The Unique Solution of Stochastic Differential Equations With. Independent Coefficients. Dietrich Ryter.

Transcription

1 The Unque Soluton of Stochastc Dffeental Equatons Wth Independent Coeffcents Detch Ryte Mdatweg 3 CH-4500 Solothun Swtzeland Phone SDE s must be solved n the ant-itô sense when the coeffcents ae ndependent. Whle the nose-nduced dft mattes fo the sample paths, t s absent n the Foe-Planc equaton, whch taes a patculaly smple fom and s nvaant unde changes of the vaables. Pue nose yelds matngale solutons. Key wods: Stochastc dffeental equatons; ntegaton sense; matngales; Foe- Planc equaton; change of vaables

2 2 I. Intoducton Stochastc dffeental (oathe ntegal) equatons ae not unquely defned n the pesence of multplcatve nose. Solvng the ntegal veson n a small tme nteval dt (wth gven ntal values) educes to an ntegal of the Wene pocess, whch s well-nown to nvolve a paamete α that specfes the evaluaton pont n the nteval ( 0 α 1; α = 0 at the begnnng Itô, α = 1/ 2 n the mddle Statonovch, and α = 1 at the end ant-itô ). It s well-nown [1] that the sum of the ncements n each nteval conveges fo each α, wth a esult dependng on α. The specfcaton of α seems thus a demandng exta tas of modelng. The ey assumpton of the pesent wo s the (functonal) ndependence of the coeffcent functons, whch holds n patcula, when the ognal dft s contavaant n a change of the vaables (t can thus not nvolve any devatves of the couplng paametes wth the nose, whch tansfom dffeently). As a consequence, the mean condtonal ncements as obtaned ethe fom the FPE o fom the SDE only agee when α = 1. A peculaole s played by the nose-nduced dft : whle t contbutes to the soluton of the SDE, t s cancelled n the FPE by othe tems. The petnent geneal theoy wll fst be summazed n the Chaptes II and III. II. The Itô fom of the geneal SDE 2.1 Genealtes Let the contnuous Maov pocess X (t) dx a ( X ) dt + b ( X ) be gven by the SDE = dw (2.1) wth smooth functons a ( x), b ( x). Summaton ove double ndces s undestood. The Wene pocesses W (t) ae ndependent and obey W ( t) W (0) > = 0 and <

3 3 2 < [ W ( t) W (0)] > t. The sense of (2.1),.e. the value of α, s not specfed at fst. = Comments: (1) (2.1) s consdeed as the a po SDE used n modelng. (2) The dft a (x ) s due to extenal felds and also to ntenal detemnstc effects le fcton, whch may depend on the nose ntensty (e.g. a local tempeatue, whch s popotonal to the b (x) ), but not on ts gadent (nvolvng devatves of the b (x) ). The last estcton s essental fo the functonal ndependence of the coeffcents a ( x), b ( x) ; t s fomally guaanteed by the postulate that a (x ) tansfoms le a contavaant vecto when x s changed, snce the devatves of the b (x) tansfom dffeently. The mpact wll not only be an nvaant FPE, but even a unque value of α, see the Chapte IV below. Suppose that [ 0, t ] s pattoned nto small ntevals of a length dt. The ey poblem s to detemne the ncements of X n [ t, t + dt] ( t beng a patton pont) wth an abtay gven v X ( t ) = x. Summng up those ncements and lettng dt 0 yelds X ( t) X (0) fo each path of W n [ 0, t ]. 2.2 Solvng the ntegal equaton n a tme nteval of a length dt Fo smplcty the lowe ndex of the tme nteval wll be dopped n the followng, and t wll be assumed that the ntal W t ) vansh (only the ncements matte). The esult of ( ths Paagaph wll be the (essentally well-nown) Itô ncement of X fo each α. The ncement X n [ t, t + dt], wth X ( t) = x, s gven by the ntegal equaton

4 4 X t+ dt ( dt) = a ( x) dt + b [ x + X ( τ )] dw ( τ ) t. (2.2) Fo small enough dt the ntegal can be solved explctly n the ode dt. To ths end t s suffcent to expand b to the fst ode, whch esults n b t+ dt m ( x) W ( dt) + b, m ( x) X ( τ ) dw ( τ ) t. Insetng the leadng pat (of the ode O ( dt ) ) t+ dt dt m mn nto the ntegal esults n X ( τ ) dw ( τ ) = b ( x) Wn ( τ ) dw ( τ ). t 0 The last ntegal nvolves α ; fo 2 = n t s well-nown to yeld [ W ( dt) + (2α 1) dt]/ 2, 2 wth the expectaton α dt and wth the α -ndependent vaance ( dt ) / 2. Fo small enough dt ths allows to eplace the ntegal by the nonandom value α dt. Snce fo n the expectaton s zeo, the esult s X ( dt) b ( x) W ( dt) + a NID ( x)α dt wth a NID m ( x) : = b, m ( x) b ( x). (2.3) Ths yelds the explct esult X = [ a ( x) + α a NID ( x)] dt + b ( x) W ( dt) + o( dt) (2.4) and theeby the equvalent Itô fom of (2.1) dx fo each α. = b ( X ) dw + [ a ( X ) +α a NID ( X )] dt (2.5) III. The Foe-Planc equaton The FPE [1-5] coespondng to (2.5) s w, t = { ( a + α ) w + (1/ 2)( D w), }, : = L w, (3.1) wth the dffuson coeffcents D ( x) : = b j ( x) b j ( x). (3.2)

