Monte-Carlo Simulation and Kernel Density Estimation of First passage time

A.C. Guidoum1 and K. Boukhetala2

2019-05-27

The fptsdekd() functions

A new algorithm based on the Monte Carlo technique to generate the random variable FPT of a time homogeneous diffusion process (1, 2 and 3D) through a time-dependent boundary, order to estimate her probability density function.

Let \(X_t\) be a diffusion process which is the unique solution of the following stochastic differential equation:

\[\begin{equation}\label{eds01} dX_t = \mu(t,X_t) dt + \sigma(t,X_t) dW_t,\quad X_{t_{0}}=x_{0} \end{equation}\]

if \(S(t)\) is a time-dependent boundary, we are interested in generating the first passage time (FPT) of the diffusion process through this boundary that is we will study the following random variable:

\[ \tau_{S(t)}= \left\{ \begin{array}{ll} inf \left\{t: X_{t} \geq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \leq S(t_{0}) \\ inf \left\{t: X_{t} \leq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \geq S(t_{0}) \end{array} \right. \]

The main arguments to ‘random’ fptsdekd() (where k=1,2,3) consist:

The following statistical measures (S3 method) for class fptsdekd() can be approximated for F.P.T \(\tau_{S(t)}\):

The main arguments to ‘density’ dfptsdekd() (where k=1,2,3) consist:

Examples

FPT for 1-Dim SDE

Consider the following SDE and linear boundary:

\[\begin{align*} dX_{t}= & (1-0.5 X_{t}) dt + dW_{t},~x_{0} =1.7.\\ S(t)= & 2(1-sinh(0.5t)) \end{align*}\]

Generating the first passage time (FPT) of this model through this boundary: \[ \tau_{S(t)}= \inf \left\{t: X_{t} \geq S(t) |X_{t_{0}}=x_{0} \right\} ~~ \text{if} \quad x_{0} \leq S(t_{0}) \]

Set the model \(X_t\):

Generate the first-passage-time \(\tau_{S(t)}\), with fptsde1d() function ( based on density() function in [base] package):

Itô Sde 1D:
    | dX(t) = (1 - 0.5 * X(t)) * dt + 1 * dW(t)
    | t in [0,1].
Boundary:
    | S(t) = 2 * (1 - sinh(0.5 * t))
F.P.T:
    | T(S(t),X(t)) = inf{t >=  0 : X(t) >=  2 * (1 - sinh(0.5 * t)) }
    | Crossing realized 966 among 1000.
 [1] 0.1339457 0.0403355 0.0185658 0.1145573 0.0091086 0.1376444
 [7] 0.6836460 0.2273584 0.0287167 0.0967543

The following statistical measures (S3 method) for class fptsde1d() can be approximated for the first-passage-time \(\tau_{S(t)}\):

[1] 0.20261
[1] 0.046399
[1] 0.11883
[1] 0.056178
       0%       25%       50%       75%      100% 
0.0069061 0.0539587 0.1188284 0.2503842 0.9993877 
[1] 5.5196
[1] 1.7513
[1] 1.0637
[1] 0.0069061
[1] 0.99939
[1] 0.011908
[1] 0.039228

The result summaries of the first-passage-time \(\tau_{S(t)}\):


Monte-Carlo Statistics of F.P.T:
|T(S(t),X(t)) = inf{t >=  0 : X(t) >=  2 * (1 - sinh(0.5 * t)) }
                        
Mean             0.20261
Variance         0.04645
Median           0.11883
Mode             0.05618
First quartile   0.05396
Third quartile   0.25038
Minimum          0.00691
Maximum          0.99939
Skewness         1.75134
Kurtosis         5.51959
Coef-variation   1.06372
3th-order moment 0.01753
4th-order moment 0.01191
5th-order moment 0.00749
6th-order moment 0.00508

Display the exact first-passage-time \(\tau_{S(t)}\), see Figure 1:

The kernel density approximation of ‘fpt1d’, using dfptsde1d() function (hist=TRUE based on truehist() function in MASS package), see e.g. Figure 2.

