# Approximate densities and random generation for first passage time in 1-D SDE Kernel density and random generation for first-passage-time (f.p.t) in 1-dim stochastic differential equations. ```r fptsde1d(object, ...) dfptsde1d(object, ...) ## Default S3 method: fptsde1d(object, boundary, ...) ## S3 method for class 'fptsde1d' summary(object, digits=NULL, ...) ## S3 method for class 'fptsde1d' mean(x, ...) ## S3 method for class 'fptsde1d' Median(x, ...) ## S3 method for class 'fptsde1d' Mode(x, ...) ## S3 method for class 'fptsde1d' quantile(x, ...) ## S3 method for class 'fptsde1d' kurtosis(x, ...) ## S3 method for class 'fptsde1d' skewness(x, ...) ## S3 method for class 'fptsde1d' min(x, ...) ## S3 method for class 'fptsde1d' max(x, ...) ## S3 method for class 'fptsde1d' moment(x, ...) ## S3 method for class 'fptsde1d' cv(x, ...) ## Default S3 method: dfptsde1d(object, ...) ## S3 method for class 'dfptsde1d' plot(x, hist=FALSE, ...) ``` ## Arguments - `object`: an object inheriting from class `snssde1d` for `fptsde1d`, and `fptsde1d` for `dfptsde1d`. - `boundary`: an `expression` of a constant or time-dependent boundary. - `x`: an object inheriting from class `dfptsde1d`. - `hist`: if `hist=TRUE` plot histogram. Based on `truehist` function. - `digits`: integer, used for number formatting. - ``...``: potentially further arguments for (non-default) methods, such as `density` for `dfptsde1d`. ## Details The function `fptsde1d` returns a random variable $tau(X(t),S(t))$ "first passage time", is defined as : $$ \tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \geq S(t) \},\quad if \quad X(t_{0}) < S(t_{0})tau(X(t),S(t))={t>=0; X(t) >= S(t)}, if X(t0) < S(t0) $$ $$ \tau_{(X(t),S(t))} = \{ t \geq 0 ; X_{t} \leq S(t) \},\quad if \quad X(t_{0}) > S(t_{0})tau(X(t),S(t))={t>=0; X(t) <= S(t)}, if X(t0) > S(t0) $$ And `dfptsde1d` returns a kernel density approximation for $tau(X(t),S(t))$ "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 - **dfptsde1d()**: gives the density estimate of fpt. - **fptsde1d()**: 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 `fptsde2d` and `fptsde3d` simulation fpt for 2 and 3-dim SDE. `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 ## Example 1: Ito SDE ## dX(t) = -4*X(t) *dt + 0.5*dW(t) ## S(t) = 0 (constant boundary) set.seed(1234) # SDE 1d f <- expression( -4*x ) g <- expression( 0.5 ) mod <- snssde1d(drift=f,diffusion=g,x0=2,M=1000) # boundary St <- expression(0) # random out <- fptsde1d(mod, boundary=St) out summary(out) # density approximate den <- dfptsde1d(out) den plot(den) ## Example 2: Stratonovich SDE ## dX(t) = 0.5*X(t)*t *dt + sqrt(1+X(t)^2) o dW(t) ## S(t) = -0.5*sqrt(t) + exp(t^2) (time-dependent boundary) set.seed(1234) # SDE 1d f <- expression( 0.5*x*t ) g <- expression( sqrt(1+x^2) ) mod2 <- snssde1d(drift=f,diffusion=g,x0=2,M=1000,type="srt") # boundary St <- expression(-0.5*sqrt(t)+exp(t^2)) # random out2 <- fptsde1d(mod2,boundary=St) out2 summary(out2) # density approximate plot(dfptsde1d(out2,bw='ucv')) ## Example 3: fptsde1d vs fptdApproximate ## Not run: f <- expression( -0.5*x+0.5*5 ) g <- expression( 1 ) St <- expression(5+0.25*sin(2*pi*t)) mod <- snssde1d(drift=f,diffusion=g,boundary=St,x0=3,T=10,N=10^4,M =10000) mod # random out3 <- fptsde1d(mod,boundary=St) out3 summary(out3) # density approximate: library("fptdApprox") # Under `fptdApprox': # Define the diffusion process and give its transitional density: OU <- diffproc(c("alpha*x + beta","sigma^2", "dnorm((x-(y*exp(alpha*(t-s)) - beta*(1 - exp(alpha*(t-s)))/alpha))/ (sigma*sqrt((exp(2*alpha*(t-s)) - 1)/(2*alpha))),0,1)/ (sigma*sqrt((exp(2*alpha*(t-s)) - 1)/(2*alpha)))", "pnorm(x, y*exp(alpha*(t-s)) - beta*(1 - exp(alpha*(t-s)))/alpha, sigma*sqrt((exp(2*alpha*(t-s)) - 1)/(2*alpha)))")) # Approximate the first passgage time density for OU, starting in X_0 = 3 # passing through 5+0.25*sin(2*pi*t) on the time interval [0,10]: res <- Approx.fpt.density(OU, 0, 10, 3,"5+0.25*sin(2*pi*t)", list(alpha=-0.5,beta=0.5*5,sigma=1)) ## plot(dfptsde1d(out3,bw='ucv'),main = 'fptsde1d vs fptdApproximate') lines(res$y~res$x, type = 'l',lwd=2) legend('topright', lty = c('solid', 'dashed'), col = c(1, 2), legend = c('fptdApproximate', 'fptsde1d'), lwd = 2, bty = 'n') ## End(Not run) ```

fptsde1d function