Forward-BackwardDoublyStochasticDifferentialEquationswithBrownianMotionsandPoissonProcess.doc

精品论文forward-backward doubly stochastic differential equations with brownian motions and poisson process zhu qingfengschool of statistics and mathematics, shandong university of finance, jinan 250014shi yufengschool of mathematics and system science, shandong university, jinan 250100abstract the existence and uniqueness for solution of backward doubly stochastic dif- ferential equations with brownian motions and poisson process and that of forward-backward doubly stochastic differential equations with brownian motions and poisson process can be obtained by means of a method of continuation. furthermore the continuity of the solutions of fbdsdep depending on parameters is also proved.key words forward-backward doubly stochastic differential equations; stochastic anal- ysis; random measure; poisson process1 introductionlet (, f , p ) be a probability space, and 0, t be a fixed arbitrarily large time duration throughout this paper. we suppose ft t0 is generated by the following three mutually independent processes:(i) let wt ; 0 t t and bt ; 0 t t be two standard brownian motions defined on(, f , p ), with values respectively in rd and in rl .(ii) let n be a poisson random measure, on r+ x, where x rl is a nonemptyopen set equipped with its borel field b(x ), with compensator nb (dx, dt) = (dx)dt, such thatne (a 0, t) = (n nb )(a 0, t)t0 is a martingale for all a b(x ) satisfying (a) . is assumed to be a -finite measure on (x, b(x ) and is called the characteristic measure.let n denote the class of p -null elements of f . for each t 0, t , we define.ft = f w f b f ntt,tt17where for any process t , f = r t ; s r t n , f = f . note that the collec-s,tt0,ttion ft , t 0, t is neither increasing nor decreasing, and it does not constitute a commonfiltration.for any n n , let m 2 (0, t ; rn ) denote the set of (classes of dp dt a.e. equal) n dimen-t2sional jointly measurable stochastic processes t ; t 0, t which satisfy: (i) e r0 |t | dt ; (ii) t , t 0, t is ft - measurable, for a.e. t 0, t . obviously m 2 (0, t ; rn ) is a hilbert space. for a given u m 2 0, t ; rd and v m 2 0, t ; rl , one can define the (standard)t0forward itos integral r us dws and the backward itos integral rm 2 (0, t ; r). (see pp2 for details.) vs dbs . they are both inin this paper we consider the following forward-backward doubly stochastic differential equations with brownian motions and poisson process (in short fbdsdep): partially supported by foundation for university by ministry of education of china, national natural science foundation of china grant 10201018, doctor promotional foundation of shandong grant 02bs127, and program for changjiang scholars and innovative research team in university. corresponding author: 精品论文 dyt = f (t, yt , yt , zt , zt , kt ) dt + g (t, yt , yt , zt , zt , kt ) dwt zt dbtr+x ht , yt , yt , zt , zt , kt (x)ne (dxdt),dy= f (t, y , y , z , z , k ) dt + g (t, y , y , z , z , k ) db z dw(1)ttt tttrtt tttttt x kt)ne (dxdt), y0= (y0 ) , yt= (yt ) .where (yt , yt , zt , zt , kt ) takes values in rn rm rnl rmd rm , f, g, h, f, g, , are mappings with appropriate dimensions which are ft -progressively measurable. our aim is to find a quintuple ft -adapted processes (yt , yt , zt , zt , kt ) satisfying the equation (1).nonlinear backward stochastic differential equations with brownian motion as noise source(bsde in short) was published by pardoux & peng pp1 in 1990, then bsde with poisson process (bsdep in short) was discussed by tang & li tl, situ st, wu w, and so on.a class of backward doubly stochastic differential equations (bdsde in short) was intro- duced by pardoux & peng pp2 in 1994. however, the theory of bdsde with poisson process (bdsdep in short) has not been investigated, which will be discussed in this paper.in 2003 peng & shi ps have introduced a type of time-symmetric forward-backward stochastic differential equations (fbsdes in short), in which both the forward equation and the backward one are types of sdes introduced by pardoux & peng pp2 under the name “backward doubly stochastic differential equations” with different directions of stochastic in- tegral. this type of fbsdes is also called as forward-backward doubly stochastic differential equations (fbdsdes in short). recently zhu, et. al z have extended the results to different dimensional fbdsdes and weaken the monotone assumptions.in this paper we consider the forward-backward doubly stochastic differential equations with brownian motions and poisson process (in short fbdsdep). in section 2, we give the preliminary existence and uniqueness result of besdep. in section 3, we successfully obtain the existence and uniqueness result for fbdsdep under some monotone assumptions. in section4, we study the problem and successfully get the desired result.2 preliminary: bdsde with poisson processwe first give the existence and uniqueness result of a solution to the following bdsdewith brownian motions and poisson process,x dyt = f (t, yt , zt , kt ) dt + g (t, yt , zt , kt ) dbt zt dwt rtk (x)ne (dxdt),(2)yt = .nwe denote the set of mappings k : 0, t x rm by f 2 (0, t ; rm ) = kt (, x) :tt0 x2k () is f predictable and e r t rkt (x)(dx)dt 0,such that+ r22 |f (t, y1 , z1 , k1 ) f (t, y2 , z2 , k2 )| c|yb |2+ |zb|2+xx |kb (x)| (dx) ,2 |g(t, y1 , z1 , k1 ) g(t, y2 , z2 , k2 )| c|yb |2 + |zb|2r |kb (x)|2 (dx) , yb = y1 y2 , zb = z1 z2 , kb = k1 k2 .then, we have the following result.theorem 2.1 assume (h2.1) holds. then, there exists a unique triple (y, z, k) nm 2 (0, t ; rm+md ) f 2 (0, t ; rm ) satisfying equation (2).proof for f m 2 (0, t ; rm ), g m 2 (0, t ; rmfiltration (gt )0tt byl ), l2(, ft , p ; rm), we set thegt = f w f b f ntt tand the gt -square integrable martingalez t z tmt = e +0f (s)ds +0g(s)dbs |gt , 0 t t .an obvious extension of itos martingale representation theorem (see iw) yields the existence of gt -progressively measurable processes zs , ks () such thathencemt = m0 +z tzs dws +0z t z0xsk (x)ne (dxds),0 t t .letz tmt = mt +tzs dws +z t zstxk (x)ne (dxds),yt = mt z tz tz tf (s)ds 0z tz tg(s)dbs0z t z= mt f (s)ds 0g(s)dbs 0 tzs dws tks (x)ne (dxds).xthen (yt , zt , kt ) is the solution of the equationzdyt = f (t) dt + g (t) dbt zt dwt xkt (x)ne (dxdt),yt = .so for each (y , z , k ) m 2 (0, t ; rm+md ) f 2 (0, t ; rm ), there exists (y , z , k ) satisfyingtttnttt dyt = f t, y , z , kdt + gt, y , z , kdb z dw rx k(x)ne (dxdt),yt = .ttt ttt ttttwe only need to prove that the mapping defined by(ut ) = ut : m 2 (0, t ; rm+md ) f 2 (0, t ; rm ) m 2 (0, t ; rm+md ) f 2 (0, t ; rm )nnis contract, where ut = (yt , zt , kt ), ut = (y , z , k ).tttwe let u 1 = (u 1 ), ubt = ut u 1 , uet = ut u 1 . using itos formula to et |ybt |2 , r, wettttgetxt0e|yb0 |2 + e res |ybs |2 + |zbs |2 + r|kbs (x)|2 (dx) dse= 2e r t es dyb , f (t, y , z , k ) f (t, y 1 , z1 , k 1 )e ds0 stttttt+e r t es |g(t, y , z , k ) g(t, y 1 , z1 , k 1 )|2 ds01 tttt s 2 2tttr2 2c t s 2+x2e r0 e|yes |+ |zes |r |kes (x)| (dx)ds+1 0 e|ybs | ds+e r t es c|ye |2 + |ze |2 + r|ke (x)|2 (dx) ds,0 ssx sthere exist c, 0 such thatte|yb0 |2 + e res ( )|yb |2 + |zb |2 + r|kb (x)|2 (dx) ds0 ssx s ce r t es |ye |2 ds+ 1 + e r t es |ze |2 + r|ke (x)|2 (dx) ds.0 s20sx s+ 1 + now choose = + 12c, and define c = 2c ,te|yb0 |2 + e res c|yb |2 + |zb |2 + r|kb (x)|2 (dx) ds1 + 0 ssx st s 2 2 2 +x2e r0 ec|yes |+ |zes |r |kes (x)| (dx)ds.then the mapping has a unique fixed point which is the unique solution of bdsdep (2).3an existence and uniqueness result of fbdsdep (1)in this section we consider the forward-backward doubly stochastic differential equation(1) with brownian motions and poisson process, wheref: 0, t rn rm rnl rmd rm rn ,g: 0, t rn rm rnl rmd rm rnd ,h: 0, t rn rm rnl rmd rm rn ,f : 0, t rn rm rnl rmd rm rm ,g : 0, t rn rm rnl rmd rm rml , : rm rn , : rn rm .2we are given an m n full-rank matrix h . let us introduce