An Application of New Method to Obtain Probability Density Function of Solution of Stochastic Differential Equations

Nihal İnce; Aladdin Shamilov

Open Access

An Application of New Method to Obtain Probability Density Function of Solution of Stochastic Differential Equations

Nihal İnce

and

Aladdin Shamilov

| Mar 31, 2020

Applied Mathematics and Nonlinear Sciences

Volume 5 (2020): Issue 1 (January 2020)

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Page range: 337 - 348

Received: Jun 15, 2019

Accepted: Aug 10, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00031

Keywords
generalized entropy optimization methods, stochastic differential equation model, Euler–Maruyama Method

© 2020 Nihal İnce et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

HMC stock path (black line) and approximate EM trajectories
X^(ti), i=1,…,39\hat X({t_i}),\,i = 1, \ldots ,39
of SDE (color lines) — HMC stock path (black line) and approximate EM trajectories X^(ti), i=1,…,39\hat X({t_i}),\,i = 1, \ldots ,39 of SDE (color lines)

HMC stock path (black line), approximate trajectories
X^(t16)\hat X({t_{16}})
(blue line) and
X^(t39)\hat X({t_{39}})
(red line) of SDE model — HMC stock path (black line), approximate trajectories X^(t16)\hat X({t_{16}}) (blue line) and X^(t39)\hat X({t_{39}}) (red line) of SDE model

Comparison of the (MinMaxEnt)m, m = 1,2,3,4 distributions for stock data
X^(t16)\hat X({t_{16}}) — Comparison of the (MinMaxEnt)m, m = 1,2,3,4 distributions for stock data X^(t16)\hat X({t_{16}})

Comparison of the (MinMaxEnt)m, m = 1,2,3,4 distributions for stock data
X^(t39)\hat X({t_{39}}) — Comparison of the (MinMaxEnt)m, m = 1,2,3,4 distributions for stock data X^(t39)\hat X({t_{39}})

Comparison of the (MaxMaxEnt)m, m = 1,2,3,4 distributions for stock data
X^(t16)\hat X({t_{16}}) — Comparison of the (MaxMaxEnt)m, m = 1,2,3,4 distributions for stock data X^(t16)\hat X({t_{16}})

Comparison of the (MaxMaxEnt)m, m = 1,2,3,4 distributions for stock data
X^(t39)\hat X({t_{39}}) — Comparison of the (MaxMaxEnt)m, m = 1,2,3,4 distributions for stock data X^(t39)\hat X({t_{39}})

Evaluation of performance of (MinMaxEnt)m, m = 1,2,3,4 distributions for X^(t39)\hat X({t.{39}})

(MinMaxEnt)_m	entropy	R²	RMSE	χ²	moment functions
(MinMaxEnt)₁	2.5274	0.6133	0.0354	0.0601	1, ln(1 + x²)
(MinMaxEnt)₂	2.5135	0.6897	0.0318	0.0452	1, x, lnx
(MinMaxEnt)₃	2.5135	0.6897	0.0318	0.0452	1, x, lnx, ln(1 + x²)
(MinMaxEnt)₄	2.5243	0.6826	0.0320	0.0495	1, x, lnx, ln²(x), ln(1 + x²)

Data set of weekly closing prices of HMC stock between 2 July 2018 to 25 March 2019

Week	Price	Week	Price	Week	Price	Week	Price	Week	Price
1	28.60	9	30.00	17	25.83	25	26.31	33	27.28
2	28.60	10	29.91	18	28.82	26	25.18	34	27.47
3	29.49	11	28.12	19	28.72	27	26.26	35	28.33
4	29.49	12	30.65	20	27.34	28	29.12	36	27.15
5	30.00	13	30.53	21	27.50	29	30.06	37	26.90
6	30.05	14	30.04	22	27.39	30	30.10	38	26.80
7	29.10	15	28.24	23	27.09	31	30.58	39	26.64
8	30.37	16	27.04	24	26.00	32	27.81

Evaluation of performance of (MaxMaxEnt)m, m = 1,2,3,4 distributions for X^(t16)\hat X({t.{16}})

(MaxMaxEnt)_m	entropy	R²	RMSE	χ²	moment functions
(MaxMaxEnt)₁	2.5572	0.2053	0.0573	0.1223	1, x²
(MaxMaxEnt)₂	2.5538	0.2353	0.0562	0.1169	1, lnx, ln²(x)
(MaxMaxEnt)₃	2.5537	0.2403	0.0560	0.1160	1, lnx, ln²(x), ln(1 + x²)
(MaxMaxEnt)₄	2.5026	0.7011	0.0350	0.0382	1, x², lnx, ln²(x), ln(1 + x²)

Reel index values of HMC stock and approximate values of X^(t16)\hat X({t.{16}}) using EM method

Week	Reel Price	EM App.	Week	Reel Price	EM App.	Week	Reel Price	EM App.
1	28.60	28.60	14	30.04	30.61	27	26.26	25.22
2	28.60	28.62	15	28.24	29.95	28	29.12	26.09
3	29.49	28.53	16	27.04	27.99	29	30.06	29.32
4	29.49	29.40	17	25.83	27.02	30	30.10	29.95
5	30.00	29.49	18	28.82	25.96	31	30.58	30.01
6	30.05	29.86	19	28.72	28.83	32	27.81	30.37
7	29.10	29.77	20	27.34	28.56	33	27.28	27.81
8	30.37	29.03	21	27.50	27.51	34	27.47	27.24
9	30.00	30.30	22	27.39	27.44	35	28.33	27.40
10	29.91	29.90	23	27.09	27.68	36	27.15	28.44
11	28.12	30.06	24	26.00	27.17	37	26.90	27.24
12	30.65	28.12	25	26.31	25.86	38	26.80	26.83
13	30.53	30.94	26	25.18	26.27	39	26.64	26.96

