The intrinsic sense of stochastic differential equations

Transcription

1 The ntnsc sense of stochastc dffeental equatons Detch Ryte Mdatweg 3 CH-4500 Solothun Swtzeland Phone A change of the vaables (establshng a nomal fom of the fowad and bacwad opeato) neutalzes the ntegaton sense, and the nvese sngles out the Statonovch sense, by means of the Itô fomula. Ths excludes a fee choce of the ntegaton sense and, n patcula, the Itô sense of the SDEs. Key wods: Stochastc dffeental equatons; ntegaton sense; change of vaables; tenso calculus

2 I. Intoducton As t s well-nown [1-5], the SDEs ae to be undestood as ntegal equatons, and the soluton s constucted by ncements n dt. The evaluaton pont n [ t, t + dt] s specfed by α ( 0 α 1), fo example at the begnnng ( α = 0, Itô) o n the mddle ( α = 1/, Statonovch). The sum of the ncements may depend on α when dt 0 ; then α defnes the sense of an SDE and belongs to ts specfcaton. Untl now t was undestood that α s a fee paamete, to be chosen by convenence o by popetes of the specfc models. A dependence on α does not exst when the couplng wth the nose (theeby the dffuson) s constant. It wll now be shown that a constant dffuson can typcally be aanged by a tansfomaton of the vaables, whch does not nvolve α (t establshes a nomal fom of the Foe-Planc opeato). The nvese tansfomaton then sngles out the value α = 1/ (Statonovch), by applyng the Itô fomula to the nduced map of the pocesses. The tansfomaton n both dectons can modfy a gven dft, when t does not tansfom as a tenso. The tansfomaton to the nomal fom has an nvese when the an of the dffuson matx D( s constant (a smooth dependence on x s assumed). It only eques quadatues, whch need not even be caed out fo povng the cucal statement. The bacgound nfomaton wll befly be summazed, patculaly n the Appendx A, whee the ncement wth any α s edeved. The essental phenomena aleady show up n one dmenson. In that case the Appendx B s tval, and the 3. educes to ts fst statement; the 4.1 shows the man fndngs n the smplest way.

3 3 II. The SDE and condtonal ncements The contnuous Maov pocess X (t) dx s supposed to obey the SDE = (.1) = a ( X ) dt + b ( X ) dw α o dx a( X ) dt + B( X ) dw α wth smooth functons a (, b (. As usual, (.1) denotes an ntegal equaton, wth the second tem nvolvng α. The Wene pocesses W (t) ae ndependent and obey < W ( t) W (0) > = 0 and < [ W ( t) W (0)] > t. In the Appendx A t wll be shown that fo gven X ( t) = x and dt 0 = X ( t + dt) x = a( dt + B( dw + α asp ( dt + o( dt), (.) whee dw = W ( t + dt) W ( t), and wth the nose-nduced o spuous dft a Sp m ( : = b, m ( b (. (.3) The tme evoluton of the pobablty densty w( x, t) of X (t) s detemned by the fowad (Foe-Planc) equaton [1-5]. Its dft s gven by the expectaton of (.) < X ( t + dt) x > = [ a( + α asp ( ] dt + o( dt), (.4) moe pecsely by a( + α a (, (.5) Sp and by the dffuson matx T D( = B( B (. (.6) The explct FPE eads w, t [ ( a + a Sp ) w + (1/ )( D w), ], = α. (.7) The dffuson matx (.6) plays a ey ole. It s obvously symmetc and nonnegatve. A constant B entals a constant D. The convese clealy holds n one dmenson. In hghe dmensons D only detemnes B O wth some matx O(x ), fo whch T OO s

