^{1}

^{2}

^{*}

^{3}

In this work, a one-step method of Euler-Maruyama (EMM) type has been developed for the solution of general first order stochastic differential equations (SDEs) using Ito integral equation as basis tool. The effect of varying stepsizes on the numerical solution is also examined for the SDEs. Two problems of first order SDEs are solved. Absolute errors for the problems are obtained from which the mean absolute errors (MAEs) are calculated. Comparison of variation in stepsizes is achieved using the MAEs. The results show that the MAEs decrease as the stepsize decreases. The strong orders of convergence and the residuals for the problem for the theoretical are respectively obtained using Least Square Fit. This work produces numerical values for the solution to the problems which differ from the existing methods of EMM type in which results are always obtained by simulation.

Modeling of physical systems by ordinary differential equations (ODEs) ignores stochastic effects. Addition of random elements into the differential equations leads to what is called stochastic differential equations (SDEs), and the term stochastic is called noise [

where

Equation (1) is a deterministic state model which can be turned to a stochastic state model by including noise term. For one dimensional noise term, consider a general first order stochastic differential Equation (SDE) of the form

where

Equation (2) can be written as

Integrating (3) from 0 to t, we have Itô integral equation

The first integral at the right hand side of Equation (4) is called Riemman integral while the second integral is called Itô or stochastic integral. Many researchers have also worked on SDEs of the form (3). Amongst these are [

The aim of this paper is to develop numerical method for solution of first order stochastic differential Equation (3). Our objectives are to develop one-step Euler-Maruyama method (EMM) for solution of SDE (3) and apply it to solve two problems in the form of first order SDEs. Absolute errors (AEs) will be determined at various point in the interval

The strong order of convergence (SOC) of the method will be determined from the results of the mean absolute errors obtained. [

There are many methods for determining the solution of SDE (3), they include, Euler-Maruyama method, Euler-Maruyama method, Runge-Kutta method, Heun method and so on. For more information about the list of methods or schemes for solution of first order SDEs (see [

One-step Euler-Maruyama method will be derived by setting

This gives

Using a conventional deterministic quadrature, the two integral terms at the right hand side can be approximated as follows:

Substituting (8) and (9) in (7), we have

Define

Equation (10) becomes

The method in Equation (11) was considered by [

Wiener increment

In this section, we will consider two problems in the form of first order stochastic differential Equation (3) to investigate the effect of varying stepsizes when finding the solution of SDEs using one step method of EMM type in Equation (11).

Problem 1.

where

The exact solution of the SDE (13) is

Problem 1 is the Black Scholes option price model with a drift function

The following stepsizes will be used to carry out our investigation,

See

t-value Error | Exact Solution | Numerical Solution | Absolute |
---|---|---|---|

0.062500 | 1.000035213412162 | 1.000035213104684 | 3.07477599e−10 |

0.125000 | 0.999992194240297 | 0.999992193320025 | 9.20271970e−10 |

0.187500 | 0.999998717493128 | 0.999998716864071 | 6.29056918e−10 |

0.250000 | 1.000012935644641 | 1.000012935226998 | 4.17643475e−10 |

0.312500 | 0.999974930990167 | 0.999974930162858 | 8.27309221e−10 |

0.375000 | 0.999978328502518 | 0.999978327981927 | 5.20591348e−10 |

0.437500 | 0.999988796767910 | 0.999988796505013 | 2.62896704e−10 |

0.500000 | 0.999951804297058 | 0.999951803662421 | 6.34636454e−10 |

0.562500 | 0.999984007130832 | 0.999984006290135 | 8.40697068e−10 |

0.625000 | 1.000014922754158 | 1.000014921748044 | 1.00611364e−9 |

The mean absolute error is 6.366694393911132e−10.

^{−4}.

See

t-value Error | Exact Solution | Numerical Solution | Absolute |
---|---|---|---|

0.062500 | 0.999989302989965 | 0.999989302508838 | 4.81126805e−10 |

0.125000 | 0.999998791743439 | 0.999998791544895 | 1.98543737e−10 |

0.187500 | 0.999969143626926 | 0.999969143306858 | 3.20068305e−10 |

0.250000 | 0.999945211010976 | 0.999945210578632 | 4.32343716e−10 |

0.312500 | 0.999984664319081 | 0.999984663809844 | 5.09236209e−10 |

0.375000 | 1.000048162051462 | 1.000048160772574 | 1.27888855e−9 |

0.437500 | 1.000031603910325 | 1.000031602873598 | 1.03672670e−9 |

0.500000 | 0.999985898306111 | 0.999985896243317 | 2.06279349e−9 |

0.562500 | 1.000027707908093 | 1.000027705706180 | 2.20191265e−9 |

0.625000 | 1.000029576187069 | 1.000029574216571 | 1.97049865e−9 |

The mean absolute error is 1.049213882442501e−9.

