As more applied scientific discipline research workers areA trying to utilize Stochastic Differential Equations ( SDEs ) in their mold, particularly when affecting Fractional Brownian Motion ( fBM ) , one common issue appears: an exact solution can non ever be found. Therefore, in this paper, we test assorted Numerical methods in work outing SDEs with standard BM that have non-linear coefficients. In add-on we extend our consequences to SDEs with fBM

Cardinal Wordss: Brownian Motion ( BM ) , fractional Brownian Motion ( fBM ) , SDEs, Numerical Approximations

Introduction

Stochastic Differential Equations ( SDEs ) affecting both Brownian Motion BM ) or fractional Brownian Motion ( fBM ) have been going more prevailing in applied mathematics and mold of assorted systems. Some illustrations of these countries, and non limited to them, are finance ( i.e Black-Scholes expression ) , webs ( i.e. informations transportation in wireless communications ) , biological science ( i.e. arrhythmia, encephalon signaling after a shot ) etc. In many of those instances, old ages of research and aggregation of empirical informations is performed in order to construct an appropriate theoretical account. More frequently than non though, the SDE that best fits the information is an SDE that does non hold a simple analytical solution. Therefore the demand appears for a consistent numerical method.

In chapter 2 we cover some brief preliminaries about BM, fBM and SDEs that are indispensable for the numerical estimates we intent to utilize. In chapter 3 we will province the three different methods tested for numerical solutions of SDEs affecting BM, present the consequences of the three methods and place the best. Once we derive the best method, we extend it to SDEs affecting fBM and compare it to an already proposed strategy ( I. Lewis ) . In chapter 4, we province our decisions.

Preliminaries

What is a Brownian Motion ( BM ) ? The award for the find of the BM belongs to the Scots phytologist Robert Brown that originally described it in 1928 [ 1 ] as he observed it in the motion of pollen atoms drifting in liquid. The first one to really build the procedure was the Missourian mathematician Norbert Wiener in 1923. Ergo the procedure itself is besides referred to as Wiener Process.

Definition 2.1 The procedure is a Brownian Motion ( BM ) if it is a procedure of independent Gaussian increases with zero first minute, i.e. a standard Brownian Gesture over is a random variable that depends continuously on and satisfies [ 2 ] :

with chance 1.

For, the random variable given by the increase is.

For, the increases and are independent.

Some basic belongingss that are easy attained by the definition above are:

, from ( 2.2 )

, from ( 2.2 ) and ( 2.5 )

Besides, for we can compose:

, that is for any we have that:

Furthermore, allow and specify. Then and

As we are be aftering to discourse Stochastic Differential Equations with Brownian Motion, we feel the demand to besides discourse the continuity of the procedure. To turn out continuity we refer to the Kolmogorov theorem as in [ 3 ] :

Theorem 1 ( Kolmogorov ‘s Continuity theorem )

Let a procedure that for all there exist such that

, for. Then there exists a uninterrupted version of X.

A cogent evidence of the theorem can be found in [ 4 ] .

For Brownian Motion, it can be shown [ 3 ] that, which by Theorem 1 we have that has a uninterrupted version. In fact, from now we will be mentioning to that uninterrupted version of.

Figure. Standard Brownian Motion Paths

As one of the purposes is to look into numerical estimates of Stochastic Differential Equations, the following natural measure is to briefly discuss integrating in footings of. Though there are multiple attacks in assorted research documents, we are interested in the one shown by D.J. Higham in [ 2 ] as in it is more lined up with numerical estimates. Another side benefit of the attack above is that it provides an interesting connexion to Classical Riemann concretion. As such, remember the left end-point Riemann amount representation of the Riemann built-in given by

, where

, or utilizing the center

First we set. Therefore we have from ( 2.7 ) that

, by telescoping series. The 2nd term drops off as it is equal to nothing. For the 3rd term, we have that:

Besides, the discrepancy of the 3rd term is of. Therefore by using bounds on both sides of ( 2.9 ) we get

, which is the Ito Integral.

By following a similar logic on ( 2.8 ) we get

, which is the Stratonovich Integral.

As explained by Oksendal in [ 3 ] , even though the two integrals look to be different, the pick of which one to be used is truly a affair depending on what belongingss the user is interested in. The more general and usual pick of normally looking into the Ito Integral is due to the fact that it is non looking into the hereafter, which is a belongings we care for in Biology. Besides Stratonovich is handled better under transmutations and particularly on SDEs on manifolds. On the other manus, the Ito integrals are martingales, hence deriving a computational advantage.

As with classical concretion, we could non perchance use the above attack every clip we need to cipher a stochastic integral. The biggest discovery in Stochastic Calculus could perchance be due to Kiyoshi Ito.

