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Math Sci (2015) 9:8792DOI 10.1007/s40096-015-0153-x

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Analytical solutions for stochastic differential equations via Martingale processes

Rahman Farnoosh1 Hamidreza Rezazadeh1 Amirhossein Sobhani1

Maryam Behboudi2

Received: 30 September 2014 / Accepted: 1 May 2015 / Published online: 30 May 2015 The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract In this paper, we propose some analytical solutions of stochastic differential equations related to Martingale processes. In the rst resolution, the answers of some stochastic differential equations are connected to other stochastic equations just with diffusion part (or drift free). The second suitable method is to convert stochastic differential equations into ordinary ones that it is tried to omit diffusion part of stochastic equation by applying Martingale processes. Finally, solution focuses on change of variable method that can be utilized about stochastic differential equations which are as function of Martingale processes like Wiener process, exponential Martingale process and differentiable processes.

Keywords Martingale process It formula Change of

variable Differentiable process Analytical solution

Introduction

The purpose of this article is to put forward some analytical and numerical solutions to solve the It stochastic differential equation (SDE):

dXt AXt; tdt BXt; tdWt;

X0 X0;

where Wt is a Wiener process and triple X; F;

is a

probability space under some conditions and special relations between drift and volatility.

Both the drift vector A :

0; T !

R and the dif-

R are considered Borel measurable and locally bounded functions. It is assumed that X0 is a non-random vector. As usual, A and B

are globally Lipschitz in R that is:

jAX; t AY; tj jBX; t BY; tj DjX Yj;

X; Y 2

R and t 2 0; T ;and result in the linear growth condition:

jAX; tj jBX; tj C1 jXj:These conditions guarantee (see [1, 2]) the Eq. (1) has a unique t-continuous solution adapted to the ltration

Ftt 0 generated by Wt and

fusion matrix a : BBT :

0; T !

& Rahman Farnoosh [email protected]

Hamidreza Rezazadeh [email protected]

Amirhossein Sobhani [email protected]

Maryam [email protected]

1 School of Mathematics, Iran University of Science and

Technology, 16844 Narmak, Tehran, Iran

2 Department of Statistics, Science and Research Branch, Islamic Azad University, Tehran, Iran

jXsj2 ds

\1: 2

It is generally accepted that, analytical solutions of partial and ordinary differential equations are so important particularly in physics and engineering, whereas most of them do not have an exact solution and even a limited number of these equations, (e.g., in classical form), have implicit solutions. Analytical methods and solutions, especially in

123

88 Math Sci (2015) 9:8792

stochastic differential equations, could be excessive fundamental in some cases therefore we draw to take a comparison and analyze computation error between them and different numerical methods. Numerous numerical methods can be applied to solve stochastic differential equations like Monte Carlo simulation method, nite elements and nite differences [2, 3]. On the other hand, due to the importance of Martingale processes and nding their representation according to Martingale representation theorem, it is struggled to express arbitrary stochastic processes as a function of Martingale processes and found numerical methods so as to solve drift-free SDEs [4].

In this paper, we resolve to represent analytical methods for stochastic differential equations, specially reputed and famous equations in pricing and investment rate models, based on Martingale processes with various examples about them which we have found in a couple of papers like [2, 57]. There are two main reasons for this approach. Firstly, the each solutions of these kind of equations are Martingale processes or analytic function of Martingale Processes. Thus, due to drift-free property, it will be caused computational error less than numerical computations with existing classic methods. Secondly, for each Martingale process (especially differentiable process), there exists a spectral expansion of two-dimensional Hermite polynomials with constant coefcients [8]. Therefore, it could be made higher the strong order of convergence with increasing the number of polynomials in this expansion. Equations are just obtained with diffusion part or drift free, by making Martingale process from other process. This method can be done by It product formula on initial process and an appropriate Martingale process. Another suitable method to convert SDEs into ODEs that we try is to omit the diffusion part of the stochastic equation.

This article is organized as follows. In Sect. 2, it is veried the making of Martingales processes by exponential Martingale process. In Sect. 3, we solve equations as a function of Martingales with prominent analytical solution, by applying change of appropriate variables method on drift-free SDEs. In Sect. 5, some analytical and numerical examples of expressed methods are demonstrated. Finally, the conclusions and remarks are brought in last section.