Since fptdApprox and DiffusionRgqd packages can very effectively handle first passage time problems for diffusions with analytically tractable transitional densities we use it to compare some of the results from the Sim.DiffProc package.

fptsde1d() vs Approx.fpt.density()

Consider for example a diffusion process with SDE:

\[\begin{align*} dX_{t}= & 0.48 X_{t} dt + 0.07 X_{t} dW_{t},~x_{0} =1.\\ S(t)= & 7 + 3.2 t + 1.4 t \sin(1.75 t) \end{align*}\] The resulting object is then used by the Approx.fpt.density() function in package fptdApprox to approximate the first passage time density:

Using fptsde1d() and dfptsde1d() functions in the Sim.DiffProc package:

Itô Sde 1D:
    | dX(t) = 0.48 * X(t) * dt + 0.07 * X(t) * dW(t)
    | t in [0,10].
Boundary:
    | S(t) = 7 + 3.2 * t + 1.4 * t * sin(1.75 * t)
F.P.T:
    | T(S(t),X(t)) = inf{t >=  0 : X(t) >=  7 + 3.2 * t + 1.4 * t * sin(1.75 * t) }
    | Crossing realized 1000 among 1000.
 [1] 5.9592 6.2240 6.0662 6.1222 6.1849 6.0208 5.9970 6.0740 6.1962
[10] 6.3749

Monte-Carlo Statistics of F.P.T:
|T(S(t),X(t)) = inf{t >=  0 : X(t) >=  7 + 3.2 * t + 1.4 * t * sin(1.75 * t) }
                         
Mean              6.51837
Variance          0.90712
Median            6.11896
Mode              6.03207
First quartile    5.95884
Third quartile    6.40436
Minimum           5.40261
Maximum           8.95967
Skewness          1.45679
Kurtosis          3.40877
Coef-variation    0.14611
3th-order moment  1.25861
4th-order moment  2.80494
5th-order moment  5.37535
6th-order moment 10.95455

By plotting the approximations:

fptsde1d() vs Approx.fpt.density()

fptsde1d() vs Approx.fpt.density()

fptsde1d() vs GQD.TIpassage()

Consider for example a diffusion process with SDE:

\[\begin{align*} dX_{t}= & \theta_{1}X_{t}(10+0.2\sin(2\pi t)+0.3\sqrt(t)(1+\cos(3\pi t))-X_{t}) ) dt + \sqrt(0.1) X_{t} dW_{t},~x_{0} =8.\\ S(t)= & 12 \end{align*}\] The resulting object is then used by the GQD.TIpassage() function in package DiffusionRgqd to approximate the first passage time density:

Using fptsde1d() and dfptsde1d() functions in the Sim.DiffProc package:

Itô Sde 1D:
    | dX(t) = theta1 * X(t) * (10 + 0.2 * sin(2 * pi * t) + 0.3 * sqrt(t) *     (1 + cos(3 * pi * t)) - X(t)) * dt + sqrt(0.1) * X(t) * dW(t)
    | t in [1,4].
Boundary:
    | S(t) = 12
F.P.T:
    | T(S(t),X(t)) = inf{t >=  1 : X(t) >=  12 }
    | Crossing realized 923 among 1000.
 [1] 2.6795 1.4390 1.8486 1.4483 1.4806 1.3327 2.5660 1.6606 3.2797
[10] 2.1307

Monte-Carlo Statistics of F.P.T:
|T(S(t),X(t)) = inf{t >=  1 : X(t) >=  12 }
                        
Mean             2.18245
Variance         0.51251
Median           2.06456
Mode             1.46359
First quartile   1.54973
Third quartile   2.65918
Minimum          1.15462
Maximum          3.99958
Skewness         0.66462
Kurtosis         2.52209
Coef-variation   0.32802
3th-order moment 0.24385
4th-order moment 0.66246
5th-order moment 0.70278
6th-order moment 1.30438

By plotting the approximations (hist=TRUE based on truehist() function in MASS package):

fptsde1d() vs GQD.TIpassage()

fptsde1d() vs GQD.TIpassage()

