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Chapter 4

Stochastic Differential Equations

4.1 Motivation and problem formulation

The time varying behavior of many physical phenomena can be described by deterministic ordinary differential equations. If we define the state of the physical system as x(t) we have

(1) [pic]

However when there are uncertainties, physical system behavior often can only be described in terms of probability and has to be described by means of a stochastic model. Therefore in this Chapter we discuss a stochastic differential equation as a model for a stochastic process Xt. We consider models of the following type:

(2) [pic]

where we have introduced a stochastic process Nt to model uncertainties in the underlying deterministic differential equation. The initial condition X0 is also assumed to be a random variable.

Example 4.1

Consider again the model for Biochemical-Oxygen Demand (BOD) in stream bodies as described in Chapter 2:

(3) [pic]

where the deterministic process B(t) is the BOD (mg/l), K1 is the reaction rate coefficient (l/day) and s1 is the source or sink along the stream.

Let us suppose that there are uncertainties associated with the source input s1. This can be modeled by adding a white noise process Nt with intensity ( to s1. The resulting stochastic model for the stochastic process Bt now becomes:

(4) [pic]

Another source of uncertainty can be the parameter K1. Adding a white noise process to this parameter results in:

(5) [pic]

Note that both stochastic models are of the general type (2).

An essential property of the stochastic model (2) is that it should be Markov. This property implies that information on the probability density of the state Xt at time t is sufficient for computing model predictions for times > t. If the model is not Markovian, information on the system state for times < t would also be required. This would make the model very impractical. As we will show in this Chapter the stochastic differential equation (2) is Markovian if Nt is a continuous Gaussian white noise process with statistics:

(6)[pic][pic]

This is one of the very few processes that guarantee the model (2) to be Markov. Using this process however introduces, as we will see in this Chapter, some mathematical difficulties in defining and solving the stochastic differential equation (2).

4.2 Wiener process

A very important stochastic process is the Wiener process or Brownian motion.

Definition: A standard Wiener process Wt = {Wt, t ( 0}, on [0, T] is a process with W0 = 0 and with stationary independent increments such that for any 0 ( s < t ( T, the increment Wt–Ws is a Gaussian random variable with mean zero and variance equal to t-s. That is, E{Wt-Ws} = 0, and Var{Wt-Ws} = t-s.

Note that formally dWt is a Gaussian random variable with mean zero and variance dt.

Exercise 4.1

It is easy to generate realizations of the Wiener process using a Gaussian random generator.

[pic]

cd c:\watbook\progsde

ex1

[pic]

Exercise 4.2

The mean and variance of the process Wt can be determined using the next program.

ex2

[pic]

The result is shown in the figure above. Here we have used 100 samples to determine the mean and variance. Recall that the standard Wiener process should have mean zero and variance t–s and this is demonstrated by the figure above. For extremely large samples, the variance is expected to be a straight line as in the definition of a standard Wiener process.

Exercise 4.3

Consider the function x(w) = ebw and the Wiener process Wt. We can now define another stochastic process according to Xt=e bWt. Using the next program, realizations of Wt and Xt are generated for b=0.3.

ex3

[pic]

4.3 Mean square calculus

Before we are able to define a stochastic differential equation properly we have to discuss stochastic limits. Consider the row of stochastic variables X1, X2, … Xn, …. An important question is now under what conditions this row Xn converge to a stochastic variable X.

Definition: A row of stochastic variables Xn converge in mean square sense to a limit X if:

[pic][pic]

This is denoted by:

[pic]

(limit in mean square sense)

This definition states that the second moment of the stochastic variable defined as the difference between Xn and X will approach zero for large n. This implies that the probability that Xn will be significantly different from X will become very small for large n.

