We assume that the unique solutions of Equation (1) and Equations (2)-(5) exist for given functions
, 
, and
.
3. Proposed Method: Overview of Spectral Method Formalism
Reduce the analysis problem to the finding of Fourier coefficients 
for the function
. Let
be an orthonormal basis of 
and let 
be an orthonormal basis of 
, then 
is the orthonormal basis of
, where
. So,
We apply the spectral method formalism [5,8] to Equation (2) and Equation (3) subject to the conditions (5), therefore
(7)
(8)
In these equations 
is the spectral characteristic of the differential operator 
subject to a function value at the initial time moment t0; 
and 
are the spectral characteristics of operators 
and 
defined by (4)
and (6), respectively, i.e., 
,
and
are 
-dimensional matrices [9] (see Appendix) with elements
is the spectral characteristic of the multiplication operator with multiplier
:
are the spectral characteristics of functions
. All these spectral characteristics are defined relative to
. Further, the 
is the column matrix with values of functions 
at the initial time moment
:
is the spectral characteristic for the probability density function 
of the initial state
.
It is defined relative to
, i.e.,
The spectral characteristic 
of the probability density function
, also called a generalized characteristic function [5,6], may be expressed as follows (
is the 
-multidimensional matrix formed by Fourier coefficients
):
(9)
The properties of the spectral characteristics for functions and linear operators in Equations (7)-(9) are described in [5,8].
As a rule [5,6], the spectral characteristic 
is expressed in terms of the spectral characteristics for differential operators and multiplication operators:
where 
and 
are the spectral characteristics of first-order and second-order differential operators 
and
, respectively; 
and 
are the spectral characteristics of multiplication operators with multipliers 
and
, respectively. These spectral characteristics are defined like a 
relative to the orthonormal basis
.
Such definition of 
is more preferred since there are analytical expressions of the spectral characteristics relative to various orthonormal functions for differential operators and multiplication operators (see [4,5]).
Equation (7) and Equation (8) are linear matrix equations for the spectral characteristics 
or linear algebraic equations for Fourier coefficients 
(for functions
). Let us consider the solution of these equations.
It follows from Equation (8) that
(10)
i.e.,
or
where
Thus,
in particular
(11)
We rewrite Equation (7) subject to Equation (11):
or
therefore
We express the spectral characteristic 
subject to Equation (9):
where 
is the 
-dimensional identity matrix. The expression in parentheses is multiplied on the right by the difference
:
i.e.,
(12)
A similar result can be obtained by multiplying on the left by
:
(13)
Equations (12) and (13), obviously, are analogues of geometric series sum. Thus,
(14)
or
(15)
are the problem solutions by the spectral method formalism.
It is easy to see that if 
(order of Erlang process) the problem reduces to the analysis of the stochastic systems with Poisson flow of jumps and
when jump part is missed:
we obtain the known solution of analysis problem for the stochastic systems with continuous trajectories [5,6]:
It is required to apply the inversion formula for finding the solution of the analysis problem [5]:
but a finite number of coefficients 
is usually defined approximately since the problem of finding all Fourier coefficients is not trivial. In this case, the infinite matrices in Equations (7)-(9) are replaced by truncated matrices. Then
where natural numbers 
are the selected orders of the truncation for the spectral characteristics.
Remarks
1) The solution of the analysis problem is possible to find in another way. To do this we express 
in terms of 
from Equation (10), then we express 
in terms of
, that makes it possible to express 
from the Equation (7). The next step is to develop the final formula for 
subject to Equation (9) and the similar transformations carried out to express Equations (14) and (15):
or
where
These expressions are equivalent to Equations (14) and (15), but Equations (14) and (15) are preferable since finding the inverse spectral characteristic 
can be avoided in this case by defining 
as the spectral characteristic of the multiplication operator with multiplier
. In particular, when the transition intensity is constant 
we have
2) A generalization of the discussed problem is to consider the transition intensity which depends on the state vector. The jump size may be described by the conditional probability density function 
that characterizes the distribution of the state vector 
after the jump. This distribution depends on the previous value
; jumps in sample paths of the process 
occur at time moment
.
In this case Equations (2) and (3) are represented as
and the operator 
(see Equation (6)) must be redefined as
Equations (7) and (8) will not change (but 
is the spectral characteristic of the multiplication operator with multiplier
, the spectral characteristic 
is calculated according to a new definition of the operator
). Therefore methods for the problem solution will not change as well as Equations (14) and (15).
4. Conclusions
We examine using of the spectral method formalism to the probabilistic analysis problem for the stochastic systems with jumps generated by Erlang flow of events. Finding of the probability density function for the state vector by the spectral method formalism are developed.
Using of Erlang flow of events allows to consider a more complex behavior for sample paths of the process
. The occurrence of jumps in sample paths can be controlled by appropriate selection of parameters such as the transition intensity 
and an order 
(for Erlang process). This option makes it quite flexible tool for modeling. Thus, for 
time intervals between jumps are described by exponential distribution law, for 
time intervals between jumps are described by Erlang distribution, which is a special case of the gamma distribution. Erlang distribution converges to the normal distribution as 
increases.
The application of the spectral method formalism allows to reduce integro-differential equations to linear algebraic equations for Fourier coefficients of the probability density function. It essentially simplifies the solution.
5. Acknowledgements
This work is supported by RFBR grant 12-08-00892-а.
Appendix
Multidimensional Matrix Operations
1) Let 
and let 
and 
be the infinite 
-dimensional matrices. The expression 
is the infinite 
-dimensional matrix
if
2) Let 
and 
be the infinite 
-dimensional and 
-dimensional matrices, respectively. The product 
is the infinite 
-dimensional matrix 
if
An infinite 
-dimensional matrix 
is said to be the identity matrix if
for each 
-dimensional matrix
. We use the notation 
to denote the product
3) Let 
be an infinite 
-dimensional matrix. An infinite 
-dimensional matrix 
is said to be the two-sided inverse of 
if
.
We use the notation 
to denote the twosided inverse of
.
4) Let 
and 
be the infinite 
-dimensional and 
-dimensional matrices, respectively. The tensor product 
is the infinite 
-dimensional matrix 
if
5) Let 
be an infinite 
-dimensional matrix. An infinite 
-dimensional matrix 
is said to be the transpose of 
if
We use the notation 
to denote the transpose of
.
NOTES