^{−5}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | 0.999993968111180 | 0.999993968007453 | 1.03727471e−10 |

0.125000 | 0.999950903398081 | 0.999950903125062 | 2.73019052e−10 |

0.187500 | 1.000018522444389 | 1.000018521838989 | 6.05399730e−10 |

0.250000 | 0.999969318836642 | 0.999969317778954 | 1.05768760e−9 |

0.312500 | 0.999995025525107 | 0.999995024550208 | 9.74898495e−10 |

0.375000 | 0.999971494315777 | 0.999971493473134 | 8.42642955e−10 |

0.437500 | 1.000023470014742 | 1.000023468767193 | 1.24754895e−9 |

0.500000 | 1.000006560271671 | 1.000006558980626 | 1.29104549e−9 |

0.562500 | 1.000016550151243 | 1.000016548430911 | 1.72033232e−9 |

0.625000 | 1.000050801848815 | 1.000050800269305 | 1.57950941e−9 |

The mean absolute error is 9.695811487020478e−10.

^{−6}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | 0.999960105863295 | 0.999960105702641 | 1.60653491e−10 |

0.125000 | 0.999967950241945 | 0.999967949695712 | 5.46233170e−10 |

0.187500 | 0.999964311166688 | 0.999964310727299 | 4.39388304e−10 |

0.250000 | 0.999983928766161 | 0.999983928116954 | 6.49207244e−10 |

0.312500 | 1.000010033769263 | 1.000010032994412 | 7.74851072e−10 |

0.375000 | 1.000060677640722 | 1.000060676842953 | 7.97768518e−10 |

0.437500 | 1.000067288106194 | 1.000067287456036 | 6.50158150e−10 |

0.500000 | 1.000122399331946 | 1.000122398556290 | 7.75655762e−10 |

0.562500 | 1.000162783152388 | 1.000162782441429 | 7.10958847e−10 |

0.625000 | 1.000180536086851 | 1.000180535284656 | 8.02195865e−10 |

The mean absolute error is 6.307070421485150e−10.

^{−7}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | 0.999972159883026 | 0.999972159601738 | 2.81287882e−10 |

0.125000 | 0.999978280985938 | 0.999978280655984 | 3.29953620e−10 |

0.187500 | 1.000027372023912 | 1.000027371636413 | 3.87499366e−10 |

0.250000 | 1.000065836108020 | 1.000065835744871 | 3.63149066e−10 |

0.312500 | 1.000101764246374 | 1.000101763882430 | 3.63944430e−10 |

0.375000 | 1.000108726423491 | 1.000108726193473 | 2.30018227e−10 |

0.437500 | 1.000148957998176 | 1.000148957704878 | 2.93298275e−10 |

0.500000 | 1.000172569073211 | 1.000172568980299 | 9.29125665e−11 |

0.562500 | 1.000206392121796 | 1.000206391996795 | 1.25001121e−10 |

0.625000 | 1.000193604241676 | 1.000193604296249 | 5.45721246e−11 |

The mean absolute error is 2.521636677244032e−10.

^{−8}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | 0.999979464841858 | 0.999979464673978 | 1.67880376e−10 |

0.125000 | 1.000036197200983 | 1.000036197026108 | 1.74874559e−10 |

0.187500 | 1.000061346273481 | 1.000061346169660 | 1.03820508e−10 |

0.250000 | 1.000101309205331 | 1.000101309176948 | 2.83835178e−11 |

0.312500 | 1.000111004081822 | 1.000111004129163 | 4.73403539e−11 |

0.375000 | 1.000054041052400 | 1.000054040994347 | 5.80526738e−11 |

0.437500 | 1.000068042560500 | 1.000068042466253 | 9.42474987e−11 |

0.500000 | 1.000067650976551 | 1.000067650858876 | 1.17674315e−10 |

0.562500 | 1.000133899286137 | 1.000133899093955 | 1.92182270e−10 |

0.625000 | 1.000143970608418 | 1.000143970444947 | 1.63471015e−10 |

The mean absolute error is 1.147927086719847e−10.

^{−9}.