4 4 unty,.e. wth any othogonal O when det O = 1 ( det O = 1 s also admtted). Ths amounts to eplacng the vecto Wene pocess W by equvalent [1]. In ths sense a constant D also entals a constant B. In the Appendx B t wll futhe be shown that OW, whch s stochastcally j a Sp : = b, j b = D, /, (.8) whch s evdent fo a dagonal B(x ), thus e.g. n one dmenson. The nose contbuton n the FPE s thus completely descbed by D(x ) and α. III. Changng vaables 3.1 Tenso popetes The vaables x may be consdeed as coodnates n the vaable space. They ae now supposed to be eplaced by z, wth a smooth and nvetble tansfom z (x ), whle the Wene pocess W (t) s unchanged. Snce x s ntnscally contavaant, t follows (n the leadng ode O ( dt ) ) that B ( x ) dw s a contavaant vecto (mnd the lowecase agument x ). The dyadc poduct B dw ( B dw ) T equals BB T dt by dw ( dw ) T = I dt, and ths shows that B B T = D s a twce contavaant tenso. Fo an ndependent poof see [5]. The tansfomaton of the dft a s not smple and wll be consdeed at the end. 3. Establshng a constant D(x ) In one dmenson D ( becomes 1 n a new vaable z ( gven by δ z = [ D( ] 1/ δ x, snce < ( dx ) > = D( dt n O (dt) ; ths s confmed by the tenso popety of D(x ). In hghe dmensons a symmetc matx D(x ) can fst be dagonalzed by a feld of othogonal matces O(x ). These otate the local coodnate axes nto the egenvectos of

5 5 D(, whch ae tangent to an othogonal net Λ of cuves. New coodnates y, gven by T δ y = Oδ x, un along these cuves. The elements of the dagonal O D O : = D d ae the egenvalues λ ( of D(x ). The an of D(x ) s supposed to be the same fo each x. Ths means that λ > 0 on some cuves of Λ, whle λ 0 on the othes. In the fst case the escalng of y by δ z : = ( D 1/ d ) δ y 1/ = λ δ y (3.1) yelds D c = 1, as n one dmenson. Fo λ = 0 = D d one may set z = y. The constant D c s dagonal, wth elements 1 o 0. The new vaables z ae thus obtaned n two steps: (1) by detemnng the egenvalues and egenvectos of D(x ) at each x, and () fo λ > 0 by ntegatng δ z 1/ : = λ δ s ( δ s beng the lne element) (3.) along the espectve cuve of Λ, whle λ. z = y when = 0 Remas: () In two dmensons a geneal nondegeneate dffuson opeato was cast nto a nomal fom by use of Beltam equatons [6]. These ae fulflled by (3.1) and (3.), and the opeato becomes the Laplacan. () The method s easly extended to moe geneal opeatos, wth (symmetc) coeffcent matces D(x ) havng egenvalues of both sgns (povded that the numbe of postve, zeo and negatve ones does not vay wth x ); t s suffcent to tae the absolute value of λ n the escalng (3.1). Ths establshes the nomal fom D ( / x x = κ /( z ) wth κ = 1,0, 1 (3.3) and ncludes, fo example, the hypebolc case. The dffuson opeato n the fowad and bacwad equatons becomes the Laplacan,

6 6 actng on the subspace whee D s nonsngula. It s mpotant to note that the tansfomed a Sp s zeo by (.8). Ths shows that a Sp s not a tenso, because any tenso vanshng n one coodnate system vanshes altogethe. Clealy a vanshng a Sp neutalzes the ntegaton sense, n vew of (.). IV. Results 4.1 The smplest case The dea s most easly seen n one dmenson and wth a ( 0 : dx = b( X ) dw wth b ( > 0 and wth an unspecfed α. (4.1) The tansfom z ( s gven by dz D x 1/ 1 = [ ( )] dx = [ b( ] dx and esults n Z( t) W ( t), snce n the new vaable b * 1 [Recall that b s a one-dmensonal vecto, wth the tansfom b * = b( dz / d 1, n accodance wth D * = ( b*) = 1]. The nvese tansfomaton x (z), wth dx / dz = b[ x( z)], expesses X (t) by Z ( t) = W ( t), and the Itô fomula entals that dx = b[ x( z)] dw + (1/ )( db / dz) dt. (4.) In vew of db ( / dz = [ db( / dx] ( dx / dz) = b'( b( (4.3) the esult s just (.) wth α = 1/ ( b ' b = asp thus mposed by the Itô fomula. ). The Statonovch sense of (4.1) s Rema: b ( < 0 s also admtted, but not a zeo pont of b (, whee z ( would not have an nvese. An altenatve (and moe elementay) appoach maes use of the FPE w t = α ( bb' w)' + (1/ )( b )''., w

