!g&IdZddlZddlmZmZddlmZGddZ Gdde Z Gdd e Z Gd d e Z Gd d e Z Gdde ZGdde ZGddZGddZ edk(rdZdZgZdevr e Zej/deZerhej2ej4edj6Zej4edj;ddZej<ddZejAede Z!erYej2ej4e!dj6Zej4e!ddZej<d!ejDjGe!dejHd"e!dzd#z z ejJZ&e'e&e d$d%d&'Z(e(jSd(d( Z*erej2ej4e*j6Zej4e*j;ddZej<d)ej2ej4ejVe*j6Zej4ejVe*j;ddZej<d*e dd+d'Z,e,jSd(d( Z-erbej2ej4e-j6Zej4e-j;ddZej<d,eddd&d-.Z.e.jSZ/e.jaddejbjed/0Z3e.jadejhdd1d2de.jkdd1e'e.jkdd1d-d13j;de.jkdd(d-d(3Z6erbej2ej4e6j6Zej4e6j;ddZej<d4eddd&d-5Z7e7jkdd6d-d(3Z8e'e8j;de'ejVe8j;ddZerbej2ej4e8j6Zej4e8j;ddZej<d7e'e7jse8dddfd-8e7jkddd-d93Z:e'd:e;d9D]Z<e'e7jse:e<d-8 eZ=e=j}dd&d;dd$d-<\Z?Z@ZAerej2ej4e?j6Zej4e?j;ddZej<d=ej2ej4eAd>d?@ej4e?jddA@ej<dBejeddgejbjdejbjdgZDeDj}dCdD ZEe'eEdjdEjdEe'eEdje'eEdj;dEj;dEe'eEdj;e'eEdje'eEdjyy)Fa6getting started with diffusions, continuous time stochastic processes Author: josef-pktd License: BSD References ---------- An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations Author(s): Desmond J. Higham Source: SIAM Review, Vol. 43, No. 3 (Sep., 2001), pp. 525-546 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/3649798 http://www.sitmo.com/ especially the formula collection Notes ----- OU process: use same trick for ARMA with constant (non-zero mean) and drift some of the processes have easy multivariate extensions *Open Issues* include xzero in returned sample or not? currently not *TODOS* * Milstein from Higham paper, for which processes does it apply * Maximum Likelihood estimation * more statistical properties (useful for tests) * helper functions for display and MonteCarlo summaries (also for testing/checking) * more processes for the menagerie (e.g. from empirical papers) * characteristic functions * transformations, non-linear e.g. log * special estimators, e.g. Ait Sahalia, empirical characteristic functions * fft examples * check naming of methods, "simulate", "sample", "simexact", ... ? stochastic volatility models: estimation unclear finance applications ? option pricing, interest rate models N)statssignalc&eZdZdZdZddZddZy) Diffusionz:Wiener Process, Brownian Motion with mu=0 and sigma=1 cyNselfs ^/var/www/dash_apps/app1/venv/lib/python3.12/site-packages/statsmodels/sandbox/tsa/diffusion.py__init__zDiffusion.