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keywords= Euler-Maruyama, Ito derivative, Ito differential, Ito differential equation, Ito formula, Ito integral, Ito lemma, ItoProcess, Ito s formula, Ito s lemma, Ito stochastic differential equation, Ito Taylor expansion, Markov process, martingale, Milstein, Milstein scheme, sde, stochastic calculus, stochastic differential equation, stochastic integral, stochastic Runge-Kutta, strong solution, weak solution;
description=ItoProcess[ a, b , x, t] represents an Ito process x(t), where \[DifferentialD]x(t) == a(t, x(t)) \[DifferentialD]t + b (t, x(t)) . \[DifferentialD]w(t). ItoProcess[ a, b, c , x, t] represents an Ito process y(t) == c(t, x(t)), where \[DifferentialD]x(t) == a(t, x(t)) \[DifferentialD]t + b (t, x(t)) . \[DifferentialD]w(t) . ItoProcess[..., x, x0 , t, t0 ] uses initial condition x(t0) == x0. ItoProcess[..., ..., ..., \[CapitalSigma]] uses a Wiener process w(t), with covariance \[CapitalSigma]. ItoProcess[proc] converts proc to a standard Ito process whenever possible. ItoProcess[sdeqns, expr, x, t, w \[Distributed] dproc] represents an Ito process specified by a stochastic differential equation sdeqns, output expression expr, with state x and time t, driven by w following the process dproc. ;

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Text of the page (random words):
sde itoprocess is a continuous time and continuous state random process if the drift a is an dimensional vector and the diffusion b an dimensional matrix the process is dimensional and driven by an dimensional wienerprocess common specifications for coefficients a and b include a scalar b scalar a scalar b vector a vector b vector a vector b matrix a stochastic differential equation is sometimes written as an integral equation the default initial time t 0 is taken to be zero and the default initial state x 0 is zero the default covariance σ is the identity matrix for a general covariance σ itoprocess canonicalizes the process by converting the diffusion matrix b to b σ 1 2 with σ 1 2 the lower cholesky factor of σ when possible a standard ito process has output consisting of a subset of differential states processes proc that can be converted to standard itoprocess form include ornsteinuhlenbeckprocess geometricbrownianmotionprocess stratonovichprocess and itoprocess converting an itoprocess to standard form automatically makes use of ito s lemma the stochastic differential equations in sdeqns can be of the form where is differentiald which can be input using dd the differentials and are taken to be ito differentials the output expression expr can be any expression involving x t and t the driving process dproc can be any process that can be converted to a standard ito process properties related to itoprocess include drift drift term diffusion diffusion matrix output output state timevariable time variable timeorigin origin of time variable statevariables state variables initialstate initial state values kolmogorovforwardequation kolmogorov forward equation fokker planck equation kolmogorovbackwardequation kolmogorov backward equation derivative ito derivative feynmankacformula pde obtained from feynman kac formula method settings in randomfunction specific to itoprocess include eulermaruyama euler maruyama order 1 2 default kloedenplatenschurz kloeden platen schurz order 3 2 milstein milstein order 1 stochasticrungekutta 3 stage rossler srk scheme order 1 stochasticrungekuttascalarnoise 3 stage rossler srk scheme for scalar noise order 3 2 itoprocess can be used with such functions as randomfunction covariancefunction pdf and expectation examples open all close all basic examples 1 define a process by its stochastic differential equation wolfram language code proc itoprocess ⅆx t x t ⅆt sqrt 1 x t 2 ⅆw t x t x 1 t w wienerprocess simulate the process wolfram language code randomfunction proc 0 5 0 01 wolfram language code listlineplot filling axis compute mean function wolfram language code mean proc t compute covariance function wolfram language code covariancefunction