Problem 2.

where

The true solution is

Problem 2 was used by [

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | −2.000030639939910 | −2.000030643064053 | 3.12414317e−9 |

0.125000 | −1.999826604550993 | −1.999826583667489 | 2.08835040e−8 |

0.187500 | −1.999771190197802 | −1.999771171017104 | 1.91806975e−8 |

0.250000 | −1.999738855905664 | −1.999738839777066 | 1.61285976e−8 |

0.312500 | −1.999549931800135 | −1.999549895620500 | 3.61796346e−8 |

0.375000 | −1.999485167195670 | −1.999485131968585 | 3.52270857e−8 |

0.437500 | −1.999441604632085 | −1.999441571887793 | 3.27442917e−8 |

0.500000 | −1.999255786341376 | −1.999255734318596 | 5.20227794e−8 |

0.562500 | −1.999277377750062 | −1.999277329159688 | 4.85903735e−8 |

0.625000 | −1.999295108719981 | −1.999295063663776 | 4.50562045e−8 |

The mean absolute error is 3.091373117491969e−8.

^{−4}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | −1.999892917380804 | −1.999892908128028 | 9.25277566e−9 |

0.125000 | −1.999846392834979 | −1.999846386521514 | 6.31346508e−9 |

0.187500 | −1.999682499484875 | −1.999682485395495 | 1.40893797e−8 |

0.250000 | −1.999535775997482 | −1.999535755103327 | 2.08941555e−8 |

0.312500 | −1.999579115423757 | −1.999579097646508 | 1.77772492e−8 |

0.375000 | −1.999694550513440 | −1.999694531148642 | 1.93647987e−8 |

0.437500 | −1.999569940134597 | −1.999569919321205 | 2.08133912e−8 |

0.500000 | −1.999357977043176 | −1.999357935211449 | 4.18317270e−8 |

0.562500 | −1.999408364465035 | −1.999408325355391 | 3.91096437e−8 |

0.625000 | −1.999339028019188 | −1.999338990093991 | 3.79251974e−8 |

The mean absolute error is 2.273717831791089e−8.

^{−5}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | −1.999906910993895 | −1.999906908359276 | 2.63461808e−9 |

0.125000 | −1.999702767364218 | −1.999702758533899 | 8.83031892e−9 |

0.187500 | −1.999830588766838 | −1.999830580080542 | 8.68629546e−9 |

0.250000 | −1.999608061287530 | −1.999608042553839 | 1.87336919e−8 |

0.312500 | −1.999610182559322 | −1.999610165808479 | 1.67508436e−8 |

0.375000 | −1.999464678473431 | −1.999464660611312 | 1.78621189e−8 |

0.437500 | −1.999545553506756 | −1.999545533746613 | 1.97601426e−8 |

0.500000 | −1.999419912678686 | −1.999419890196399 | 2.24822863e−8 |

0.562500 | −1.999374919082127 | −1.999374891267540 | 2.78145871e−8 |

0.625000 | −1.999402649038457 | −1.999402624540933 | 2.44975247e−8 |

The mean absolute error is 1.680524275293749e−8.

^{−6}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | −1.999805341404871 | −1.999805337513111 | 3.89176003e−9 |

0.125000 | −1.999753891446814 | −1.999753882755321 | 8.69149241e−9 |

0.187500 | −1.999668007898771 | −1.999667999884824 | 8.01394706e−9 |

0.250000 | −1.999651870477482 | −1.999651860214105 | 1.02633777e−8 |

0.312500 | −1.999655185128882 | −1.999655173868698 | 1.12601839e−8 |

0.375000 | −1.999732080887753 | −1.999732070781591 | 1.01061621e−8 |

0.437500 | −1.999676933687320 | −1.999676925131407 | 8.55591287e−9 |

0.500000 | −1.999767219155966 | −1.999767210691419 | 8.46454751e−9 |

0.562500 | −1.999813341382202 | −1.999813334738500 | 6.64370181e−9 |

0.625000 | −1.999791597710221 | −1.999791590169479 | 7.54074203e−9 |

The mean absolute error is 8.343182744674494e−9.