__init__< Nc|dz|z }tj|d|}tj|tjj ||fz}tj |d}||_||fS)z*generate sample of Wiener Process ?size)nplinspacesqrtrandomnormalcumsumdW)r nobsTdtnrepltrWs r simulateWzDiffusion.simulateW?sjsU4Z KKAt $ WWR[))t})= = IIbO!t rcp|j||||\}}|||}|jd} || |fS)zmget expectation of a function of a Wiener Process by simulation initially test example from rrrrr)r"mean) r funcrrrrr!r UUmeans r expectedsimzDiffusion.expectedsimIsB ~~415~A1 AJq %{rdrNr)__name__ __module__ __qualname____doc__r r"r)r rr rr9s rrc&eZdZdZdZddZddZy)AffineDiffusionz differential equation: :math:: dx_t = f(t,x)dt + \sigma(t,x)dW_t integral: :math:: x_T = x_0 + \int_{0}^{T}f(t,S)dt + \int_0^T \sigma(t,S)dW_t TODO: check definition, affine, what about jump diffusion? cyrr r s r r zAffineDiffusion.__init__drrNc|j||||\}}|j|j|zz}tj|d}|j d} || |fS)Nr$rr)r"_drift_sigrrr%) r rrrrr!r dxxxmeans r simzAffineDiffusion.simgs`~~415~A1kkmdiikAo- YYr!_q %{rc &||z}| |j}||dz|z }|j||||\}}|j} tj|d|}||z} ||z } tj || f} |} || dddf<tj d| D]s}tj| ddtj ||dz zdz||zfd}| |j| z|j| |zz} | | dd|f<u| S)a7 from Higham 2001 TODO: reverse parameterization to start with final nobs and DT TODO: check if I can skip the loop using my way from exactprocess problem might be Winc (reshape into 3d and sum) TODO: (later) check memory efficiency for large simulations Nrr$rr)r7) xzeror"rrrzerosarangesumr4r5)r r;rrrrTratior!r rDtLXemXtempjWincs r simEMzAffineDiffusion.simEMps&f} =JJE :3tB~~415~A1 WW KKAt $ BY Khhay!AaC1Q A66"Qryy1aqAAB1EDDKK%K00499u93E3LLEC!H   rr*)Nr+rNr)r,r-r.r/r r9rFr rr r1r1Ss  "rr1c$eZdZdZdZddZdZy)ExactDiffusionztDiffusion that has an exact integral representation this is currently mainly for geometric, log processes cyrr r s r r zExactDiffusion.__init__rrcBtj|||z|}tj|j |z}tjj ||f}|j ||j||zz}tjdgd| g|S)z`ddt : discrete delta t should be the same as an AR(1) not tested yet rr) rrexplambdrr _exactconst _exactstdrlfilter) r r;rddtrr expddtnormrvsincs r exactprocesszExactDiffusion.exactprocesss KKT#Xt , c)*))""t "5v&)?')II~~rdRL#66rctj|j |z}||z|j|z}|j |}t j ||SNlocscale)rrLrMrNrOrnormr r;r expntmeantstdts r exactdistzExactDiffusion.exactdistsR {Q'  0 0 77~~e$zze400rNr)r,r-r.r/r rUr`r rr rIrIs  7"1rrIc0eZdZdZdZdZdZddZdZy) ArithmeticBrownianz2 :math:: dx_t &= \mu dt + \sigma dW_t c.||_||_||_yrr;musigmar r;rgrhs r r zArithmeticBrownian.__init__  rc|jSrrgr argskwdss r r4zArithmeticBrownian._drifts wwrc|jSrrhrms r r5zArithmeticBrownian._sigs zzrNc,| |j}tj|||z|}tjj ||f}|j |j tj|z|zz}|tj|dzS)z7ddt : discrete delta t not tested yet rr) r;rrrrr4_sigmarr)r rr;rQrr rSrTs r rUzArithmeticBrownian.exactprocesss} =JJE KKT#Xt ,))""t "5kkDKK"''#,6@@ryyQ'''rctj|j |z}|j|z}|jtj |z}t j||SrW)rrLrMr4rsrrr[r\s r r`zArithmeticBrownian.exactdistsN {Q' a{{RWWQZ'zze400r)Nrrb) r,r-r.r/r r4r5rUr`r rr rdrds    (1rrdc"eZdZdZdZdZdZy)GeometricBrownianzGeometric Brownian Motion :math:: dx_t &= \mu x_t dt + \sigma x_t dW_t $x_t $ stochastic process of Geometric Brownian motion, $\mu $ is the drift, $\sigma $ is the Volatility, $W$ is the Wiener process (Brownian motion). c.