7 7 Its equvalent n the z vaable s u, t = (1/ ) u, zz (by b * 1) and does not contan α (see also (.8) ). The nvese tansfom wll thus specfy α. Indeed, by obsevng that u dz = wdx (gvng u = w dx / dz = b w ) and / z = ( dx / dz) / x = b / x, t follows that w t = (1/ ) [ bb' w + ( b )'] ', whch mples that α = 1/. (That appoach s less, w appopate fo genealzng). These fndngs emnd the well-nown fact [1] that the Statonovch ntegal (only) obeys the ules of conventonal analyss; stochastc analyss and the FPE ae essentally paallel. 4. Hghe dmensons The Itô fomula can fomally be obtaned by tang the Taylo expanson to the second ode n dw and by dw dw = δ dt. Ths gves the clue fo the extenson of the above esult to hghe dmensons. It s supposed that both z (x ) and x (z ) exst, see the pecedng 3.. The followng agument apples n the subspace whee D(x ) s nonsngula. Snce D c s unty thee, one can modfy the tansfomed B to become unty as well, accodng to the dea outlned afte (.7). Then the analogue of (4.) s m dx = b [ x( z)] dw + (1/ ){ b [ x( z)]/ z } dw m dw (4.4) wth m n n m n b [ x( z)]/ z = ( b / x )( x / z ) = ( b / x ) b nm. (4.5) Obsevng that dw m dw = δ dt leads to (.), (.3) wth α = 1/. m 4.3 A nonzeo dft The dft a (x ) becomes a * ( z ) n the new vaables, and the new SDE eads dz = a *( Z) dt + I dw, (4.6) c whee the dagonal I c s unty n the subspace whee D(x ) s nonsngula, and zeo

8 8 othewse. Ths s unque and can be solved n pncple (as well as the espectve fowad and bacwad equatons). Fo the extenson of the 4. consde the ncement of X (t) m m = ( x / z ) dz + (1/ )( x / z z dz dz. dx ) Insetng (4.6), and poceedng as above n (4.4) and (4.5), esults n dx = [ a **( + (1/ ) asp ( ] dt + B( dw + o( dt) (4.7) whee a **( s the nvese tansfom of a * ( z). Clealy a **( a( when a (x ) s a tenso (so a * = ( z / x ) a ). Othewse a **( s a sot of pojecton of a (x ), whee possble devatves of D(x ), thus of B(x ), ae set equal to zeo. Ths can be seen by the example a( : = β a ( wth some constant β, whch yelds a * 0 and theeby a * * = 0 Sp (note that β = 1/ would ental the Itô sense). A gven dft s thus educed to ts tenso pat. V. Comments Ths wo was ncted by the dea that the numbe of solutons fo models descbed by a SDE cannot depend on the choce of the coodnates. Ths would ndeed be the case n the exstng concept: statng fom a system wth a constant dffuson (thus wth a unque soluton) a nonlnea change of the vaables entals an α -dependence and theeby a contnuum of solutons. The new aguments athe dscad the fee choce of α and exhbt the geneal value α = 1/ (when α eally mattes). Condtons fo that esult wee wea assumptons on the dffuson matx: a constant an and a smooth dependence on the agument x. Ths new fndng s puely mathematcal and must not be confused wth attempts to detemne α by extenal aguments (as n the Itô o Statonovch dlemma [7]). It athe paallels the appoach by Wong and Zaa [8], but does not nvoe any exta