proc s t wolfram language code plot3d s 0 5 t 0 5 colorfunction rainbow scope 19 basic uses 10 define a wiener process with drift and diffusion from the stochastic differential equation sde wolfram language code itoprocess μ σ x 0 t directly convert from the parametric process wolfram language code itoprocess wienerprocess μ σ define a process where wolfram language code itoprocess μ σ c x t x 0 t use differential notation to define the same process wolfram language code itoprocess ⅆx t μ ⅆt σ ⅆw t c x t x 0 t w wienerprocess define a vector process with output wolfram language code itoprocess v x 0 1 x x v x0 v0 t 0 using differential notation wolfram language code itoprocess ⅆx t v t ⅆt ⅆv t x t ⅆt ⅆw t x t x v x0 v0 t w wienerprocess define a vector process where wolfram language code itoprocess μ σ x x 2 x 0 t using differential notation wolfram language code itoprocess ⅆx t μⅆt σ ⅆw t x t x t 2 x 0 t w wienerprocess define a vector process where wolfram language code itoprocess y x 1 0 0 1 x y 0 0 t using differential notation wolfram language code itoprocess ⅆx t y t ⅆt ⅆw1 t ⅆy t x t ⅆt ⅆw2 t x t y t x y 0 0 t w1 wienerprocess w2 wienerprocess define a process driven by two correlated wiener processes wolfram language code σ 1 3 5 3 5 1 wolfram language code b subscript σ 1 subscript σ 2 proc itoprocess 1 x t b x t x 1 2 t 0 σ the canonicalized process has diffusion matrix equal to with the diffusion matrix before canonicalization wolfram language code bnew proc 1 2 wolfram language code bnew b choleskydecomposition σ define a scalar process corresponding to the sde wolfram language code itoprocess ⅆx t subscript w 1 t ⅆsubscript w 2 t x t x 0 t subscript w 1 wienerprocess subscript w 2 wienerprocess define vector process and corresponding to the sde and wolfram language code itoprocess ⅆx t subscript w 1 t ⅆsubscript w 2 t ⅆy t subscript w 1 t ⅆsubscript w 1 t x t y t x y 0 0 t subscript w 1 wienerprocess subscript w 2 wienerprocess define a process corresponding to the 2d correlated wiener process wolfram language code noise𝒫 σ1_ σ2_ ρ_ refine itoprocess 0 0 σ1 0 0 σ2 w1 w2 0 0 t 1 ρ ρ 1 1 ρ 1 define vector process driven by correlated 2d wiener process wolfram language code itoprocess ⅆs t μ s t ⅆt sqrt r t s t ⅆn1 t ⅆr t θ μ r t ⅆt sqrt r t ⅆn2 t s t r t s r s0 r0 t n1 n2 noise𝒫 subscript σ 1 subscript σ 2 ρ simulate itoprocess paths using different methods wolfram language code proc itoprocess ⅆx t v t ⅆt ⅆv t x t ⅆt ⅆn t x t x v 1 0 t n wienerprocess simulation methods and their corresponding orders wolfram language code methods eulermaruyama milstein stochasticrungekutta kloedenplatenschurz stochasticrungekuttascalarnoise wolfram language code orders order 1 2 order 1 order 1 order 3 2 order 3 2 specify the simulation method as an option in randomfunction wolfram language code paths table randomfunction proc 0 2 pi 0 05 6 method m m methods wolfram language code grid partition mapthread listlineplot 1 plotlabel column 2 3 imagesize 160 paths methods orders upto 3 spacings 2 process properties extraction 2 define an ito process by its stochastic differential equation wolfram language code proc itoprocess ⅆx t μ x t ⅆt sqrt 1 x t 2 ⅆw t x t x 1 t w wienerprocess available ito process properties wolfram language code proc properties drift and diffusion of the process wolfram language code proc drift diffusion kolmogorov forward equation wolfram language code proc kolmogorovforwardequation traditionalform inactive is used here to avoid expanding the partial derivatives use activate to expand the expression wolfram language code activate traditionalform kolmogorov backward equation wolfram language code proc kolmogorovbackwardequation traditionalform compute the ito derivative of a function the output is a list consisting of drift and diffusion terms wolfram language code mu sig proc derivative f x t t wolfram language code traditionalform mu wolfram language code traditionalform sig the property feynmankacformula gives a pde whose solution satisfies the conditional expectation and terminal condition wolfram