@@ -14,7 +14,7 @@ f = (u,p,t) -> 1.01u
1414σ = (u,p,t) -> 0.87 u
1515(ff:: typeof (f))(:: Type{Val{:analytic}} ,u0,p,t,W) = u0.* exp .(0.63155 t+ 0.87 W)
1616
17- doc"""
17+ @doc doc"""
1818```math
1919du_t = βudt + αudW_t
2020```
4343 end
4444end
4545(ff:: typeof (f))(:: Type{Val{:analytic}} ,u0,p,t,W) = u0.* exp .(0.63155 * t+ 0.87 * W)
46- doc"""
46+ @doc doc"""
47478 linear SDEs (as a 4x2 matrix):
4848
4949```math
@@ -73,7 +73,7 @@ prob_sde_2Dlinear_stratonovich = SDEProblem(f,σ,ones(4,2)/2,(0.0,1.0))
7373f = (u,p,t) -> - .25 * u* (1 - u^ 2 )
7474σ = (u,p,t) -> .5 * (1 - u^ 2 )
7575(ff:: typeof (f))(:: Type{Val{:analytic}} ,u0,p,t,W) = ((1 + u0). * exp .(W)+ u0- 1 ). / ((1 + u0). * exp .(W)+ 1 - u0)
76- doc"""
76+ @doc doc"""
7777```math
7878du_t = \\ frac{1}{4}u(1-u^2)dt + \\ frac{1}{2}(1-u^2)dW_t
7979```
@@ -89,7 +89,7 @@ prob_sde_cubic = SDEProblem(f,σ,1/2,(0.0,1.0))
8989f = (u,p,t) -> - 0.01 * sin .(u).* cos .(u).^ 3
9090σ = (u,p,t) -> 0.1 * cos .(u).^ 2
9191(ff:: typeof (f))(:: Type{Val{:analytic}} ,u0,p,t,W) = atan .(0.1 * W + tan .(u0))
92- doc"""
92+ @doc doc"""
9393```math
9494du_t = -\\ frac{1}{100}\s in(u)\c os^3(u)dt + \\ frac{1}{10}\c os^{2}(u_t) dW_t
9595```
@@ -107,7 +107,7 @@ f = (u,p,t) -> p[2]./sqrt.(1+t) - u./(2*(1+t))
107107p = (0.1 ,0.05 )
108108(ff:: typeof (f))(:: Type{Val{:analytic}} ,u0,p,t,W) = u0./ sqrt .(1 + t) + p[2 ]* (t+ p[1 ]* W). / sqrt .(1 + t)
109109
110- doc"""
110+ @doc doc"""
111111Additive noise problem
112112
113113```math
137137end
138138(ff:: typeof (f))(:: Type{Val{:analytic}} ,u0,p,t,W) = u0./ sqrt (1 + t) + sde_wave_βvec.* (t+ sde_wave_αvec.* W). / sqrt (1 + t)
139139
140- doc"""
140+ @doc doc"""
141141A multiple dimension extension of `additiveSDEExample`
142142
143143"""
@@ -154,7 +154,7 @@ end σ ρ β
154154 du[i] = 3.0 # Additive
155155 end
156156end
157- doc"""
157+ @doc doc"""
158158Lorenz Attractor with additive noise
159159
160160```math
@@ -173,7 +173,7 @@ prob_sde_lorenz = SDEProblem(f,σ,ones(3),(0.0,10.0),(10.0,28.0,2.66))
173173f = (u,p,t) -> (1 / 3 )* u^ (1 / 3 ) + 6 * u^ (2 / 3 )
174174σ = (u,p,t) -> u^ (2 / 3 )
175175(ff:: typeof (f))(:: Type{Val{:analytic}} ,u0,p,t,W) = (2 t + 1 + W/ 3 )^ 3
176- doc"""
176+ @doc doc"""
177177Runge–Kutta methods for numerical solution of stochastic differential equations
178178Tocino and Ardanuy
179179"""
@@ -346,7 +346,7 @@ function stiff_quad_f_strat(::Type{Val{:analytic}},u0,p,t,W)
346346 (tmp* exp_tmp + u0 - 1 )/ (tmp* exp_tmp - u0 + 1 )
347347end
348348
349- doc"""
349+ @doc doc"""
350350The composite Euler method for stiff stochastic
351351differential equations
352352
@@ -365,7 +365,7 @@ Higher α or β is stiff, with α being deterministic stiffness and
365365"""
366366prob_sde_stiffquadito = SDEProblem (stiff_quad_f_ito,stiff_quad_g,0.5 ,(0.0 ,3.0 ),(1.0 ,1.0 ))
367367
368- doc"""
368+ @doc doc"""
369369The composite Euler method for stiff stochastic
370370differential equations
371371
@@ -384,7 +384,7 @@ Higher α or β is stiff, with α being deterministic stiffness and
384384"""
385385prob_sde_stiffquadstrat = SDEProblem (stiff_quad_f_strat,stiff_quad_g,0.5 ,(0.0 ,3.0 ),(1.0 ,1.0 ))
386386
387- doc"""
387+ @doc doc"""
388388Stochastic Heat Equation with scalar multiplicative noise
389389
390390S-ROCK: CHEBYSHEV METHODS FOR STIFF STOCHASTIC
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