forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathornstein_uhlenbeck.html
284 lines (246 loc) · 7.58 KB
/
ornstein_uhlenbeck.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
<html>
<head>
<title>
ORNSTEIN_UHLENBECK - Approximate Solution of a Stochastic Differential Equation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
ORNSTEIN_UHLENBECK <br> Approximate Solution of a Stochastic Differential Equation
</h1>
<hr>
<p>
<b>ORNSTEIN_UHLENBECK</b>
is a C++ library which
approximates solutions of the Ornstein-Uhlenbeck
stochastic differential equation (SDE) using the Euler method
and the Euler-Maruyama method,
and creating graphics files for processing by gnuplot.
</p>
<p>
The Ornstein-Uhlenbeck stochastic differential equation has the form:
<pre>
dx(t) = theta * ( mu - x(t) ) dt + sigma dW,
x(0) = x0.
</pre>
where
<ul>
<li>
<b>theta</b> is a nonnegative decay rate;
</li>
<li>
<b>mu</b> is a mean value for x;
</li>
<li>
<b>sigma</b> measures the strength of the stochastic perturbation.
</li>
</ul>
and the equation is to be integrated over the interval [0,<b>tmax</b>].
</p>
<p>
The starting value <b>x0</b> represents a deviation from the mean value
<b>mu</b>.
The decay rate <b>theta</b> determines how fast <b>x(t)</b> will move
back towards its mean value. The coefficient <b>sigma</b> determines
the relative magnitude of stochastic perturbations.
</p>
<p>
In general, the solution starts at <b>x0</b> and over time moves towards
the value <b>mu</b>, but experiences random "wobbles" whose size is
determined by <b>sigma</b>. Increasing <b>theta</b> makes the solution
move towards the mean faster.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>ORNSTEIN_UHLENBECK</b> is available in
<a href = "../../c_src/ornstein_uhlenbeck/ornstein_uhlenbeck.html">a C version</a> and
<a href = "../../cpp_src/ornstein_uhlenbeck/ornstein_uhlenbeck.html">a C++ version</a> and
<a href = "../../f77_src/ornstein_uhlenbeck/ornstein_uhlenbeck.html">a FORTRAN77 version</a> and
<a href = "../../f_src/ornstein_uhlenbeck/ornstein_uhlenbeck.html">a FORTRAN90 version</a> and
<a href = "../../m_src/ornstein_uhlenbeck/ornstein_uhlenbeck.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/black_scholes/black_scholes.html">
BLACK_SCHOLES</a>,
a C++ library which
implements some simple approaches to
the Black-Scholes option valuation theory,
by Desmond Higham.
</p>
<p>
<a href = "../../cpp_src/brownian_motion_simulation/brownian_motion_simulation.html">
BROWNIAN_MOTION_SIMULATION</a>,
a C++ program which
simulates Brownian motion in an M-dimensional region.
</p>
<p>
<a href = "../../cpp_src/colored_noise/colored_noise.html">
COLORED_NOISE</a>,
a C++ library which
generates samples of noise obeying a 1/f^alpha power law.
</p>
<p>
<a href = "../../cpp_src/gnuplot/gnuplot.html">
GNUPLOT</a>,
C++ programs which
illustrate how a program can write data and command files
so that gnuplot can create plots of the program results.
</p>
<p>
<a href = "../../cpp_src/pink_noise/pink_noise.html">
PINK_NOISE</a>,
a C++ library which
computes a "pink noise" signal obeying a 1/f power law.
</p>
<p>
<a href = "../../cpp_src/sde/sde.html">
SDE</a>,
a C++ library which
illustrates the properties of stochastic differential equations (SDE's), and
common algorithms for their analysis, including the Euler method,
the Euler-Maruyama method, and the Milstein method,
by Desmond Higham;
</p>
<p>
<a href = "../../cpp_src/stochastic_rk/stochastic_rk.html">
STOCHASTIC_RK</a>,
a C++ library which
applies a Runge Kutta (RK) scheme to a stochastic differential equation.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Desmond Higham,<br>
An Algorithmic Introduction to Numerical Simulation of
Stochastic Differential Equations,<br>
SIAM Review,<br>
Volume 43, Number 3, September 2001, pages 525-546.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "ou.cpp">ou.cpp</a>, the source code.
</li>
<li>
<a href = "ou.hpp">ou.hpp</a>, the include file.
</li>
<li>
<a href = "ou.sh">ou.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "ou_prb.cpp">ou_prb.cpp</a>,
a sample calling program.
</li>
<li>
<a href = "ou_prb.sh">ou_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "ou_prb_output.txt">ou_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<p>
<b>OU_EULER</b> is data created by the OU_EULER routine:
<ul>
<li>
<a href = "ou_euler_data.txt">ou_euler_data.txt</a>,
solution data.
</li>
<li>
<a href = "ou_euler_commands.txt">ou_euler_commands.txt</a>,
gnuplot commands to plot the data
</li>
<li>
<a href = "ou_euler.png">ou_euler.png</a>,
a PNG image of the data.
</li>
</ul>
</p>
<p>
<b>OU_EULER_MARUYAMA</b> is data created by the OU_EULER_MARUYAMA routine:
<ul>
<li>
<a href = "ou_euler_maruyama_data.txt">ou_euler_maruyama_data.txt</a>,
solution data.
</li>
<li>
<a href = "ou_euler_maruyama_commands.txt">ou_euler_maruyama_commands.txt</a>,
gnuplot commands to plot the data
</li>
<li>
<a href = "ou_euler_maruyama.png">ou_euler_maruyama.png</a>,
a PNG image of the data.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>OU_EULER</b> applies the Euler method to the Ornstein-Uhlenbeck SDE.
</li>
<li>
<b>OU_EULER_MARUYAMA</b> applies Euler-Maruyama to the Ornstein-Uhlenbeck SDE.
</li>
<li>
<b>R8_UNIFORM_01</b> returns a unit pseudorandom R8.
</li>
<li>
<b>R8VEC_LINSPACE</b> creates a vector of linearly spaced values.
</li>
<li>
<b>R8VEC_NORMAL_01</b> returns a unit pseudonormal R8VEC.
</li>
<li>
<b>R8VEC_UNIFORM_01</b> returns a unit pseudorandom R8VEC.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 21 January 2013.
</i>
<!-- John Burkardt -->
</body>
<!-- Initial HTML skeleton created by HTMLINDEX. -->
</html>