The l.i.m. operator satisfies the following rules (see problems):

(7) [pic]

(8) [pic]

(9) [pic]

Using the l.i.m. operator we are now able to define the derivative X't of a stochastic process Xt :

[pic]

and, by generalizing the classical Rieman Stieltjes integral, the integral Y over a stochastic process Xt:

[pic]

where we divide the interval [a,b] into many small intervals [ti,ti+1] and choose ti' somewhere in this interval.

Example 4.2

Consider the Wiener process Wt. We now have:

[pic]

so that:

[pic]

From this Example we see that the formal derivative of the Wiener process is the Gaussian continuos white noise process:

[pic]

or

[pic]

It is now convenient to rewrite the stochastic differential equation (2) in term of the Wiener process:

(10) [pic]

or:

(11) [pic]

The only problem left is the definition of the second integral in (11).

Exercise 4.4

Consider the stochastic differential equation:

[pic]

where a and b are constants. This equation is well defined. The exact solution is:

[pic]

This process is called the generalized Wiener process. Using the next program stochastic tracks of Xt can be generated for a=0.2 and b=0.3 and for zero initial condition.

ex4

[pic]

4.4 Ito stochastic integrals

Solving the stochastic differential equation (10) requires the evaluation of a stochastic integral of the type:

[pic]

where Gs is a general stochastic process and Ws is a Wiener process.

Example 4.3

Consider the deterministic integral:

[pic]

The classical Rieman Stieltjes definition for this integral is:

[pic]

where we divide the interval [t0,t] into many small intervals [ti,ti+1] and choose ti' in this interval. Consider now the stochastic integral:

[pic]

where Wt is a Wiener process. An obvious definition for this stochastic integral would be:

[pic]

where we again divide the interval [t0,t] into many small intervals [ti,ti+1] and choose ti' in this interval. This expression can be rewritten according to:

[pic]

This can be verified by rearranging the right hand side of this equation. Using the following properties of the Wiener process (see problems):

[pic]

[pic]

[pic]

it is possible to derive the exact solution in this case:

[pic]

From the result of Example 4.3 we see that unlike in the deterministic case this limit is not uniquely defined. The choice of ti' is important for the final result of the integral. Therefore we need another definition for a stochastic integral. The Japanese mathematician Ito proposed the first and the most important definition:

Definition: The Ito integral is defined as:

[pic]

Using the Ito definition the evaluation point ti' is chosen at the beginning of the interval.

For Ito integrals we have the following results (see problems):

(12) [pic]

(13) [pic]

Example 4.4

Interpreting the integral from Example 4.3 in Ito sense results in:

[pic]

The result is not what we intuitively may expect. Compared to the corresponding deterministic result an additional term ( t t0)/2 is obtained.

The Ito definition is not the only way to treat stochastic integrals. Stratonovitch has introduced another definition.

Definition: The Stratonovitch integral is defined as:

[pic]

Using the Stratonovitch definition the evaluation point ti' is chosen exactly in the middle of the interval.

For Stratonovitch integrals, the relations (12) and (13) do not hold. This is one of the reasons why the Ito definition is used more often then the Stratonovitch definition.

Example 4.5

Interpreting the integral from Example 4.3 in Stratonovitch sense results in:

[pic]

From this Example we see that the Stratonovitch result is in agreement with the corresponding deterministic results.

4.5 Stochastic differential equations

Having defined the Ito integral we are now able to define the Ito stochastic differential equation (SDE):

[pic]

or:

[pic]

where the stochastic integral has to be interpreted in Ito sense. By introducing a vector Wiener process Wt consisting of n independent Wiener processes Wti, i=1, … ,n, it is easy to generalize the SDE just defined to the vector case.

Using the definition of the Ito integral it is possible to derive a numerical scheme for solving an Ito stochastic differential equation. For a small time step [pic] we have:

(14) [pic]

This is called the Euler scheme. This scheme is consistent with the Ito definition of an integral and can only be used for an Ito differential equation.

From the Euler approximation we can also see that Xt is a Markov process. Additional information about Xs for s ................
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