FPT for 2-Dim SDE’s

The following \(2\)-dimensional SDE’s with a vector of drift and a diagonal matrix of diffusion coefficients:

\[\begin{equation}\label{eq:09} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t}) dt + g_{x}(t,X_{t},Y_{t}) dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t}) dt + g_{y}(t,X_{t},Y_{t}) dW_{2,t} \end{cases} \end{equation}\]

\(W_{1,t}\) and \(W_{2,t}\) is a two independent standard Wiener process. First passage time (2D) \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})\) is defined as:

\[ \left\{ \begin{array}{ll} \tau_{S(t),X_{t}}=\inf \left\{t: X_{t} \geq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \leq S(t_{0}) \\ \tau_{S(t),Y_{t}}=\inf \left\{t: Y_{t} \geq S(t)|Y_{t_{0}}=y_{0} \right\} & \hbox{if} \quad y_{0} \leq S(t_{0}) \end{array} \right. \] and \[ \left\{ \begin{array}{ll} \tau_{S(t),X_{t}}= \inf \left\{t: X_{t} \leq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \geq S(t_{0}) \\ \tau_{S(t),Y_{t}}= \inf \left\{t: Y_{t} \leq S(t)|Y_{t_{0}}=y_{0} \right\} & \hbox{if} \quad y_{0} \geq S(t_{0}) \end{array} \right. \]

Assume that we want to describe the following Stratonovich SDE’s (2D):

\[\begin{equation}\label{eq016} \begin{cases} dX_t = 5 (-1-Y_{t}) X_{t} dt + 0.5 Y_{t} \circ dW_{1,t}\\ dY_t = 5 (-1-X_{t}) Y_{t} dt + 0.5 X_{t} \circ dW_{2,t} \end{cases} \end{equation}\]

and \[ S(t)=\sin(2\pi t) \]

Set the system \((X_t , Y_t)\):

Generate the couple \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})\), with fptsde2d() function::

Stratonovich Sde 2D:
    | dX(t) = 5 * (-1 - Y(t)) * X(t) * dt + 0.5 * Y(t) o dW1(t)
    | dY(t) = 5 * (-1 - X(t)) * Y(t) * dt + 0.5 * X(t) o dW2(t)
    | t in [0,1].
Boundary:
    | S(t) = sin(2 * pi * t)
F.P.T:
    | T(S(t),X(t)) = inf{t >=  0 : X(t) <=  sin(2 * pi * t) }
    |   And 
    | T(S(t),Y(t)) = inf{t >=  0 : Y(t) >=  sin(2 * pi * t) }
    | Crossing realized 1000 among 1000.
         x       y
1  0.12680 0.50495
2  0.13276 0.50512
3  0.14800 0.50806
4  0.12442 0.50139
5  0.13105 0.50873
6  0.14841 0.50785
7  0.14134 0.49099
8  0.14555 0.51156
9  0.13435 0.51354
10 0.11551 0.49764

The following statistical measures (S3 method) for class fptsde2d() can be approximated for the couple \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})\):

[1] 0.13377 0.50309
[1] 0.000185360 0.000028352
[1] 0.13266 0.50319
[1] 0.13091 0.50198
$x
      0%      25%      50%      75%     100% 
0.094199 0.124116 0.132663 0.142240 0.186445 

$y
     0%     25%     50%     75%    100% 
0.48659 0.49997 0.50319 0.50659 0.52154 
[1] 3.3985 3.2624
[1]  0.378860 -0.084235
[1] 0.101825 0.010589
[1] 0.094199 0.486586
[1] 0.18645 0.52154
[1] 0.0000001170023 0.0000000026277
[1] 0.00034078 0.06410149

The result summaries of the couple \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})\):


Monte-Carlo Statistics for the F.P.T of (X(t),Y(t))
    | T(S(t),X(t)) = inf{t >=  0 : X(t) <=  sin(2 * pi * t) }
    |    And
    | T(S(t),Y(t)) = inf{t >=  0 : Y(t) >=  sin(2 * pi * t) }
                  T(S,X)   T(S,Y)
Mean             0.13377  0.50309
Variance         0.00019  0.00003
Median           0.13266  0.50319
Mode             0.13091  0.50198
First quartile   0.12412  0.49997
Third quartile   0.14224  0.50659
Minimum          0.09420  0.48659
Maximum          0.18645  0.52154
Skewness         0.37886 -0.08423
Kurtosis         3.39853  3.26240
Coef-variation   0.10182  0.01059
3th-order moment 0.00000  0.00000
4th-order moment 0.00000  0.00000
5th-order moment 0.00000  0.00000
6th-order moment 0.00000  0.00000