Lemma 2.1 ( Ito ‘s Lemma ) [ 3 ]

Let be an Ito procedure given by

Let. Then is once more an Ito procedure and

, where is computed harmonizing to the regulations

and

The Ito Lemma, or otherwise known as the Ito expression, is the equivalent of a alteration of variable expression. One could reasonably easy notice from the construction of the expression that it stems from a Taylor series enlargement to the 2nd partial derived function in footings of the stochastic procedure.

As an illustration, we would wish to corroborate the consequence ( 2.12 ) , i.e. evaluate. Therefore we set and. Then and by Ito ‘s expression we get

, which leads to the same reply as ( 2.12 ) , viz.

Preliminaries for fractional Brownian Motion ( fBM )

Our probe will non be limited to the Brownian Motion and to SDEs with BM. We are interested in widening our consequences to the fractional Brownian gesture every bit good to SDEs with fBM. Harmonizing to [ 6 ] , the procedure has been defined in 1940 by Kolmogorov in [ 7 ] and its belongingss, i.e. self similarity and long term dependance, were developed by Mandelbrot and Van Ness in [ 8 ] . Another of import subscriber was the British hydrologist Harold Edwin Hurst [ 9 ] . In his surveies on the Nile River, he observed through 800 old ages worth of empirical informations, that the H2O degrees had a long term dependence and self similarity. To depict that dependence, he estimated a parametric quantity, allow us name H, based on his informations.

Definition 2.2 We define a Gaussian procedure with uninterrupted sample waies as a standard fractional Brownian Motion ( fBM ) with Hurst parametric quantity if it satisfies:

, for all.

Merely by merely looking at look ( 2.19 ) , it is obvious that we should see a trichotomy on the value of the power in the right manus side, more peculiarly at the value:

For, , therefore is the standard B.M.

For the increases are positively correlated

For the increases are negatively correlated

As we mentioned supra, two really of import belongingss of fBM are self similarity and long term dependance.

Definition 2.3 A procedure is said to be self similar with parametric quantity if for each

It is reasonably easy to see that for the procedure we can compose

Therefore fBM is a self similar procedure with parametric quantity H and

Besides, sing long scope dependance, allow.

Then for

and therefore the procedure is long scope dependant.

Figure. Fractional Brownian Motion Paths with H=0.7

Besides, we are interested in the undermentioned theorem as a tool for work outing SDEs affecting fBM:

Theorem 2.1 if is with derived functions to order two, so

a.s.

If we let so we have the usual Ito expression.

Numeric Approximation and Simulations

The chief range of our work Is to develop tolls and methods that can be used to numerically stand for Brownian Motion waies, fractional Brownian Motion waies and SDEs with either BM or fBM. The intent of imitating the first two is so that we can utilize them as inputs in the SDEs in both instances of existent expressed solutions and numerical estimates. The intent to imitate SDEs comes as we can come close numerically their solutions in instances where an expressed solution can non be found. The plans used for this paper can be found in Appendix A. We will get down by specifying our mistake measuring expression.

Definition 3.1 ( Error expressions )

Let be the existent values of X and the numerical approximated values of Ten at clip points. Then

is the absolute mistake,

is the comparative mistake, and

is the mean mistake

We use different signifiers of mistake measurings so that we are susceptible to misdirecting consequences.

Next we deal with our attack to imitate the different procedures. The basic and common rule is to discretize the procedure as we are utilizing Matlab. Get downing with the standard Brownian Motion, we use its belongingss, i.e. the fact that it is a Gaussian procedure whose increases follow a normal distribution with average 0 and discrepancy equal to the time-step. Therefore we use a build-in random figure generator that provides us with a and we scale by, where is the time-step. For our work we considered equidistant dividers, i.e. , where T is the stopping clip and N is the figure of time-steps desired. Besides, we normally investigate our procedures on in order to cut down as much complexness and cost on the plan. As expected, we produce different waies of the Brownian Motion even if we preserve all the invariables ( Figure 1 ) . Though the writer ‘s original codification was successful, the codification suggested in [ 2 ] by Higham is slender and really efficient.

We besides employ the belongingss of the fractional Brownian gesture in order to imitate its waies. The undermentioned stairss are needed [ 10 ] :

Form an NxN matrix A whose entries are given by ( 2.19 ) , i.e the covariance of the procedure.

Measure the square root of A utilizing the Cholesky decomposition method.

Generate a 1xN vector V whose entries are from a standard Gaussian distribution

Apply to v.

A sample of five fBM waies with parametric quantity H=0.7 can be seen in Figure 2.

As we now have tools to imitate both BM and fBM, we proceed to discourse the estimates of SDEs. We start by look intoing three methods for Stochastic Differential Equations affecting standard Brownian Motion as defined in [ 5 ] . The best acting method will be applied to Stochastic Differential Equations with fractional Brownian Motion. So, the undertaking is to come close the stochastic procedure fulfilling the SDE:

on and initial value

For simpleness intents we set and. So we get

.