language code proc feynmankacformula traditionalform additional arguments can be provided for the generalized situations with an additional argument the property feynmankacformula gives a pde whose solution satisfies the conditional expectation and the same terminal condition wolfram language code proc feynmankacformula α x t traditionalform with a third argument the property feynmankacformula gives a pde whose solution satisfies the conditional expectation and the same terminal condition wolfram language code proc feynmankacformula α x t β x t traditionalform define heston model with itoprocess wolfram language code heston itoprocess μ s t κ θ ν t ν t 1 2s t 0 0 σ ν t 1 2 s ν subscript s 0 subscript ν 0 t 1 1 2 1 2 1 drift wolfram language code heston drift diffusion matrix after canonicalization wolfram language code heston diffusion kolmogorov forward equation wolfram language code heston kolmogorovforwardequation traditionalform kolmogorov backward equation wolfram language code heston kolmogorovbackwardequation traditionalform ito derivative formula wolfram language code mu sig heston derivative f s t ν t t wolfram language code mu traditionalform wolfram language code sig traditionalform special ito processes 5 an ito process corresponding to the wienerprocess wolfram language code itoprocess wienerprocess μ σ an ito process corresponding to the geometricbrownianmotionprocess wolfram language code itoprocess geometricbrownianmotionprocess μ σ x0 an ito process corresponding to the brownianbridgeprocess wolfram language code itoprocess brownianbridgeprocess σ subscript t 1 a subscript t 2 b an ito process corresponding to the ornsteinuhlenbeckprocess wolfram language code itoprocess ornsteinuhlenbeckprocess μ σ θ x0 an ito process corresponding to the coxingersollrossprocess wolfram language code itoprocess coxingersollrossprocess μ σ θ x0 process slice properties 2 define jacobi diffusion process wolfram language code jacobi𝒫 x0_ itoprocess ⅆx t 1 2 x t ⅆt sqrt x t 1 x t ⅆw t x t x x0 t w wienerprocess compute low order cumulants of time slice distribution wolfram language code table cumulant jacobi𝒫 x0 t r r 1 4 find the limit of infinite time horizon wolfram language code limit t compare with cumulants of the uniform distribution wolfram language code table cumulant uniformdistribution r r 1 4 define a vector process given by a system of linear sdes wolfram language code proc σ1_ σ2_ x0_ y0_ itoprocess ⅆx t x t y t ⅆt σ1 ⅆw t ⅆy t 2x t ⅆt σ2ⅆw t x t y t x y x0 y0 t w wienerprocess find the probability density function of the time slice distribution wolfram language code pdf proc 1 1 1 2 1 4 t x y compute cross covariance of and wolfram language code cov covariance proc subscript σ 1 subscript σ 2 x0 y0 t 1 2 infinite time horizon limit exists only if wolfram language code collect cov e _ simplify applications 11 computing properties 3 compute cross covariance of the ornstein uhlenbeck process and its underlying wiener process wolfram language code joint𝒫 itoprocess ⅆx t θ μ x t ⅆt σ ⅆw t x t w t x x0 t w wienerprocess wolfram language code covariance joint𝒫 t 1 2 compute moments of the process where is the standard wiener process wolfram language code itoprocess ⅆx t w t 2ⅆt x t x 0 t w wienerprocess wolfram language code table moment t r r 0 8 vector ito process driven by scalar noise 1d oscillator driven by white noise wolfram language code proc itoprocess ⅆx t v t ⅆt ⅆv t x t ⅆt ⅆn t x t x v 1 0 t n wienerprocess simulate process paths wolfram language code path randomfunction proc 0 2 pi 0 05 12 method stochasticrungekutta wolfram language code listlineplot path compute mean and variance functions wolfram language code mf t_ mean proc t wolfram language code vf t_ simplify variance proc t plot mean function and the standard deviation band together with generated paths wolfram language code show plot mf t sqrt vf t mf t sqrt vf t mf t t 0 2pi filling 1 2 listlineplot path plotrange all martingales 3 determine values of and for which the process is a martingale where is the standard wiener process wolfram language