Modeling with Itô Stochastic Differential Equations 2.4-2.6 E. Allen presentation by T. Perälä 27.0.2009 Postgraduate seminar on applied mathematics 2009
Outline Hilbert Space of Stochastic Processes ( 2.4) Computer Generation of Stochastic Processes ( 2.) Examples of Stochastic Processes ( 2.6)
2.4 A Hilbert Space of Stochastic Processes Motivation: Hilbert space is needed for discussing stochastic integrals and stochastic differential equations For example, we can show convergence of a sequence of stochastic processes in Hilbert space. In particular, Cauchy sequences in the Hilbert space will converge in the space. In this part of the talk, we first describe a metric space consisting of elementary stochastic processes. This space will then be completed to a Hilbert space and the set of elementary stochastic processes will be dense in the Hilbert space
Metric Space of Elementary Stochastic Processes Consider continuous stochastic processes defined on the interval and probability space. Let be an elementary stochastic process which is a random step function defined on. That is, it is of form where is a partition of and is the characteristic function. Random step function on [0,], Gaussian white noise 2 0!!2 0 0. It is assumed that the random variable for each, in particular, for each. Now, the metric space is defined as
Metric Space of Elementary Stochastic Processes On, the inner product is defined as and the norm as Compare to inner product in, which was, and the norm. The space is a metric space with the metric. However, is not complete since not all Cauchy sequences converge in. This space can be completed by adding to it additional stochastic processes. The complete space is denoted as and is dense in. That is, given and given there is a such that.
Some Useful Results Suppose, for example, that a stochastic process satisfies, for some positive constants and, the inequalities and Then, and forms a Cauchy sequence in that converges to. Indeed the norm is bounded, In addition, Fubini s theorem (F) states that For, the Cauchy-Schwarz inequality (C-S),, is very useful and written explicitly has the form Thus, for example, applying C-S and F Furthermore, the triangle inequality, is explicitly
Simple Random Step Functions Sometimes it is useful to apply even more elementary set of stochastic processes than those in. Let be the set of simple random step functions. That is, has the form As simple functions are dense in, is also dense in. Next we will have some examples to clarify Hilbert space of stochastic processes emphasis:. But before that, for
Example 2.8. A Converging Seq. of Stochastic Processes Define the stochastic process as and is a Wiener process on. Clearly. Also, Since, the sequence of stochastic processes converges to in.
Example 2.9. Another Converging Seq. of Stochastic Proc s Let be a Wiener process on. Define the stochastic process in the following way: Then Thus, converges to in as.
Example 2.0. Integration of a Function of a Poisson Proc. Consider where is a Poisson process with intensity on the interval. Suppose that experiences unit increases at the times on and let and. Then can be written in the form Then, it is easily seen that Furthermore, it is interesting that Also, as, then and.
Example 2.. A Commonly Used Approx. to Wiener Proc. Consider the interval and let where. Let be a Wiener process. Define the continuous piecewise linear stochastic process on this partition of by for and. Notice that for and is continuous on. Also, Thus,, i.e.,.
Example 2.. A Commonly Used Approx. to Wiener Proc. For a large value of, the graph of a sample path of corresponding sample path of. is indistinguishable from the graph of the The graph of a Wiener process trajectory is often represented by plotting for a large value of. In the figure, two sample paths is used to generate the 000 values are plotted where the recurrence relation η i N(0, )
Some Properties of Stochastic Processes in Hilbert Space First, if, then which implies that for almost every. In addition, if converges to a stochastic process, then. Hence, for almost every. That is, convergence in Hilbert space implies convergence in probability on. Specifically, if for almost every. then for any Now, suppose that and satisfies for any for a constant. Then a bound on can be found and is continuous on with respect to the norm. First, if, then and Thus, is bounded by. Also, it is easy to see that is continuous on with respect to the norm. Given, then whenever.
Some Properties of Stochastic Processes in Hilbert Space Finally, it is useful to present some terminology used in the literature regarding the Hilbert space applied later in this seminar. that is Let be a Wiener process defined on a probability space. Let be a family of sub- -algebras of satisfying if, is -measurable, and is independent of. is the -algebra of events generated by the values of the Wiener process until time. A stochastic process is said to be adapted to if is independent of a Wiener increment Furthermore, the Hilbert space is the set of nonanticipative stochastic processes such that satisfies.