Change of measure and Martingale process

In this section under some conditions, we intend to make a Martingale process from a random one in L2 R 0; T , where T is called maturity time. The ex-

ponential Martingale process associated with kt is de

ned as follows:

Zkt exp Z

: 3

It can be indicated by It formula that Zkt is a Martingale

due to the drift-free property:

dZkt kZktdWt; Zkt0 1: 4 Theorem 1 Suppose that stochastic processes Xt verify in differential equation:

dXt lXt; tdt rXt; tdWt; 5

and let kt : lXt; t=rXt; t: Therefore, XZkt is a

Martingale process.

Proof With attention to real function kt, we have: dXlX;tdtrX;tdWt ktrX;tdtrX;tdWt;

dZktZktkdWt:

By utilizing It product formula, we get:

dXZkt XdZkt ZktdX dXdZkt

kXZktdWt lX; tZktdt rX; tZktdWt

krX; tZktdt:

According to theorem assumption, we obtain:

dXZkt ZktXk rX; tdWt: 6 It emphasizes that XZkt is a P-Martingale. h

Therefore, kt lX;trX;t is the sufcient condition for

following SDEs equivalence:

dX lX; tdt rX; tdWt , dXZkt ZktXkt rX; tdWt:

Consequently, by solving the obtained equation in Eq. (6), we obtain the following result when Zk0 1:

XZkt Z

0 ks dWs

0 k2s ds

ZktXks rX; t dWt X0: 8

By taking mathematical expectation from both sides of Eq. (8):

EPXZkt X0 ) EPX X0Zkt 1: 9 In addition, to compute the variance of this stochastic process:

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Math Sci (2015) 9:8792 89

EPXZkt2 X20 E Z

by It

o isometry

X20 Z

Zks2Xks rX; t2 ds

8 <

2 u00b2t AuY; t;

u0bt BuY; t:

Zks2E Xks rX; t2

h i

ds:

2 B0

: 14

var XZkt Zkt2 var X

Thus, it concludes that:

bt B

2 BB0 A ) A

Zks2E Xks rX; t2

h i

ds: 10

Applying (6) and using numerical approximation by EM method, we have:

DXiZkti ZktiXikti riDWi: Xti1Zkti1 XtiZkti ZktiXtikti riDWi:

Xti1 Zkti1 1ZktiXti Xtikti riDWi: Direct calculations would lead to the conclusion that:

Rti Zkti1 1Zkti exp Z

ti1

ti ksdWs

Finally, the equation o

oY A

0 is necessary condition to solve an equation via change of variable in (12)

B0 oBoX .

Case 2 Consider the exponential Martingale process SDE (3):

dY ktYdWt;

Y0 Y0:

B 12 B0

Applying It formula for uY X, to (15), we acquire: u0kY Bu; t ktY

So the following Milstein recursive method is inferred as a good numerical method to nd Xti1:

Xti1 RtiXti Xtikti riDWi

8 <

ti1

jk2sjds

Bu or u0

Bu;

2 u00k2Y2 Au; t:

So from the last equality, we have B0

kt 2AB kt.

Therefore, o

B0u 2ktAB

2 R2tikti Xtikti ri

D2Wi Dti:

0 is necessary condition to

solve SDE, with this change of variable.

Case 3 Consider the well-known equation:

dY atYdt btYdWt;

Y0 Y0:

Which is BlackScholes equation with exact solution

Y0 exp Z

In example 1, we compare this method with usual Milstein method in the case that a stochastic differential equation contains drift and volatility both parts and indicate that this method could be better in some cases.

Change of variable method

This section intends to analyze the change of variable method like [9], to get explicitly the solution of arbitrary SDE:

dX AX; tdt BX; tdWt; X0 x:By nding appropriate variables uY X and their con

ditions so that Y is the answer of a well-known SDEs related to Martingale processes.

dY f X; tdt gX; tdWt; y0 y:For more explanation and different conditions under which they are possible, we could see [5, 10]. Now we consider following various cases.