Using the Ito expression to ( 3.5 ) we have that

We now introduce the three methods:

Definition 3.2 ( Euler Method )

For on the interval, the Euler estimate is a uninterrupted clip stochastic procedure fulfilling the iterative strategy:

More specifically in our instance that we wish to use the method to ( 3.6 ) , we get:

Definition 3.3 ( Heun Method )

For on the interval, the Heun method is fulfilling the iterative strategy:

, where

More specifically in our instance that we wish to use the method to ( 3.6 ) , we get:

, where

The rule behind the Heun method is really much alike to the Euler one, with the difference that alternatively of the procedure being evaluated at the end points, the trapezoid regulation is being used.

Definition 3.4 ( Milstein Method )

For on the interval, the Milstein estimate is a uninterrupted clip stochastic procedure fulfilling the iterative strategy:

More specifically in our instance that we wish to use the method to ( 3.6 ) , we get:

The Milstein method is in a sense an “ evolutionary ” signifier of the Euler method. The basic difference is that one excess term is included in the method. Another of import comment is that the Ito-Taylor enlargement is used in order to deduce this method, hence supplying an order 1.0 strong Taylor strategy. Next we compare the three methods with the existent solution diagrammatically.

Figure Simulations for N=1000 and

Figure. Simulations for

Figure. Simulations for

Table. Table of Absolute Mistakes

Table. Table of Relative Mistakes

As shown by graphs 3-5 we get the thought that the Heun method is non appropriate for SDEs whatsoever. In fact, the strategy seems to diverge one time BM is involved. Therefore it is wholly abandoned for our intents. In comparing the two staying methods, even though both seem to follow the existent solution, the Milstein strategy seems to hold a much smaller divergence from the existent solution ( Tables 1 & A ; 2 ) . The consequence is non surprising as both Euler and Milstein can be derived by using the Taylor multinomial enlargement to the SDE, with the difference that the Milstein strategy is of higher order. The one chief concern normally with higher order strategies, is the how computationally expensive it can be. Truth is though, that even a criterion place computing machine can easy run the plans in affair of seconds. As such, we further prove the Milstein strategy against the existent solutions of two more non-linear SDEs, viz. :

, that has as an expressed solution

Besides we test the SDE

, whose solution is

Our following measure is to widen our consequences to supply a method that works in SDEs with fBM. We besides compare numerically our method with an N-step method suggested by Ian Lewis in [ 6 ] . As with the Milstein method for SDEs affecting Brownian Motion, we apply the Taylor multinomial to the general signifier of SDE with fBM. Our consequence and suggested method is given by:

One comment for our method is that if we set we get expression ( 3.13 ) which is the Milstein method for SDEs affecting standard Brownian Gesture.

Proof:

The Milstein Scheme for standard Brownian gesture can be produced by adding the term to the Euler method. In similar attack we have

Measuring the last term we have:

Substituting back in ( 3.20 ) we get

For the numerical simulation, we consider the SDE

with

Its solution is given by

Next we run a comparing of the drawn-out Milstein strategy to the existent solution of the SDE with. The result is really encouraging.

Figure. SDE with fBM utilizing the drawn-out Milstein Method Table 5. Average Mistakes

In a caput to head comparing with the method suggested in [ 6 ] , we resulted in an absolute mistake of nothing. After farther probe it seems that the two strategies are in fact the same strategy. The chief difference is that the suggested method in this paper is a much simpler look and non dependent on summing ups of ternary integrals.

Decisions

We believe that our methods for imitating Brownian Motion and fractional Brownian Motion is reasonably strong due to the fact that they are derived straight from the belongingss of the procedures. Sing SDEs with Brownian Motion, we reject the Heun method and take to either usage either Euler or Milstein method. The Milstein method is slightly closer to the exact solution, but the Euler method might be more appropriate for finer dividers on t. Finally we suggest that for SDEs affecting fBM, the drawn-out Milstein method should be used.

R. Brown, A brief history of microscopical observations made in the months of June, July and August, 1827, on the atoms contained in the pollen of workss ; and on the general being of active molecules in organic and inorganic organic structures. ” Phil. Mag. 4, 161-173, 1828.

D.J. HIGHAM, An algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, September 2001, Vol. 43, Number 3, pp. 525-546

B.Oksendal, Stochastic Differential Equations, An Introduction with Applications, Fifth Edition, Springer, 1998.

D.W. Strook & A ; S.R.S Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, 1979, p51.

P. Kloeden AND E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992

I. Lewis, One Stage Approximation of SDEs with Noise Modeled as fractional Brownian Motion, Honors Undergraduate Thesis supervised by B. Pasik-Duncan, 2005

A.N. Kolmogorov, Wienersche Spiralan and einige andere interessante Kurven im Hilbertschen Raum, C.R. ( doklady ) Acad. Sci. Urss ( N.S. ) , 26, 1940, pp. 115-118

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