Document Type : Research Paper

**Authors**

Department of Mathematics, Iran University of Science and Technology, Tehran, Iran

10.22034/amfa.2020.1873599.1260

**Abstract**

The multidimensional exponential Levy equations are used to describe many stochastic phenomena such as market fluctuations. Unfortunately in practice an exact solution does not exist for these equations. This motivates us to propose a numerical solution for n-dimensional exponential Levy equations by block pulse functions. We compute the jump integral of each block pulse function and present a Poisson operational matrix. Then we reduce our equation to a linear lower triangular system by constant, Wiener and Poisson operational matrices. Finally using the forward substitution method, we obtain an approximate answer with the convergence rate of O(h). Moreover, we illustrate the accuracy of the proposed method with a 95% confidence interval by some numerical examples.

**Keywords**

[1] Merton, R.C., *Option pricing when underlying stock returns are discontinuous*, Journal of Financial Economic, 1976, 2, P. 125-144.

[2] Kou, S. G., *A jump diffusion model for option pricing*, Management Science, 2002, 48, P. 1086–1101.

[3] Glasserman, P. and Kou, S.G., *The term structure of simple forward rates with jump risk*, Mathematical finance, 2003, 13(3), P. 383–410.

[4] Cont, R. and Tankov, P., *Financial Modelling with Jump Processes*, Financial Mathematics Series, Chapman and Hall/CRC, 2004.

[5] Brummelhuis, R., Chan, RTL., *A radial basis function scheme for option pricing in exponential Levy models*, Applied Mathematical Finance, 2014, 21 (3). P. 238-269.

[6] Chen, X. and Wan, J., *Option pricing for Time-change Exponential levy Model under MEMM*, Acta Mathematicae Applicatae Sinica, English Series, 2007, 23 (4). P. 651-664.

[7] Izadikhah, M., *Improving the Banks Shareholder Long Term Values by Using Data Envelopment Analysis Model*, Advances in Mathematical Finance and Applications,2018, **3**(2), P. 27-41. Doi: 10.22034/AMFA.2018.540829

[8] Jiang, Z.H. and Schaufelberger, W., *Block Pulse Functions and Their Applications in Control Systems*, Springer-Verlag, 1990.

[9] Prasada Rao, G., *Piecewise Constant Orthogonal Functions and their Application to Systems and Control*, Springer, Berlin, 1983.

[10] Maleknejad, K., Khodabin, M., Rostami, M., *A numerical method for solving m-dimensional stochastic Ito–Volterra integral equations by stochastic operational matrix*, Computers and Mathematics with Applications, 2012, 63, P. 133–143.

[11] Maleknejad, K., Khodabin, M., Rostami, M., *Numerical Solution of Stochastic Volterra Integral Equations By Stochastic Operational Matrix Based on Block Pulse Functions*, Mathematical and Computer Modeling, 2012, 55, P. 791–800.

[12] Khodabin, M., Maleknejad, K., Rostami, M., *Numerical approach for solving stochastic Volterra-Fredholm integral equations by stochastic operational matrix*, Computers and Mathematics with Applications, 2012, 64, P. 1903–1913.

[13] Maleknejad, K., Sohrabi, S., Rostami, Y., *Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials*, Applied Mathematics and Computation, 2007, 188, P. 123–128.

[14] Maleknejad, K., Rahimi, B., *Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind*, Communications in Nonlinear Science and Numerical Simulation 2011, 16, P. 2469–2477.

[15] Maleknejad, K., Rahimi, B., *Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind*, Communications in Nonlinear Science and Numerical Simulation 2011, 16, P. 2469–2477.

[16] Maleknejad, K., Safdari, H., Nouri, M., *Numerical solution of an integral equations system of the first kind by using an operational matrix with block **pulse functions*, International Journal of Systems Science, 2011, 42, P. 195–199.

[17] Zhang, X., *Stochastic Volterra equations in Banach spaces and stochastic partial differential equation*, Journal of Functional Analysis, 2010, 258, P. 1361–1425.

[18] Manafian, J. and Bolghar, P., *Numerical solutions of nonlinear 3**‐**dimensional Volterra integral**‐**differential equations with 3D**‐**block**‐**pulse functions, *Mathematical Methods in the Applied Sciences, 2018, 41 (12), P. 4867-4876.

[19] Safavi, M., Khajehnasiri, A.A., Angelamaria Cardone, A., *Numerical solution of nonlinear mixed Volterra-Fredholm integro-differential equations by two-dimensional block-pulse functions*, Cogent Mathematics and Statistics, 2018, 2 (1), P. 1-13.

[20] Yonthanthum, W., Rattana, A., Razzaghi, M., An approximate method for solving fractional optimal control problems by the hybrid of block‐pulse functions and Taylor polynomials, Optimal Control Applications and Methods, 2018, 39 (2), P. 873-887.

[21] Lee, J. and Kim, Y., *Sampling methods for NTR processes*, in Proceedings of International Workshop for Statistics (SRCCS ’04), Lecture Note, Seoul, Korea, 2004.

Spring 2020

Pages 247-259

**Receive Date:**20 July 2019**Revise Date:**28 January 2020**Accept Date:**28 January 2020