# Approximate densities and random generation for first passage time in 3-D SDE's Kernel density and random generation for first-passage-time (f.p.t) in 3-dim stochastic differential equations. ```r fptsde3d(object, ...) dfptsde3d(object, ...) ## Default S3 method: fptsde3d(object, boundary, ...) ## S3 method for class 'fptsde3d' summary(object, digits=NULL, ...) ## S3 method for class 'fptsde3d' mean(x, ...) ## S3 method for class 'fptsde3d' Median(x, ...) ## S3 method for class 'fptsde3d' Mode(x, ...) ## S3 method for class 'fptsde3d' quantile(x, ...) ## S3 method for class 'fptsde3d' kurtosis(x, ...) ## S3 method for class 'fptsde3d' skewness(x, ...) ## S3 method for class 'fptsde3d' min(x, ...) ## S3 method for class 'fptsde3d' max(x, ...) ## S3 method for class 'fptsde3d' moment(x, ...) ## S3 method for class 'fptsde3d' cv(x, ...) ## Default S3 method: dfptsde3d(object, pdf=c("Joint","Marginal"), ...) ## S3 method for class 'dfptsde3d' plot(x,display="rgl",hist=FALSE, ...) ``` ## Arguments - `object`: an object inheriting from class `snssde3d` for `fptsde3d`, and `fptsde3d` for `dfptsde3d`. - `boundary`: an `expression` of a constant or time-dependent boundary. - `pdf`: probability density function `Joint` or `Marginal`. - `x`: an object inheriting from class `dfptsde3d`. - `digits`: integer, used for number formatting. - `display`: display plots. - `hist`: if `hist=TRUE` plot histogram. Based on `truehist` function. - ``...``: potentially arguments to be passed to methods, such as `density` for marginal density and `sm.density` for joint density. ## Details The function `fptsde3d` returns a random variable $(tau(X(t),S(t)),tau(Y(t),S(t)),tau(Z(t),S(t)))$ "first passage time", is defined as : $$ \tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \geq S(t) \},\quad if \quad X(t_{0}) < S(t_{0})tau(X(t),S(t))={t>=0; X(t) >= S(t)}, if X(t0) < S(t0) $$ $$ \tau_{(Y(t),S(t))} = \{ t \geq 0 ; Y_{t} \geq S(t) \},\quad if \quad Y(t_{0}) < S(t_{0})tau(Y(t),S(t))={t>=0; Y(t) >= S(t)}, if Y(t0) < S(t0) $$ $$ \tau_{(Z(t),S(t))} = \{ t \geq 0 ; Z_{t} \geq S(t) \},\quad if \quad Z(t_{0}) < S(t_{0})tau(Z(t),S(t))={t>=0; Z(t) >= S(t)}, if Z(t0) < S(t0) $$ and: $$ \tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \leq S(t) \},\quad if \quad X(t_{0}) > S(t_{0})tau(X(t),S(t))={t>=0; X(t) <= S(t)}, if X(t0) > S(t0) $$ $$ \tau_{(Y(t),S(t))} = \{ t \geq 0 ; Y_{t} \leq S(t) \},\quad if \quad Y(t_{0}) > S(t_{0})tau(Y(t),S(t))={t>=0; Y(t) <= S(t)}, if Y(t0) > S(t0) $$ $$ \tau_{(Z(t),S(t))} = \{ t \geq 0 ; Z_{t} \leq S(t) \},\quad if \quad Z(t_{0}) > S(t_{0})tau(Z(t),S(t))={t>=0; Z(t) <= S(t)}, if Z(t0) > S(t0) $$ And `dfptsde3d` returns a marginal kernel density approximation for $(tau(X(t),S(t)),tau(Y(t),S(t)),tau(Z(t),S(t)))$ "first passage time". with $S(t)$ is through a continuous boundary (barrier). An overview of this package, see `browseVignettes('Sim.DiffProc')` for more informations. ## Returns - **dfptsde3d()**: gives the marginal kernel density approximation for fpt. - **fptsde3d()**: generates random of fpt. ## References Argyrakisa, P. and G.H. Weiss (2006). A first-passage time problem for many random walkers. **Physica A**. 363 , 343--347. Aytug H., G. J. Koehler (2000). New stopping criterion for genetic algorithms. **European Journal of Operational Research**, 126 , 662--674. Boukhetala, K. (1996) Modelling and simulation of a dispersion pollutant with attractive centre. ed by Computational Mechanics Publications, Southampton ,U.K and Computational Mechanics Inc, Boston, USA, 245--252. Boukhetala, K. (1998a). Estimation of the first passage time distribution for a simulated diffusion process. **Maghreb Math.Rev**, 7 (1), 1--25. Boukhetala, K. (1998b). Kernel density of the exit time in a simulated diffusion. **les Annales Maghrebines De L ingenieur**, 12 , 587--589. Ding, M. and G. Rangarajan. (2004). First Passage Time Problem: A Fokker-Planck Approach. **New Directions in Statistical Physics**. ed by L. T. Wille. Springer. 31--46. 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. 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. Gardiner, C. W. (1997). **Handbook of Stochastic Methods**. Springer-Verlag, New York. ## Author(s) A.C. Guidoum, K. Boukhetala. ## See Also `fptsde1d` for simulation fpt in sde 1-dim. `fptsde2d` for simulation fpt in sde 2-dim. `FPTL` for computes values of the first passage time location (FPTL) function, and `Approx.fpt.density` for approximate first-passage-time (f.p.t.) density in package "fptdApprox". `GQD.TIpassage` for compute the First Passage Time Density of a GQD With Time Inhomogeneous Coefficients in package "DiffusionRgqd". ## Examples ```r ## 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) ## x0 = 0, y0 = -2, z0 = 0, and barrier -3+5*t. ## W1(t), W2(t) and W3(t) three independent Brownian motion set.seed(1234) # SDE's 3d fx <- expression(4*(-1-x)*y, 4*(1-y)*x, 4*(1-z)*y) gx <- rep(expression(0.2),3) mod3d <- snssde3d(drift=fx,diffusion=gx,M=500) # boundary St <- expression(-3+5*t) # random out <- fptsde3d(mod3d,boundary=St) out summary(out) # Marginal density denM <- dfptsde3d(out,pdf="M") denM plot(denM) # Multiple isosurfaces ## Not run: denJ <- dfptsde3d(out,pdf="J") denJ plot(denJ,display="rgl") ## End(Not run) ```

fptsde3d function