some notationsu = (y, y, z, z) ,a (t, u, k) = h t f, h f, h t g, h g (t, u, k) .where h g = (h g1 h gd ). we also assume that(h3.1) f, f, g, g, h and , are lipschitz: there exist constants c 0 and 0 0, 1 + 2 0, 1 + 2 0, 2 + 1 0. moreover we have 1 0, 1 0 (resp. 2 0, 2 0 ) when m n (resp.n m).then we have the main result of this section.theorem 3.1 we assume (h3.1)(h3.3), then equation (1) has a unique adapted solutionn(yt , yt , zt , zt , kt ) in m 2 0, t ; rn+m+nl+md f 2 (0, t ; rm ), t 0, t .sproof (uniqueness part) we first prove the uniqueness. let us = (ys , ys , zs , zs ), u 0 =(y0 , y 0 , z0 , z0 ) and (us , ks ), (u 0 , k0 ) be two solutions of (1). we set ub = (y y0 , y y 0 , z ssssss) = (b y , b z), k = (k k0 ). using itos formula to hh b , yb i, we getz0 , z z0y, bz, b bdyssee hh b, (y) (y0 )i eh ( (y ) (y 0 ) , ybyt t t0 0 0= e r t ha(s, u , k ) a(s, u 0 , k0 ), ub ids + e r t rhhbh, kb (x)i(dx)ds0 sssss0 x2s22 rt 2 2 ty | + |h z |ds eh t yb + h t zb + rh t kb (x)(dx)ds. 1 e r0then|h bsbs2 0s 2s x s2 2 rt 2 2 ty | + |h z |ds + eh t yb + h t zb + rh t kb (x)(dx) ds1 e r0|h bsyt 2bs22 0s s x s+1 e |h b |+ 2 e h t yb0 0.yt 22zt | 0. in thiswe first treat the case when m n; then 1 0, 1 0, |h b | 0, |h bcase we have b 0, b 0. thus y y0 , zs z0 . in particular, (yt ) = (y0 ). thus fromytzts s s ttheorem 2.1, it follows that ys y 0 , zs z0 and ks k0 .ssswe now discuss the second case when m 0, 2 0, thus ys y 0 , zs z0 ,ssand ks k0 . in particular, (y0 ) = (y 0 ). thus from theorem 2.1, it follows that ys y0 ,s0 ssand zs z0 .similarly to above two cases, the result can be obtained easily in the case m = n.in order to get the existence part of the theorem, we need the following lemma 3.2 andlemma 3.3 according to different dimensions of y and y .case 1 m n. then 1 0, 1 0. we consider following family of fbdsdepparametrized by 0, 1: dyt = f (t, ut , kt ) + f0 (t) dt zt dbt + g (t, ut , kt ) + g0 (t) dwt+rtxht , u , kt (x)+ h0 (t , x)ne (dxdt),dyt = f (t, ut , kt ) (1 ) 1 h yt + f0 (t) dt zt dwtr(3)+ g (t, ut , kt ) (1 ) 1 h zt + g0 (t) dbt x kt (x)ne (dxdt), y0 = (y0 ) + , yt = (yt ) + (1 ) h yt + ,nwhere ut = (yt , yt , zt , zt ) and (f0 , f0 , g0 , g0 ) m 2 0, t ; rm+n+ml+nd , h0 f 2 (0, t ; rm ), l2 (, f0 , p ; rn ) and l2 (, ft , p ; rm ) are given arbitrarily .when = 1, the existence of equation (3) implies that of equation (1). when = 0, fromthe existence and uniqueness of bdsdep, we can easily know that equation (3) has a uniquesolution. we need the following lemma.lemma 3.2 we assume m n, (h3.1)(h3.3). if for an 0 0, 1), l2 (, f0 , p ; rn ),n l2 (, ft , p ; rm ), (f0 , f0 , g0 , g0 ) m 2 0, t ; rm+n+ml+nd , h0 f 2 (0, t ; rm ), thereexists a solution (y0 , y 0 , z0 , z0 , k0 ) of (3), then there exists a positive constant 0 , suchthat for each 0 , 0 + 0 , there exists a solution (y , y , z , z , k ) of fbdsdep (3).proof since for each l2 (, f0 , p ; rn ), l2 (, ft , p ; rm ), (f0 , f0 , g0 , g0 ) n (0, t ; rm 2 (0, t ; rm+n+ml+nd ), h0 f 2 m), = 0 , there exists a (unique) solution of (3),nthus, for each (u , k ) = y, y , z, z, k m 2 0, t ; rn+m+nl+md f 2 (0, t ; rm ), therenexists a unique (u, k) = (y, y, z, z, k) m 2 0, t ; rn+m+nl+md f 2(0, t ; rm ) satisfyingthe following fbdsdep: dyt = 0 f (t, ut , kt ) + f t, ut , k t + f0 (t) dt zt dbt+ 0 g (t, ut , kt ) + g t, ut , k t + g0 (t) dwtr e+ x0 ht , ut , kt (x)+ ht , ut , kt (x)+ h0 (t )n (dxdt),dyt = 0 f (t, ut , kt ) (1 0 ) 1 h yt + f t, ut , k t + 1 h yt + f0 (t) dt+ 0 g (t, ut , kt ) (1 0 )1 h zt + g t, ut , k t + 1 h zt + g0 (t) dbtrxzt dwt kt (x)ne (dxdt),0 y0 = 0 (y0 ) + (y ) + , yt= 0 (yt) + (1 0 ) h yt+ ( (yt) h yt) + ,here is independent of the positive constant 0 no more than 1.we want to prove that the mapping defined by0(ut , kt , yt , y0 ) = i0 + (ut , k t , yt , y ) :n (0, t ; rm 2 0, t ; rn+m+nl+md f 2m m 2 0, t ; rn+m+nl+md f 2) l2m(, f0 , p ; rn2) l2n(, ft , p ; rm )2 mis contract.n (0, t ; r) l(, f0 , p ; r) l(, ft , p ; r )let (u 0 , k