Display the exact first-passage-time \(\tau_{S(t)}\), see Figure 5:

The marginal density of \((\tau_{(S(t),X_{t})}\) and \(\tau_{(S(t),Y_{t})})\) are reported using dfptsde2d() function, see e.g. Figure 6.

A contour and image plot of density obtained from a realization of system \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})})\).

A \(3\)D plot of the Joint density with:

Return to fptsde2d()

FPT for 3-Dim SDE’s

The following \(3\)-dimensional SDE’s with a vector of drift and a diagonal matrix of diffusion coefficients:

\[\begin{equation}\label{eq17} \begin{cases} dX_t = f_{x}(t,X_{t},Y_{t},Z_{t}) dt + g_{x}(t,X_{t},Y_{t},Z_{t}) dW_{1,t}\\ dY_t = f_{y}(t,X_{t},Y_{t},Z_{t}) dt + g_{y}(t,X_{t},Y_{t},Z_{t}) dW_{2,t}\\ dZ_t = f_{z}(t,X_{t},Y_{t},Z_{t}) dt + g_{z}(t,X_{t},Y_{t},Z_{t}) dW_{3,t} \end{cases} \end{equation}\] \(W_{1,t}\), \(W_{2,t}\) and \(W_{3,t}\) is a 3 independent standard Wiener process. First passage time (3D) \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})\) is defined as:

\[ \left\{ \begin{array}{ll} \tau_{S(t),X_{t}}=\inf \left\{t: X_{t} \geq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \leq S(t_{0}) \\ \tau_{S(t),Y_{t}}=\inf \left\{t: Y_{t} \geq S(t)|Y_{t_{0}}=y_{0} \right\} & \hbox{if} \quad y_{0} \leq S(t_{0}) \\ \tau_{S(t),Z_{t}}=\inf \left\{t: Z_{t} \geq S(t)|Z_{t_{0}}=z_{0} \right\} & \hbox{if} \quad z_{0} \leq S(t_{0}) \end{array} \right. \] and \[ \left\{ \begin{array}{ll} \tau_{S(t),X_{t}}= \inf \left\{t: X_{t} \leq S(t)|X_{t_{0}}=x_{0} \right\} & \hbox{if} \quad x_{0} \geq S(t_{0}) \\ \tau_{S(t),Y_{t}}= \inf \left\{t: Y_{t} \leq S(t)|Y_{t_{0}}=y_{0} \right\} & \hbox{if} \quad y_{0} \geq S(t_{0}) \\ \tau_{S(t),Z_{t}}= \inf \left\{t: Z_{t} \leq S(t)|Z_{t_{0}}=z_{0} \right\} & \hbox{if} \quad z_{0} \geq S(t_{0}) \\ \end{array} \right. \]

Assume that we want to describe the following SDE’s (3D): \[\begin{equation}\label{eq0166} \begin{cases} dX_t = 4 (-1-X_{t}) Y_{t} dt + 0.2 dW_{1,t}\\ dY_t = 4 (1-Y_{t}) X_{t} dt + 0.2 dW_{2,t}\\ dZ_t = 4 (1-Z_{t}) Y_{t} dt + 0.2 dW_{3,t} \end{cases} \end{equation}\] and \[ S(t)=-1.5+3t \]

Set the system \((X_t , Y_t , Z_t)\):

Generate the triplet \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})\), with fptsde3d() function::

