Martingale representation for degenerate diffusions

Abstract

Let $(W,H,\mu )$ be the classical Wiener space on ${\rm IR}^{d}$. Assume that $X=(X_{t})$ is a diffusion process satisfying the stochastic differential equation $dX_{t}=\sigma (t,X)dB_{t}+b(t,X)dt$, where $\sigma :[0,1]{\rm ×}C([0,1],{\rm IR}^{n})\rightarrow{\rm IR}^{n}\bigotimes{{\rm IR}^{d}}$, $b:[0,1]{\rm ×}C([0,1],{\rm IR}^{n})\rightarrow{\rm IR}^{n}$, B is an ${\rm IR}^{d}$-valued Brownian motion. We suppose that the weak uniqueness of this equation holds for any initial condition. We prove that any square integrable martingale M w.r.t. to the filtration $({\rm F}_{t}(X),t\in [0,1])$ can be represented as $$M_{t}=E[M_{0}]+\int\limits_{0}^{t}{P_{s}}(X)\alpha_{s}(X).dB_{s}$$ where $\alpha (X)$ is an ${\rm IR}^{d}$-valued process adapted to $({\rm F}_{t}(X),t\in [0,1])$, satisfying $E\int_{0}^{t}{(}a(X_{s})\alpha_{s}(X),\alpha_{s}(X))ds<\infty$, $a=\sigma^{\bigstar}\sigma$ and $P_{s}(X)$ denotes a measurable version of the orthogonal projection from ${\rm IR}^{d}$ to $\sigma\mathop{\left({X_{s}}\right)}^{\bigstar}({\rm IR}^{n})$. In particular, for any $h\in H$, we have $$ (0.0) %Translator MathMagic Pro Win v8.6, LaTeX converter, 2020.2.6 11:43 \begin{array}{r} {{E}{[}\mathit{\rho}{(}\mathit{\delta}{h}{)|}{\mathcal{F}}_{1}{(}{X}{)]}{=}{\mathrm{exp}}\left({\mathop{\int}\limits_{0}\limits^{1}{}}\right.\left({{P}_{s}}\right.\left({X}\right){\dot{h}}_{s}{,}{dB}_{s}\left.{{\dot{h}}_{s},{dB}_{s}}\right)}\\ {{-}\frac{1}{2}\hspace{0.25em}\mathop{\int}\limits_{0}\limits^{1}{}{|}{P}_{s}\left({X}\right){\dot{h}}_{s}{|}^{2}{ds}\left.{{\dot{h}}_{s}{|}^{2}ds}\right){,}} \end{array} %MathMagic MMF.7h]O5*00_ESKK]]64?d2oH(N:H0Fm|YMiTeFU5BX9ATBL`6R?:R9VP[e9IFE=ZVSOno(L4T)Kfj15[ERJloLM^K|L4QV)ennF:dGTfbnFPhN1m_YNSK9E^]X|C_m|]Qm)W`HGQo_Qfl)Ml?OoCPI3KKI?;^JAK?O_^a)Qo^k2`VRek?e1]`S6`mM?1*PNCk9IY6:QdTlE4:9N2SU(j)OZAAd|kOGEi?i(QX=cX?1MS?;|_Wb9FjmfB`WReWdjV5o76kfYm?QkQ)H?i`R4BlFdogMjL_afg;oAkh3R6FgF3G4:QO[B[bjnOSb^=ooF^Y(Ubj4|he`(QLWgI^k)3_LkQn681^^kfmgMjG6am?k;lL3e(K]deikfEMeCmVbYfkiE)7bZLYUCnVbYgKIGkc|[UkfUjnJiHO=EMnYmiB_WRYO?EFnjRUOmIB_n|]GgNF[o_9ecnW[W_9e|gbBo`VGSXZUb]LbU[4=LY36f^I[3F]Y0S00G32bZ511TJ0R[1f;i=4X2B05h((J6US6^OOW2=LRGf8J(Z`E(l4dk=SW0=)**F7;WCmC5XGL*FeR[0?bQ5C`CmTVU8P]P6BI:6*W[75kX`(P5|HQ||*4RWG20b(?DQL8VDP:T=JBDh8GcE9*b41J0(fZDi`2IIT;Cd6i:SOUjg]b3SA?*4]6R:Hc:80^3a20hJMSJl5eD^f[7B]LNahkIL3`50bVh4(0XeRYAW(c`d6NPRT((HF;DQVjXNPOc`Xa:B_42]iJ?0f;J*Aobho26^bCXQd|GAI5ECKQ?F0a2aGFWZgCZ](B`MJ|5A9EDIY`4Q;J_P2fRYXT;18NPRkbB3aEG:2df|GaRl7aAW1|OhOkQh_16F;F1FBi2aIOk)UH:c[Lgh*emF58fN?f)|CbO7^Omf61=5OaA_Ai5a*XZD[cV(15)*lldjAEJZUPJdIoZ]R)ZJjHC*e[kmBbPdTCk^9HajBNPiAMHE:8fV`A|PiEKG1Y=WV4hL1F1bA5KBP9Ed)NdbD5Jd*YADeG6hfB1U)YdTA]ZL`_ba9JkYQ`h69C0BCUXPaIHhD6I7WjTTJT:X3RP:j(0W1:J4JZ4_6cTC*VOHUHJdS5;PjYnGf:aV*iH:AVehN4*BTZ*n[?8|(`:FeYFS|M6YHUTC*]:eBSQ)IUdLfB1VH9E(d*LbTXXHV9VYNCJkS[RYPZScElidRDFTVX[XE?d58STYiobj3Eh)F3g8NoWXHWJPe4e453A17F*F]9d_X4KD9Im6GUBZdS[J]YoI)njM=|o0eI4Tne(n_eJYE5HPbGY1PWl2]a0Gd:8[Zeg?kLN8?9IK93Y][?XkU2MaQg?;cV2Wa`WAb?ml4)WeL_3gNkhkLL)l3;oJOMJImS3gRm_j7g^EbBP^CekWSHoGBcEb5K;65fnQ0*9SokNS[^0UK(Ha9TV|T^Pl``fCC8;9(m3k:TRc3G9OAm;(RdRf3AIdh_5X`eNZ7P]=6K1)N=gQkZa2WKIXkN5d[VU:|cYgbK)IFfVJ(GPPIcm3;*H:ka8Q24GNfSNo]7fbkcY=OLeIWC_|6LCQ_(6M5Rc|PfLdIaiXb^(fM(VcUSflbIY(fLL*gVMSM`eN3C;cB8YgmP4P2|1Jd9JAaS^f?=g:2HiP4:C;0d60B(YeM[603;nn?]kVJHkKnN`Tc(?g:L3noYjV[E]8(9*SlM2g03Yl|O:CR|=ZlF4M8fGfHAQ]_l|7Yc)MW(D7W6j(]|]|c`_eW03|KFhoKbLC^=IR?h|gdgfVkPD0B1285(=89OB3;j3Uo_hSR6UX4O6LO_XaNS;FbD577Tj7`)c]?XKNjeODnA_0_NYNonjfObQS|2^FOHRgU:icbL69g;^=|i_SeRo=LQRX1aPg4d:5VHJGAMCnX1W?ZlA)GeM]AUe18I(9|dI0Y^NHo4dkVe(g8*8a(O;m]YPD0E6OaW*M]U67*1o*IVRTB7dUiaQY]^2SWJ`?edm0o?1Q3fB=L1o5mLOjN]7J|?YAlOA_n2j]jHWDcWI`?7(?P;)cbRm*.mmf $$