9 9 appoxmatons o lmtng pocedues. As a bypoduct, a vaable tansfom was specfed, whch establshes a standad fom fo qute geneal lnea dffeental opeatos of the second ode. Outloo: In [9] t s shown that multplcatve nose excludes the Maov popety of the soluton X (t). Ths contadcts wth [1,] and eques an extenson of the exstng theoy. The futhe fndng that α = 1 admts an appoxmate Maov popety may favo that case fo pactcal applcatons. Appendx A The ntegal equaton (.1) s to be solved n [ t, t + dt], wth X ( t) = x. The exact ncement X obeys X t+ dt t+ dt ( dt) = a [ x + X ( τ )] dτ + b [ x + X ( τ )] dw ( τ ) t t. (A.1) Fo small enough dt the fst ntegal yelds a ( x ) dt, and the second one can be solved explctly. To ths end t s suffcent to expand b to the fst ode, whch esults n b t+ dt m ( W ( dt) + b, m ( X ( τ ) dw ( τ ) t (wth W ( 0) = 0 snce only the ncements matte). The fst tem s the leadng pat of O ( dt ), and successve appoxmaton amounts to nset t nto the ntegal, whch esults n t+ dt m X ( τ ) dw ( τ ) = b t mn ( dt 0 W ( τ ) dw n ( τ ). The last ntegal nvolves α. Fo = n t s well-nown to yeld [ W ( dt) + (α 1) dt]/, wth the expectaton α dt and wth the α -ndependent vaance ( dt ) /. Fo small enough dt ths allows to eplace the ntegal by the nonandom value α dt. Snce fo n the

10 10 expectaton s zeo, the esult s X ( dt) b ( W ( dt) + a Sp ( α dt, wth the nose-nduced o spuous dft a Sp m ( : = b, m ( b (. (A.) Ths yelds the explct esult X = [ a ( + α a Sp ( ] dt + b ( W ( dt) + o( dt) (A.3) and theeby the equvalent Itô fom of (.1) dx = [ a ( X ) + α a Sp ( X )] dt + b ( X ) dw ( α = 0) (A.4) fo each α. Appendx B The spuous dft a Sp can always be expessed n tems of the dffuson matx D(x ). Fo a dagonal matx B (of the elements b ) - thus n one dmenson - t s obvous that j b, j b = D, /, (B.1) and fo a symmetc B the same follows by dagonalzng B. Each asymmetc B can be symmetzed on substtutng W (t) by an equvalent W *( t) gven by dw : = O dw * : Wth B *: = B O ths entals B dw = B * dw *. When B s squae, one can fnd a O whch yelds a symmetc B * by whch (B.1) holds agan; a ectangula B can be completed by zeos. Ths shows that (B.1) holds n geneal (but only by stochastc equvalence when B s not symmetc) : j a Sp : = b, j b = D, /. (B.). Acnowledgment The autho wshes to than H. Spohn fo valuable comments on an eale veson.

11 11 Refeences [1] L. Anold, Stochastsche Dffeentalglechungen (Oldenboug, München, 1973), and Stochastc Dffeental Equatons: Theoy and Applcatons (Wley, New Yo, 1974) [] I.I. Gchman and A.W. Soochod, Stochastsche Dffeentalglechungen (Aademe- Velag, Beln, 1971) [3] B. Osendal, Stochastc dffeental equatons (Spnge, Beln, Hedelbeg, New Yo, [4] C. W. Gadne, Handboo of Stochastc Methods (Spnge, Beln, Hedelbeg 1994) [5] H. Rsen, The Foe-Planc Equaton (Spnge, Beln, 1989) nd ed. [6] R. Couant and D. Hlbet, Methods of Mathematcal Physcs II (Intescence/Wley, New Yo/ London 196) [7] N.G. van Kampen, Stochastc Pocesses n Physcs and Chemsty (Elseve/Noth- Holland, Amstedam, 3 d ed. 008). [8] E. Wong and M. Zaa, Ann. Math. Statst. 36, 1560 (1965) [9] D. Ryte, Stochastc dffeental equatons: loss of the Maov popety by multplcatve nose axv.og/