2. Computer Generation of Stochastic Processes Let s consider how stochastic processes can be computationally simulated using pseudo-random numbers. First, consider simulation of a nonhomogeneous Markov chain (MC) on where and is a discrete random variable for each time. Specifically,. The time dependent transition probability matrix is Consider generation of one trajectory or sample path. At time,. To find, are first computed for. Next, a pseudo-random number uniformly distributed on is generated.. Then, is calculated so that Finally, is set equal to. To find, are computed for. Then, uniformly distributed on is generated and is calculated so that Then is set equal to. These steps are repeated times to give on realization of the discrete stochastic process.
Example of Computer Generated Realization of MC Two homogenous Markov chains, 0 20 40 60 80 00!2! 0 2 0 20 40 60 80 00!2! 0 2 P = P = 2 2 0 0 0 0 2 2 0 0 0 0 2 2 0 0 0 0 2 2 2 0 0 0 2 M = { 2,, 0,, 2}
Continuous Case Now consider generating a trajectory for a continuous Markov process. Generally, trajectories of continuous processes are determined at a discrete set of times. Specifically, a trajectory is calculated at the times where. Then, may be approximated between these points using, for example, piecewise linear approximation. Next some examples that illustrate this behavior are presented.
Example 2.2. Simulation of a Poisson Process Consider a Poisson process with intensity. Recall that equals the number of observations in time where the probability of one observation in time is equal to. From Example.4, we remember that Consider now simulating this continuous stochastic process at the discrete times where. Let and the random numbers are chosen so that Then, are Poisson distributed with intensity at the discrete times. Notice that to find given uniformly distributed on, one uses the relation
Simulation of a Poisson Process with Matlab Poisson process, h = λ = λ = 0 20 0 0 0 2 4 6 8 0 40 20 00 80 60 40 20 0 0 2 4 6 8 0
Example 2.3. Simulation of a Wiener Proc. Sample Path Consider the Wiener process on. Consider simulating this continuous stochastic process at the discrete times where. Let where and are normally distributed numbers with mean and variance. As in the previous example, each sample path of the continuous stochastic process is computed at the discrete times. Thus, To estimate, at a time for any, a continuous linear interpolant can be used as was shown in Example 2.. In particular,
Computer Generation of Wiener Process Sample Path Wiener process on [0, 0] N = 0 0 0!!0!!20 0 0 N = 00 0 20 30 40 0 N = 000 0 0 0!!!0! 0 0 20 30 40 0!0 0 0 20 30 40 0
Example 2.4. Simulation of Wiener by a Discrete Process Let. Define the discrete stochastic process on the partition in the following way. Let and let the transition probabilities be assuming that. Then, as explained in 2.2, the probability distribution for satisfies the forward Kolmogorov equations where. For small, approximately equals where satisfies the partial differential equation Solving this partial differential equation gives In particular, for, is approximately normally distributed with mean and variance. Furthermore, is approximately normally distributed with mean and variance and is independent of. Indeed, approximates a Wiener process on the partition.
2.6 Examples of Stochastic Processes Stochastic processes are common in physics, meteorology, and finance. Stochastic process occur whenever dynamical systems experience random (uncertain) influences A classical example is radioactive decay where atoms of unstable isotopes transform to other isotopes. Suppose that there are initially atoms of a radioactive isotope. Let be the number of atoms at time. Let be the decay constant of the isotope. This means that the probability that an atom transforms in small time interval is equal to. Consider finding the expected number of atoms at time, i.e.,. Let be the probability that there are atoms at time. Then, considering the possible transitions in time interval, one obtains Thus, letting,
Radioactive Decay Continued The expected number of atoms can now be computed as This leads to Hence, And as the solution to this differential equation we obtain for the expected number of atoms at time
Population Biology Population biology is rich in stochastic processes. The birth-death process, in itself, is a random process. Also, variability in the environment introduces additional random influences which are time and spatial varying. As a result, growth of a population exhibits random behavior.
Weather Climatic quantities can be considered stochastic processes Too many different influences to make an accurate deterministic model Annual precipitation in Lubbock, Texas exhibits a Wiener-like behavior
Stock Prices Stock trading involves many uncertainties For example stock prices are thus modeled as a stochastic process