||_||_||_yrrfris r r zGeometricBrownian.__init__rjrc*|d}|j|zSNr7rlr rnror7s r r4zGeometricBrownian._drifts Iww{rc*|d}|j|zSryrqrzs r r5zGeometricBrownian._sig IzzA~rN)r,r-r.r/r r4r5r rr rvrvs  rrvc<eZdZdZdZdZdZdZd dZdZ dZ y ) OUprocesszOrnstein-Uhlenbeck :math:: dx_t&=\lambda(\mu - x_t)dt+\sigma dW_t mean reverting process TODO: move exact higher up in class hierarchy c<||_||_||_||_yr)r;rMrgrh)r r;rgrMrhs r r zOUprocess.__init__s   rcD|d}|j|j|z zSry)rMrgrzs r r4zOUprocess._drifts" IzzTWWq[))rc*|d}|j|zSryrqrzs r r5zOUprocess._sigr|rctj|j |z}||z|jd|z zz|jtj d||zz dz |jz z|zzSNr@)rrLrMrgrhr)r r;r rSr]s r exactzOUprocess.exactsq  {Q' 1U7 33 RWWae mR%7 %BCCgMN Orctj|||z|}tj|j |z}tj|j |z}tjj ||f}ddlm} |jd|z z|jtjd||zz dz |jz z|zz} | jdgd| g| S)zddt : discrete delta t should be the same as an AR(1) not tested yet # after writing this I saw the same use of lfilter in sitmo rr)rrrr) rrrLrMrrscipyrrgrhrrP) r r;rrQrr r]rRrSrrTs r rUzOUprocess.exactprocesss KKT#Xt , {Q' c)*))""t "5 1V8$ RWWav or%9$**%DEEOPv~~rdRL#66rctj|j |z}||z|jd|z zz}|jtj d||zz dz |jz z}ddlm}|j||S)Nrrr)rrX) rrLrMrgrhrrrr[)r r;r r]r^r_rs r r`zOUprocess.exactdist1sz {Q' 1U7 33zzBGGQuU{]B$6tzz$ABBuzze400rct|dz }tjtj||ddf}tjj ||ddd\}}}}|\} } ||dz z } tj |  |z } tj| dztj | zd| dzz z |z } | d| z z }|| | fS)Nassumes data is 1d, univariate time series formula from sitmo rNrcondrrb)lenr column_stackoneslinalglstsqlogr)r datarrexogparestresranksingconstslopeerrvarrMrhrgs r fitlszOUprocess.fitls:s 4y{ tCRy9:"$))//$QR/"KT4 ud2gr!" RVVE]2QuaxZ@CD ag 5%rNra) r,r-r.r/r r4r5rrUr`rr rr r~r~s+  *O7(1  rr~cBeZdZdZdZdZdZdfd ZdZdZ xZ S) SchwartzOnezthe Schwartz type 1 stochastic process :math:: dx_t = \kappa (\mu - \ln x_t) x_t dt + \sigma x_tdW \ The Schwartz type 1 process is a log of the Ornstein-Uhlenbeck stochastic process. cJ||_||_||_||_||_yr)r;rgkapparMrh)r r;rgrrhs r r zSchwartzOne.__init__Us%    rcfd|z |j|jdzdz |jz z zS)Nrrbr)rgrhrr r]s r rNzSchwartzOne._exactconst\s0%DGGdjj!mb&8$**&DDEErcr|jtjd||zz dz |jz zSr)rhrrrrs r rOzSchwartzOne._exactstd_s0zzBGGQuU{]B$6tzz$ABBBrctj|}t|j|||||}tj |S)z/uses exact solution for log of process rQr)rrsuper __class__rUrL)r r;rrQrlnxzerolnxrs r rUzSchwartzOne.exactprocessbs?