5 5 The last tem of L ncludes a dft pat, n vew of ( = +. By D w), D, w D w, a NID = D, / 2, (3.3) see the Appendx, (3.1) can thus be ewtten as w = { [ a + ( α 1) a ] w + (1/ 2) D }. (3.4), t NID w Note that fo α = 1 the nose-nduced dft falls away, and the dffuson paametes (3.2) ae only dffeentated once. The last popety was obseved n [6,7] fo some physcal systems wth themal equlbum. By use of the pobablty cuent J : = [ a + ( α 1) anid ] w (1/ 2) D w (3.5) the FPE can be vewed as the consevaton law of pobablty, snce t amounts to, + J = 0. Fo α = 1 t genealzes Fc s law fo patcles movng wth the exta w t nonandom velocty a (x ). IV. Condtonal ncements, and the value of α The squae bacet n (3.4) o (3.5) s the total (o effectve) dft: a + (α 1) : = a TOT. (4.1) Consstently, the mean ncement when X ( t) = x at some tme t, s gven by < dx > = a ( x dt, (4.2) = X t x TOT ) ( ) as s easly vefed by computng L z w dz wth the delta-functon w = δ ( z x) ( D w does not contbute). Note that dx > s not n geneal gven by ( a + α a ) dt NID as (2.5) mght suggest; < X ( t ) =x see the Comment (5) below fo an ntutve explanaton.

6 6 On the othe hand, the evdent meanng of (2.1) s dx ) = a ( x) dt + b ( x dw X ( t ) = x (4.3) whch entals that < dx >= a ( x) dt + b ( x) < dw > = a ( x) dt. (4.4) = X ( t) x Ths only agees wth (4.2) and (4.1) when α = 1 f ( x) 0. (4.5) Comments: (3) It s not possble to consde (x) as a pat of a (x ), n vew of the Comment (2) n the Chapte II : n vew of (3.3) t tansfoms qute dffeently, snce a twce contavaant tenso. (4) (4.3) s not based on a specfc α, and t seves fo modelng by (2.1). D s (5) It s possble to let a ( x) 0, and thus to consde the mpact of pue nose. Somewhat supsngly, X (t) s then a matngale (wth α = 1, and not n the Itô case), espectve of (x). Ths can be undestood as follows: (x) eally dves X n the decton of nceasng dffuson, but the lage dffuson thee causes a depleton and thus a bacflow whch compensates the afflux. Accodngly, the sample paths computed by dx = b ( x) dw + a NID ( x) dt exhbt an ntal tendency to follow (x), but also an enhanced pobablty of etun. Stll fo a ( x) 0, a gven ntal pobablty densty w(x ) s meely flattened and spead out n tme, snce the cuent D ); moeove, J = D w goes aganst w ( w J 0, as fo constant J = 0 whee w = 0, so that extema of w ae not shfted n x. [An example: Ovedamped Bownan moton [2] n a medum wth tempeatue T (x ) and wth eflectng boundaes always leads to a constant patcle densty].

7 7 V. Summay and concludng emas The unque value α = 1 fo solvng an SDE was found to be a consequence of the assumed ndependence of the coeffcent functons. The coespondng geneal FPE eads, = [ a w+ (1/ 2) D w]. (5.1) w t It does not nvolve a possble nose-nduced dft (n contast to the soluton of the SDE), no the second devatves of D. It s nvaant unde changes of x, snce w s a scala densty, a a contavaant vecto (by assumpton) and D a twce contavaant tenso. Mnd that the pesent aguments apply fo stctly contnuous and Maovan models (by the popetes of the Wene pocesses), see e.g. [5] fo somewhat dffeent models. The pesent assumptons povde a vald appoxmaton n vaous physcal models wth themal equlbum [6,7]. They ae howeve only patally met n [8] (dealng wth the Itô o Statonovch dlemma ) and n the theoy [9], whch s elated wth α = 1/ 2. Appendx The nose-nduced dft a N can be expessed n tems of the dffuson matx D, see (3.2). Fo a dagonal matx B (of the elements b ) t s obvous that j b, j b = D, / 2, (A.1) and fo symmetc B the same follows by dagonalzng B. Each non-symmetc B can be symmetzed on substtutng W (t) by an equvalent W *( t) gven by dw : = O dw * : Note that fo any T 1 1T T = B B = B * O O B, B *: = B O the dffuson matx D s peseved, D * when O s othogonal ( det O = 1 s also admtted). Ths entals B dw = B * dw *. When B s squae, one can fnd a O whch yelds a symmetc B * by whch (A.1) holds agan; a ectangula B can be completed by zeos. Ths shows that (A.1) holds n geneal (but

8 8 only by stochastc equvalence when B s not symmetc) : j a N : = b, j b = D, / 2. (A.2) Refeences [1] L. Anold, Stochastsche Dffeentalglechungen (Oldenboug, München, 1973), and Stochastc Dffeental Equatons: Theoy and Applcatons (Wley, New Yo, 1974) [2] H. Rsen, The Foe-Planc Equaton (Spnge, Beln, 1989) 2nd ed. [3] I.I. Gchman and A.W. Soochod, Stochastsche Dffeentalglechungen (Aademe- Velag, Beln, 1971) [4] B. Osendal, Stochastc dffeental equatons (Spnge, Beln, Hedelbeg, New Yo, 1992) [5] C. W. Gadne, Handboo of Stochastc Methods (Spnge, Beln, Hedelbeg 1985) [6] H. Gabet and M.S. Geen, Phys.Rev.A19, 1747 (1979) [7] Yu.L. Klmontovch, Physca A 163, 515 (1990) [8] N.G. van Kampen, Stochastc Pocesses n Physcs and Chemsty (Elseve/Noth- Holland, Amstedam, 3 d ed. 2008). [9] E. Wong and M. Zaa, Ann. Math. Statst. 36, 1560 (1965)