^{−7}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | −1.999841496174841 | −1.999841492043785 | 4.13105594e−9 |

0.125000 | −1.999784874982539 | −1.999784870148436 | 4.83410290e−9 |

0.187500 | −1.999857131369450 | −1.999857126530599 | 4.83885132e−9 |

0.250000 | −1.999897512575721 | −1.999897508515622 | 4.06009937e−9 |

0.312500 | −1.999930285134005 | −1.999930281503511 | 3.63049413e−9 |

0.375000 | −1.999876177386879 | −1.999876175259421 | 2.12745799e−9 |

0.437500 | −1.999921851349309 | −1.999921848982462 | 2.36684672e−9 |

0.500000 | −1.999917674573231 | −1.999917674818573 | 2.45341969e−10 |

0.562500 | −1.999944122996693 | −1.999944123256830 | 2.60136801e−10 |

0.625000 | −1.999830784974864 | −1.999830786914828 | 1.93996397e−9 |

The mean absolute error is 2.843435109589621e−9.

^{−7}.

See

t-value | Exact Solution | Numerical Solution | Absolute Error |
---|---|---|---|

0.062500 | −1.999863407270136 | −1.999863404946251 | 2.32388553e−9 |

0.125000 | −1.999958592655718 | −1.999958590662368 | 1.99335082e−9 |

0.187500 | −1.999959037106798 | −1.999959036084589 | 1.02220876e−9 |

0.250000 | −2.000003915981924 | −2.000003916122282 | 1.40357947e−10 |

0.312500 | −1.999957999627532 | −1.999958000650131 | 1.02259934e−9 |

0.375000 | −1.999712179640226 | −1.999712178747598 | 8.92628638e−10 |

0.437500 | −1.999679195925699 | −1.999679194613133 | 1.31256606e−9 |

0.500000 | −1.999603058636733 | −1.999603056922618 | 1.71411507e−9 |

0.562500 | −1.999726729199231 | −1.999726727095061 | 2.10416973e−9 |

0.625000 | −1.999681957561421 | −1.999681955778895 | 1.78252613e−9 |

The mean absolute error is 1.430840801397437e−9.

^{−9}.

In this section, it will be wise to determine the strong order of convergence (SOC) and the residual of the method as this property is important to examine how good or how accurate our approximations are. SOC is a property of solution that is indispensable in examining the accuracy of any numerical method for solution of SDEs.

The issues of convergence and SOC of SDEs have also been examined by [

Stepsize | Problem 1 |
---|---|

MAE of One Step Method EMM Type | |

2^{−4} | 6.36669439e−10 |

2^{−5} | 1.04921388e−9 |

2^{−6} | 9.69581149e−10 |

2^{−7} | 6.30707042e−10 |

2^{−8} | 2.52163668e−10 |

2^{−9} | 1.14792709e−10 |

Stepsize | Problem 2 |
---|---|

MAE of One Step Method EMM Type | |

2^{−4} | 3.09137312e−8 |

2^{−5} | 2.27371783e−8 |

2^{−6} | 1.68052428e−8 |

2^{−7} | 8.34318274e−9 |

2^{−8} | 2.84343511e−9 |

2^{−9} | 1.43084080e−9 |

One-step Method of EMM Type | ||
---|---|---|

Problem 1 | Problem 2 | |

SOC | 0.5471 | 0.9193 |

RESIDUAL | 1.0975 | 0.6067 |

In this paper effort has been made to discuss the derivation of one-step method of Euler-Maruyama type. This method was applied to two problems in the form of first order SDEs. The method was used to determine the numerical solution of the two problems. Absolute errors were calculated using the numerical approximation and the corresponding exact solution. Mean absolute errors were also determined. To determine the accuracy of the method, strong order of convergence and residuals were obtained for each problem.

In this paper, two problems in the form of first order SDEs have been considered. One-step method of Euler- Maruyama type for solution of general first order SDEs has been derived. The absolute errors between the exact solution and numerical solution can be observed. The mean absolute error for varying stepsizes has been determined. The result shows that the mean absolute error generally decreases as the stepsizes decreases. The accuracy of the method is determined by finding the strong order of convergence of the method. The result shows that the strong order of convergence is 0.5471 and the residual is 1.0975 for Problem 1 while the strong

order of convergence is 0.9193 and the residual is 0.6067 for Problem 2. The effect of the varying stepsizes can also be seen by observing the behaviour of the exact solution and numerical solution using graphical method as indicated in Figures 1-12. The results are obtained using MATLAB as supporting tool.

Sunday Jacob Kayode,Akeem Adebayo Ganiyu,Adegoke Sule Ajiboye, (2016) On One-Step Method of Euler-Maruyama Type for Solution of Stochastic Differential Equations Using Varying Stepsizes. Open Access Library Journal,03,1-15. doi: 10.4236/oalib.1102247