Itô Sde 3D:
    | dX(t) = 4 * (-1 - X(t)) * Y(t) * dt + 0.2 * dW1(t)
    | dY(t) = 4 * (1 - Y(t)) * X(t) * dt + 0.2 * dW2(t)
    | dZ(t) = 4 * (1 - Z(t)) * Y(t) * dt + 0.2 * dW3(t)
    | t in [0,1].
Boundary:
    | S(t) = -1.5 + 3 * t
F.P.T:
    | T(S(t),X(t)) = inf{t >=  0 : X(t) <=  -1.5 + 3 * t }
    |   And 
    | T(S(t),Y(t)) = inf{t >=  0 : Y(t) >=  -1.5 + 3 * t }
    |   And 
    | T(S(t),Z(t)) = inf{t >=  0 : Z(t) <=  -1.5 + 3 * t }
    | Crossing realized 1000 among 1000.
         x        y       z
1  0.55604 0.024212 0.74398
2  0.57666 0.023595 0.76097
3  0.52223 0.024165 0.76671
4  0.51569 0.025876 0.81817
5  0.54366 0.020797 0.72875
6  0.52437 0.025312 0.78193
7  0.54578 0.024291 0.85570
8  0.55933 0.023881 0.74460
9  0.53802 0.021959 0.81993
10 0.52354 0.026541 0.79178

The following statistical measures (S3 method) for class fptsde3d() can be approximated for the triplet \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})\):

[1] 0.531421 0.023273 0.783244
[1] 0.0001797741 0.0000016317 0.0010413796
[1] 0.531635 0.023249 0.784739
[1] 0.532304 0.023071 0.786603
$x
     0%     25%     50%     75%    100% 
0.48934 0.52283 0.53163 0.53961 0.57670 

$y
      0%      25%      50%      75%     100% 
0.018989 0.022437 0.023249 0.024127 0.027370 

$z
     0%     25%     50%     75%    100% 
0.66295 0.76184 0.78474 0.80541 0.87300 
[1] 3.3214 2.9400 3.0973
[1]  0.078856  0.055384 -0.234933
[1] 0.025243 0.054913 0.041222
[1] 0.489340 0.018989 0.662954
[1] 0.57670 0.02737 0.87300
[1] 0.000000107559478 0.000000000007843 0.000003365691761
[1] 0.0800593358 0.0000002987 0.3801585251

The result summaries of the triplet \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})\):


Monte-Carlo Statistics for the F.P.T of (X(t),Y(t),Z(t))
    | T(S(t),X(t)) = inf{t >=  0 : X(t) <=  -1.5 + 3 * t }
    |    And
    | T(S(t),Y(t)) = inf{t >=  0 : Y(t) >=  -1.5 + 3 * t }
    |    And
    | T(S(t),Z(t)) = inf{t >=  0 : Z(t) <=  -1.5 + 3 * t }
                  T(S,X)  T(S,Y)   T(S,Z)
Mean             0.53142 0.02327  0.78324
Variance         0.00018 0.00000  0.00104
Median           0.53163 0.02325  0.78474
Mode             0.53230 0.02307  0.78660
First quartile   0.52283 0.02244  0.76184
Third quartile   0.53961 0.02413  0.80541
Minimum          0.48934 0.01899  0.66295
Maximum          0.57670 0.02737  0.87300
Skewness         0.07886 0.05538 -0.23493
Kurtosis         3.32143 2.93997  3.09733
Coef-variation   0.02524 0.05491  0.04122
3th-order moment 0.00000 0.00000 -0.00001
4th-order moment 0.00000 0.00000  0.00000
5th-order moment 0.00000 0.00000  0.00000
6th-order moment 0.00000 0.00000  0.00000

Display the exact first-passage-time \(\tau_{S(t)}\), see Figure 9:

The marginal density of \(\tau_{(S(t),X_{t})}\) ,\(\tau_{(S(t),Y_{t})}\) and \(\tau_{(S(t),Z_{t})})\) are reported using dfptsde3d() function, see e.g. Figure 10.