Case 1 Consider the following SDE:

dY atdt btdWt: 12

Applying It formula for uY X, to (12), we get:

u0at

Applying It formula for uY X, to (17), we get:

u0atY

0 bsdWs Z

0 as

2 b2s

8 <

2 u00b2tY2 Au; t;

u0Ybt Bu; t btY

Bu:

For this reason, u0

Bu and we have:

2 B0u bt cu; t: 19

It means that o

ou cu; t 0, is a necessary condition to

solve the initial stochastic differential equation by this change of variable.

Case 4 Another appropriate and prominent case is as follows:

dYt f Yt; tdt ctYtdWt;

Y0 Y0:

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90 Math Sci (2015) 9:8792

This kind of equations, applying It formula on Xt YtZctt 1, is converted to a ordinary differential

equations.

Theorem 2 The stochastic differential equations in (20) given by continuous functions f : R

R and C : R !

R can be written as:

dYtZctt 1 Zctt 1f Yt; tdt; 21 where Zctt is an exponential Martingale process.

(See Oksendal [1], Chapter 5, Exercise 17]). To be more

precise, using change of variable V XZctt 1, it is

enough to solve

X0t Zctt 1f XtZctt;

X0 X0:

Example 2 Consider the following SDE that is named BlackScholes equation.

dX ltXdt rtXdWt:

Using (6), we have:

dXZkt ZktXk rtdWt ZktXk XrtdWt

XZktk rtdWt:

From this equality we could conclude that XZkt, is the exponential Martingale Zkrt. Finally, X Zkt 1Zkrt exp R

t0 rt dWs R

t 0

lt r2 ds .

( 22

Applying It formula for uY Mt, in (20) we get:

dMt M0tdY

2 M00tdY2:

f Y; tM0t

This is the exact solution of BlackScholes equation.

Example 3 Consider the following stochastic model

4 t2X2dt tX3=2dWt;

X0 0:

8 <

It can be checked that for this equation the necessary condition holds for this equation. According to (13), we have u0bt tu3=2. Since u is just a function of Y, we

should get bt t, u 4Y2 and atbt 0 (or at 0). Thus, dY tdWt and Y R

t0 sdWs Y0, and ultimately

X uY 4 R

8 <

12 M00tc2tY2 AMt; t; 1 ctYM0t BMt; t; uY0 M0: 2

BMt. Besides,

if the new stochastic differential equation is related to a Martingale process, we have AMt; t 0 and:

f Y; t

c2tY

According to (23), we have BMt; t ct

t0 sdWs Y0

is the exact solution

(Fig. 1).

Example 4 Consider the following SDE model

Again, applying It formula for /Mt Vt to Martingale

equation contributes to

dMt BMt; tdWt ct

8 <

First of all, we check the necessary condition in case 2:

B0u

2c2trX2r 1 c2tXrdtctXrdWt; r6 1

X00:

BMt0 1:

BMtdWt;

we can achieve to a novel group of stochastic differential equation that its solution is as a function of a Martingale process.

Examples

Example 1 Consider the following SDE

dX at X

p dt bt X

ctrur 1

c2tru2r 1 c2tur

ctur

ctkt:

Utilizing the rst equation in Eq. (16), u0ktY ctur. Hence, lnY u r1 r1, that r6 1, Y01 and u10.

Therefore, the exact solution is as follows:

X uY 1 r Z

In a particular case, if r 12, we reach the following model:

c2t

0 cs; dWt

0 c2sds

p dWt;

( 25

from (9), we can get immediately EX X0Zkt 1 such

that k atbt : The graphs of various numerical solutions of

this example by Milstein method, proposed formula (11) that is drift free and Taylor method of order 2 introduced as exact solution.

p dt ct X

X0 X0:

4 c2t X

dWt;

Example 5 Consider the following SDE model:

1 4

0 ctdWt

0 c2sds

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Math Sci (2015) 9:8792 91

a b

Fig. 1 a The graphs of the numerical solutions of Example 1 by Milstein method, proposed formula (11) and Taylor method of order 2 introduced as exact solution. In this example, it is considered that

drift, volatility and initial condition respectively are: at 0:2; bt 1:0, X0 1:5, maturity time T 2 and number

of points N 30. b The graphs of absolute error

First of all, we check the necessary condition in Case 3:

cu; t u

2 2u bt

dX X3dt X2dWt; X0 1:

8 <

: 28 Applying It formula for Xt eZt, to the last equation, we

obtain the following drift-free stochastic equation:

dXt Xt ln2XtdWQt;

Xt0 1:

dZt

ln 22

Z2t

2 ln 2Ztdt ln 2 ZtdWQt;

0 0:

2 :

( 29

according to (23), we have Yu0 u ln2u. Consequently,

Y ln2X2, X uY 12 e2Y.