&&-DNND6udSX6Yvvc{rctj|j |z}tj||z|j |z}|j |}t j||SrW)rrLrMrrNrOrlognormr\s r r`zSchwartzOne.exactdistis[ {Q'u %(8(8(??~~e$}}d33rct|dz }tjtj|tj|ddf}tj j |tj|ddd\}}}}|\} } ||dz z } tj|  |z } tj| | zdtjd| z|zz z } | dtj| |zz z | dzdz | z z}tj|dk(r|d }tj| dk(r| d } || | fS) rrNrrrrbrr) rrrrrrrrrLshape)r rrrrrrrrrrrrrhrgs r rzSchwartzOne.fitlsps2 4y{ bffT#2Y.?@A"$))//$tABx8HPR/"ST4 ud2gr!!BFF2e8B;,?*?@A avby)) *UAXb[-> > 88B<$ AB 88E?T !!HE5%rra) r,r-r.r/r rNrOrUr`r __classcell__)rs@r rrJs(FC4 rrceZdZdZddZy)BrownianBridgecyrr r s r r zBrownianBridge.__init__rrc\|dz}|dz|z }tj|||z |}tj|||}||z d||z z g} d|||z z z } ||||z z z} |tj|d|z z|z z} |tj|||z |z z||z z z} tj||f}||dddf<| tjj ||fz}t d|D]+}|dd|dz f| |z| |z|dd|fz|dd|f<-||| fS)Nrrrr)rrrr<rrrange)r x0x1rrrQrhrr wmwmiwm1susr7rvsis r simulatezBrownianBridge.simulatesW !V VD[ KKCFD ) KKC &eQquW CE l"c!e*o BGGAqsGCK( ( 2772s1uRx=#a%01 1 HHeT] #!A#   uTl 33q 9Aq1uXc!f_s1v-AaC8AacF 9!RxrN)rrr)r,r-r.r rr rr rrs  rrcJeZdZdZej j fdZddZy)CompoundPoissonzCnobs iid compound poisson distributions, not a process in time ct|t|k7r tdt||_||_t j ||_y)Nz9lambd and randfn need to have the same number of elements)r ValueErrornobjrandfnrasarrayrM)r rMrs r r zCompoundPoisson.__init__s@ u:V $XY YJ  ZZ& rcR|j}tj|||f}tjj |j ddddf|||f}t |D]}|j|}|dddd|f}|||tj|f} td| j| j| jdtj|dddftj||dz f} | |dddd|f<d||dk(<||fS)Nrz rvs.sum()rrr)rrr<rpoissonrMrrmaxprintr>rrr=) r rrrr7Niorandfncncrxios r rzCompoundPoisson.simulates yy HHeT4( ) II  djjd15U4rsBFF1s1u9-ri)rrzBrownian Motion - exp g @rg{Gz?rrfr+zGeometric Brownianz$Geometric Brownian - log-transformedg?zArithmetic Browniang?)r;rgrMrh) rrgY@rzOrnstein-Uhlenbeck)r;rgrrh2z Schwartz One)rrz true: mu=1, kappa=0.5, sigma=0.1c)rrQrhzBrownian Bridger theoretical)label simulatedzBrownian Bridge - Variancei Nr)Hr/numpyrrrrmatplotlib.pyplotpyplotpltrr1rIrdrvr~rrrr,doplotrexampleswr"wsfigureplotrtmpr%titler&r)usrr[rLinfaverrrgbrFgbsrababsouousrrrouerrUouessososrsos2rrbbrbbsr rstdlegendcpcpsr>rr rr rsy1d4?i?B 1_1B 1 1F2K K \8 .8 x0J$ z F EH  K [[U[ +  CJJL#((2a577#C#((2a5::a=A6C CIIA B- ]]4c] 7  CJJL#((2a577#C#((2a5A.C CII- . r!uvrvva1gbj'99266B e RD <hhCsh+  CJJL#((355/C#((388A;!4C CII* + CJJL#((6266#;==)C#((6266#((1+.!q#C8  CJJL#((466"C#((499Q<15C CII* + qQc =ooa#o6 chhqk fbffSXXa[!"  CJJL#((355/C#((388A;!4C CIIn % bhhs1Q3x3h'(q#A6 01q ,A "((47c(* + , [[C3Bc[J Q  CJJL#((355/C#((388A;!4C CII' ( CJJL CHHR= 1 CHHSWWQZ{ 3 CII2 3 CJJL !A!1!1"))2B2B C DB ++5q+ )C #a&**R.  R ! #a&**, #a&++b/  r "# #a&++- #a&++ #a&**,mr