code x𝒫 itoprocess 0 1 exp α w t β t w 0 t convert to the standard form wolfram language code st𝒫 itoprocess x𝒫 zero drift coefficient of the standard form is a necessary condition for to be a martingale wolfram language code reduce st𝒫 1 1 0 α β scalar ito process driven by vector wiener process wolfram language code itoprocess 0 1 1 1 x 0 t 0 define the same process via a stochastic equation wolfram language code itoprocess ⅆx t ⅆw1 t ⅆw2 t ⅆw3 t x t x 0 t w1 wienerprocess w2 wienerprocess w3 wienerprocess construct a scalar process driven by two wiener processes wolfram language code proc1 itoprocess ⅆx t w1 t sqrt w1 t 2 w2 t 2 ⅆw1 t w2 t sqrt w1 t 2 w2 t 2 ⅆw2 t x t x x0 t w1 wienerprocess w2 wienerprocess quadratic variation of wolfram language code integrate simplify proc1 derivative x t 2 1 t 0 t by l é vy characterization is a brownian motion the mean of the process is the same as the initial state wolfram language code mean proc1 t modeling 2 the dynamics of a free particle under the effect of thermal fluctuation can be modeled by the langevin equation of motion where is the standard wienerprocess and is the strength of the thermal noise here it is assumed that can only depend on and focus on the equation of velocity there are two common ways to integrate the equation of motion ito formulation and stratonovich formulation they can be defined via wolfram language code ito itoprocess ⅆv t v t ⅆt σ v t ⅆb t v t v vi t b wienerprocess wolfram language code str stratonovichprocess ⅆv t v t ⅆt σ v t ⅆb t v t v vs t b wienerprocess when is a constant the two formulations are identical and lead to the same stationary distribution as wolfram language code limit pdf ito t σ v t sqrt 2 σ1 v simplify t infinity wolfram language code limit pdf str t σ v t sqrt 2 σ1 v simplify t infinity if is velocity dependent then due to the nature of the wienerprocess has nonzero quadratic variation and the two formulations lead to different results convert stratonovich formulation to the equivalent ito formulation wolfram language code istr itoprocess str the drift under stratonovich formulation is different from the drift under ito formulation wolfram language code ito drift wolfram language code istr drift the gompertz curve is typically used in the modeling of a growth process such as tumor growth by assuming gaussian noise in the logarithm of the growth process you can write the model as a stochastic differential equation wolfram language code proc itoprocess ⅆx t α exp α t log x0 k x t ⅆt σ x t ⅆw t x t x x0 t w wienerprocess mean of the process is the usual gompertz curve wolfram language code mean proc t slice distribution of the process at time obeys lognormaldistribution wolfram language code pdf proc t x simulate the process with and from to wolfram language code proc1 proc x0 1 k 2 σ 0 1 α 0 3 wolfram language code sample randomfunction proc1 0 10 0 01 visualize the sampled path wolfram language code listlineplot sample filling axis generate a thousand samples with the same conditions then visualize the paths and slice data at wolfram language code samples randomfunction proc1 0 10 0 1 1000 wolfram language code graphicsrow listlineplot samples imagesize 250 plotrange all aspectratio 3 4 basestyle directive thin opacity 0 5 plotrangepadding 0 25 5 5 histogram samples slicedata 10 automatic pdf ito process representations 3 use itoprocess to represent the standard wienerprocess wolfram language code wiener itoprocess ⅆb t ⅆw t b t b 0 t w wienerprocess is a martingale use ito lemma to compute the derivative of wolfram language code m exp t 2 sin b t drift diffusion wiener derivative m create a coxingersollrossprocess and represent it with itoprocess wolfram language code cir coxingersollrossprocess 5 1 1 2 10 proc itoprocess cir obtain kolmogorov forward equation wolfram language code forward activate proc kolmogorovforwardequation x t p solve the equation numerically in with a localized initial condition at and dirichlet boundary conditions wolfram language code f x_ exp 100 x 10 2 sqrt 200 2pi plot f x x 0 30 plotra...