From (24), we have f Y2 and consequently, the

exact solution of corresponding SDE is X 12 e2Y such that

its related stochastic equation is:

dY Ytdt YtdWQt;

ln2

2 :

From (18), we should have u0btY u2. Therefore, if

bt 1, we can get immediately u 1lnY and at

b2t

2 12, so that Y is the solution of following equation.

2 Ydt YdWt;

8 >

Therefore, according to geometric Brownian motion process, the exact solution is determined

Y 1e exp R

t0 dWt

eWt 1, and nally exact solution is equal to X

1 1 Wt.

1 e :

8 <

As we know, the exact solution of this linear stochastic differential equation is as follows:

ln2

2 exp WQt

Example 6 Consider the stochastic model as follows:

dZt

Z2t

2 ln 2Ztdt ln 2 ZtdWt;

8 <

First, by applying Girsanov theorem so that

WQt Wt ln2

0 0:

: 30

Finally, the exact solution of this example is:

2 t, we reach the following equation:

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92 Math Sci (2015) 9:8792

Zt lnXt ln

2 e2Yt

2Yt ln 2 ln 2 exp WQt

Conclusions and remarks

In this paper, a couple of analytical solutions of some determined set of stochastic differential equations was indicated via making the Martingale process from a stochastic process. Converting stochastic differential equations to ordinary ones as another suitable method was posed. Indeed, it is tried to omit diffusion part of stochastic equation by applying Martingale processes. In addition, change of variable method on SDEs related to Martingale process-eswas discussed. Last of all with some examples, we analyzed and obtained its exact solutions and in some cases their solutions compared with other numerical methods.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http:// creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

References

1. ksendal, B. K.: Stochastic Differential Equations: An Introduction with Applications, 4th edn. Springer, Berlin (1995)

2. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1999)

3. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525546 (2001)

4. Pascucci, A.: PDE and Martingale Methods in Option Pricing. Bocconi & Springer Series, vol. 2. Springer, Milan (2011)

5. Lamperti, J.: A simple construction of certain diffusion processes.J. Math.Kyoto Univ. 4, 161170 (1964)6. Skiadas, C.H.: Exact solutions of stochastic differential equations: Gompertz, generalized logistic and revised exponential. Methodol Comput. Appl. Probab. 12, 261270 (2010)

7. Kouritsin, M.A., DeliOn, L.: explicit solutions to stochastic differential equations. Stoch. Anal. Appl. 18(4), 571580 (2000)8. Udriste, C., Damian, V., Matei, L., Tevy, I.: Multitime differentiable stochastic process, diffusion PDEs, Tzitzeica hypersur-faces. UPB Sci. Bull. Ser. A 74(1), 310 (2012)

9. Evans, L.C.: An Introduction to Stochastic Differential Equations. American Mathematical Society, Providence (2013)

10. McKean, H.: Stochastic Integrals. Academic Press, New York (1969)

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Islamic Azad University 2015

Abstract

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In this paper, we propose some analytical solutions of stochastic differential equations related to Martingale processes. In the first resolution, the answers of some stochastic differential equations are connected to other stochastic equations just with diffusion part (or drift free). The second suitable method is to convert stochastic differential equations into ordinary ones that it is tried to omit diffusion part of stochastic equation by applying Martingale processes. Finally, solution focuses on change of variable method that can be utilized about stochastic differential equations which are as function of Martingale processes like Wiener process, exponential Martingale process and differentiable processes.

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Title

Analytical solutions for stochastic differential equations via Martingale processes

Author

Farnoosh, Rahman; Rezazadeh, Hamidreza; Sobhani, Amirhossein; Behboudi, Maryam

Pages

87-92

Publication year

2015

Publication date

May 2015

Publisher

Springer Nature B.V.

ISSN

20081359

e-ISSN

22517456

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/s40096-015-0153-x

ProQuest document ID

1690606911

Islamic Azad University 2015

Analytical solutions for stochastic differential equations via Martingale processes

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