# Approximate densities and random generation for first passage time in 2-D SDE's Kernel density and random generation for first-passage-time (f.p.t) in 2-dim stochastic differential equations. ```r fptsde2d(object, ...) dfptsde2d(object, ...) ## Default S3 method: fptsde2d(object, boundary, ...) ## S3 method for class 'fptsde2d' summary(object, digits=NULL, ...) ## S3 method for class 'fptsde2d' mean(x, ...) ## S3 method for class 'fptsde2d' Median(x, ...) ## S3 method for class 'fptsde2d' Mode(x, ...) ## S3 method for class 'fptsde2d' quantile(x, ...) ## S3 method for class 'fptsde2d' kurtosis(x, ...) ## S3 method for class 'fptsde2d' skewness(x, ...) ## S3 method for class 'fptsde2d' min(x, ...) ## S3 method for class 'fptsde2d' max(x, ...) ## S3 method for class 'fptsde2d' moment(x, ...) ## S3 method for class 'fptsde2d' cv(x, ...) ## Default S3 method: dfptsde2d(object, pdf=c("Joint","Marginal"), ...) ## S3 method for class 'dfptsde2d' plot(x,display=c("persp","rgl","image","contour"), hist=FALSE, ...) ``` ## Arguments - `object`: an object inheriting from class `snssde2d` for `fptsde2d`, and `fptsde2d` for `dfptsde2d`. - `boundary`: an `expression` of a constant or time-dependent boundary. - `pdf`: probability density function `Joint` or `Marginal`. - `x`: an object inheriting from class `fptsde2d`. - `digits`: integer, used for number formatting. - `display`: display plots. - `hist`: if `hist=TRUE` plot histogram. Based on `truehist` function. - ``...``: potentially further arguments for (non-default) methods. arguments to be passed to methods, such as `density` for marginal density and `kde2d` fro joint density. ## Details The function `fptsde1d` returns a random variable $(tau(X(t),S(t)),tau(Y(t),S(t)))$ "first passage time", is defined as : $$ \tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \geq S(t) \},\quad if \quad X(t_{0}) < S(t_{0})tau(X(t),S(t))={t>=0; X(t) >= S(t)}, if X(t0) < S(t0) $$ $$ \tau_{(Y(t),S(t))} = \{ t \geq 0 ; Y_{t} \geq S(t) \},\quad if \quad Y(t_{0}) < S(t_{0})tau(Y(t),S(t))={t>=0; Y(t) >= S(t)}, if Y(t0) < S(t0) $$ and: $$ \tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \leq S(t) \},\quad if \quad X(t_{0}) > S(t_{0})tau(X(t),S(t))={t>=0; X(t) <= S(t)}, if X(t0) > S(t0) $$ $$ \tau_{(Y(t),S(t))} = \{ t \geq 0 ; Y_{t} \leq S(t) \},\quad if \quad Y(t_{0}) > S(t_{0})tau(Y(t),S(t))={t>=0; Y(t) <= S(t)}, if Y(t0) > S(t0) $$ And `dfptsde2d` returns a kernel density approximation for $(tau(X(t),S(t)),tau(Y(t),S(t)))$ "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 - **dfptsde2d()**: gives the kernel density approximation for fpt. - **fptsde2d()**: 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. `fptsde3d` for simulation fpt in sde 3-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) = 5*(-1-Y(t))*X(t) * dt + 0.5 * dW1(t) ## dY(t) = 5*(-1-X(t))*Y(t) * dt + 0.5 * dW2(t) ## x0 = 2, y0 = -2, and barrier -3+5*t. ## W1(t) and W2(t) two independent Brownian motion set.seed(1234) # SDE's 2d fx <- expression(5*(-1-y)*x , 5*(-1-x)*y) gx <- expression(0.5 , 0.5) mod2d <- snssde2d(drift=fx,diffusion=gx,x0=c(2,-2),M=100) # boundary St <- expression(-1+5*t) # random fpt out <- fptsde2d(mod2d,boundary=St) out summary(out) # Marginal density denM <- dfptsde2d(out,pdf="M") denM plot(denM) # Joint density denJ <- dfptsde2d(out,pdf="J",n=200,lims=c(0.28,0.4,0.04,0.13)) denJ plot(denJ) plot(denJ,display="image") plot(denJ,display="image",drawpoints=TRUE,cex=0.5,pch=19,col.pt='green') plot(denJ,display="contour") plot(denJ,display="contour",color.palette=colorRampPalette(c('white','green','blue','red'))) ```

fptsde2d function