Reel index values of HMC stock and approximate values of X^(t39)\hat X({t.{39}}) using EM method

Week	Reel Price	EM App.	Week	Reel Price	EM App.	Week	Reel Price	EM App.
1	28.60	28.60	14	30.04	30.64	27	26.26	25.09
2	28.60	28.49	15	28.24	30.43	28	29.12	26.53
3	29.49	28.57	16	27.04	28.06	29	30.06	29.21
4	29.49	29.70	17	25.83	26.80	30	30.10	30.17
5	30.00	29.57	18	28.82	25.97	31	30.58	30.23
6	30.05	30.20	19	28.72	29.02	32	27.81	30.66
7	29.10	30.44	20	27.34	28.59	33	27.28	28.01
8	30.37	29.03	21	27.50	27.32	34	27.47	27.33
9	30.00	30.56	22	27.39	27.83	35	28.33	27.21
10	29.91	29.54	23	27.09	27.56	36	27.15	28.47
11	28.12	29.93	24	26.00	26.94	37	26.90	27.12
12	30.65	28.18	25	26.31	26.02	38	26.80	26.67
13	30.53	30.47	26	25.18	26.39	39	26.64	26.65

Evaluation of performance of (MinMaxEnt)m, m = 1,2,3,4 distributions for X^(t16)\hat X({t.{16}})

(MinMaxEnt)_m	entropy	R²	RMSE	χ²	moment functions
(MinMaxEnt)₁	2.5532	0.2348	0.0562	0.1169	1, ln(1 + x²)
(MinMaxEnt)₂	2.5009	0.7036	0.0349	0.0379	1, x, lnx
(MinMaxEnt)₃	2.5008	0.7047	0.0348	0.0378	1, x, x², ln²(x)
(MinMaxEnt)₄	2.5010	0.7027	0.0349	0.0380	1, x, lnx, ln²(x), ln(1 + x²)

Evaluation of performance of (MaxMaxEnt)m, m = 1,2,3,4 distributions for X^(t39)\hat X({t.{39}})

(MinMaxEnt)_m	entropy	R²	RMSE	χ²	moment functions
(MaxMaxEnt)₁	2.5303	0.5900	0.0365	0.0630	1, x²
(MaxMaxEnt)₂	2.5278	0.6174	0.0352	0.0595	1, lnx, ln²(x)
(MaxMaxEnt)₃	2.5296	0.6678	0.0328	0.0519	1, x, x², ln(1 + x²)
(MaxMaxEnt)₄	2.5299	0.6437	0.0339	0.0555	1, x, x², ln²(x), ln(1 + x²)

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An Application of New Method to Obtain Probability Density Function of Solution of Stochastic Differential Equations

Published Online: Mar 31, 2020

Page range: 337 - 348

Received: Jun 15, 2019

Accepted: Aug 10, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00031

Keywords
generalized entropy optimization methods, stochastic differential equation model, Euler–Maruyama Method

© 2020 Nihal İnce et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Evaluation of performance of (MinMaxEnt)m, m = 1,2,3,4 distributions for X^(t39)\hat X({t.{39}})

Data set of weekly closing prices of HMC stock between 2 July 2018 to 25 March 2019

Evaluation of performance of (MaxMaxEnt)m, m = 1,2,3,4 distributions for X^(t16)\hat X({t.{16}})

Reel index values of HMC stock and approximate values of X^(t16)\hat X({t.{16}}) using EM method

Reel index values of HMC stock and approximate values of X^(t39)\hat X({t.{39}}) using EM method

Evaluation of performance of (MinMaxEnt)m, m = 1,2,3,4 distributions for X^(t16)\hat X({t.{16}})

Evaluation of performance of (MaxMaxEnt)m, m = 1,2,3,4 distributions for X^(t39)\hat X({t.{39}})

An Application of New Method to Obtain Probability Density Function of Solution of Stochastic Differential Equations

Published Online: Mar 31, 2020

Page range: 337 - 348

Received: Jun 15, 2019

Accepted: Aug 10, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00031

Keywordsgeneralized entropy optimization methods, stochastic differential equation model, Euler–Maruyama Method

© 2020 Nihal İnce et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Evaluation of performance of (MinMaxEnt)m, m = 1,2,3,4 distributions for X^(t39)\hat X({t.{39}})

Data set of weekly closing prices of HMC stock between 2 July 2018 to 25 March 2019

Evaluation of performance of (MaxMaxEnt)m, m = 1,2,3,4 distributions for X^(t16)\hat X({t.{16}})

Reel index values of HMC stock and approximate values of X^(t16)\hat X({t.{16}}) using EM method

Reel index values of HMC stock and approximate values of X^(t39)\hat X({t.{39}}) using EM method

Evaluation of performance of (MinMaxEnt)m, m = 1,2,3,4 distributions for X^(t16)\hat X({t.{16}})

Evaluation of performance of (MaxMaxEnt)m, m = 1,2,3,4 distributions for X^(t39)\hat X({t.{39}})

Keywords
generalized entropy optimization methods, stochastic differential equation model, Euler–Maruyama Method