Marginal density for the F.P.T of X(t)
    | T(S,X) = inf{t >= 0 : X(t) <= -1.5 + 3 * t}

Data: out[, "x"] (1000 obs.);   Bandwidth 'bw' = 0.0028298

       x                f(x)       
 Min.   :0.48085   Min.   : 0.002  
 1st Qu.:0.50693   1st Qu.: 0.564  
 Median :0.53302   Median : 4.417  
 Mean   :0.53302   Mean   : 9.575  
 3rd Qu.:0.55910   3rd Qu.:17.384  
 Max.   :0.58519   Max.   :34.315  

Marginal density for the F.P.T of Y(t)
    | T(S,Y) = inf{t >= 0 : Y(t) >= -1.5 + 3 * t}

Data: out[, "y"] (1000 obs.);   Bandwidth 'bw' = 0.00028503

       y                 f(y)       
 Min.   :0.018133   Min.   :  0.02  
 1st Qu.:0.020656   1st Qu.:  3.27  
 Median :0.023179   Median : 50.70  
 Mean   :0.023179   Mean   : 98.99  
 3rd Qu.:0.025702   3rd Qu.:177.25  
 Max.   :0.028225   Max.   :317.12  

Marginal density for the F.P.T of Z(t)
    | T(S,Z) = inf{t >= 0 : Z(t) <= -1.5 + 3 * t}

Data: out[, "z"] (1000 obs.);   Bandwidth 'bw' = 0.007299

       z                 f(z)       
 Min.   :0.018133   Min.   :  0.02  
 1st Qu.:0.020656   1st Qu.:  3.27  
 Median :0.023179   Median : 50.70  
 Mean   :0.023179   Mean   : 98.99  
 3rd Qu.:0.025702   3rd Qu.:177.25  
 Max.   :0.028225   Max.   :317.12  

For an approximate joint density for \((\tau_{(S(t),X_{t})},\tau_{(S(t),Y_{t})},\tau_{(S(t),Z_{t})})\) (for more details, see package sm or ks.)

Return to fptsde3d()

Further reading

  1. snssdekd() & dsdekd() & rsdekd()- Monte-Carlo Simulation and Analysis of Stochastic Differential Equations.
  2. bridgesdekd() & dsdekd() & rsdekd() - Constructs and Analysis of Bridges Stochastic Differential Equations.
  3. fptsdekd() & dfptsdekd() - Monte-Carlo Simulation and Kernel Density Estimation of First passage time.
  4. MCM.sde() & MEM.sde() - Parallel Monte-Carlo and Moment Equations for SDEs.
  5. TEX.sde() - Converting Sim.DiffProc Objects to LaTeX.
  6. fitsde() - Parametric Estimation of 1-D Stochastic Differential Equation.

References

  1. Boukhetala K (1996). Modelling and Simulation of a Dispersion Pollutant with Attractive Centre, volume 3, pp. 245-252. Computer Methods and Water Resources, Computational Mechanics Publications, Boston, USA.

  2. Boukhetala K (1998). Estimation of the first passage time distribution for a simulated diffusion process. Maghreb Mathematical Review, 7, pp. 1-25.

  3. Boukhetala K (1998). Kernel density of the exit time in a simulated diffusion. The Annals of The Engineer Maghrebian, 12, pp. 587-589.

  4. Guidoum AC, Boukhetala K (2019). Sim.DiffProc: Simulation of Diffusion Processes. R package version 4.4, URL https://cran.r-project.org/package=Sim.DiffProc.

  5. Pienaar EAD, Varughese MM (2016). DiffusionRgqd: An R Package for Performing Inference and Analysis on Time-Inhomogeneous Quadratic Diffusion Processes. R package version 0.1.3, URL https://CRAN.R-project.org/package=DiffusionRgqd.

  6. Roman, R.P., Serrano, J. J., Torres, F. (2008). First-passage-time location function: Application to determine first-passage-time densities in diffusion processes. Computational Statistics and Data Analysis. 52, 4132-4146.

  7. Roman, R.P., Serrano, J. J., Torres, F. (2012). An R package for an efficient approximation of first-passage-time densities for diffusion processes based on the FPTL function. Applied Mathematics and Computation, 218, 8408-8428.


  1. Department of Probabilities & Statistics, Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail ()

  2. Faculty of Mathematics, University of Science and Technology Houari Boumediene, BP 32 El-Alia, U.S.T.H.B, Algeria, E-mail ()