where $ %Translator MathMagic Pro Win v8.6, LaTeX converter, 2020.2.6 11:52 \mathit{\rho}{(}\mathit{\delta}{h}{)}{=}{\mathrm{exp}}{(}\mathop{\int}\nolimits_{0}\nolimits^{1}{(}{\dot{h}}_{s}{,}{dB}_{s}{)}{-}\frac{1}{2}\hspace{0.25em}{|}{H}{|}_{H}^{2}{)} %MathMagic MMF.7h|V4`00QESKT]Y64?d2oX57DJFUY^NR6OV=aLBV|X0;I3]E8*o49SJE_CP|M^a|n?M(]fJTUY0fRgOYdcgCUc(m3O9f__aY]Ei(R_UZ)GPJK:O[fJAH[I?5k_Ai|O]dn31lLg`H_SoL3kniLCHJK8]iLC=;IWmmgId)3oMGh5G_I^^=gijHM6SCXO2JUi=RU|QdV:E3:JA8Q`0_S7bQdL7|UcLgToTb6*g)Pl5f(b^:nO8EQ]i|UY?5;7Wk^3l)=o_CjG3obBmo?2DRGBbVno_Cen)?iOk_(X9G*kMJ]]BbE:]J_K[mn)ZhgomIfGBG;KPc;GM*Z[?^h3H]3WOkaj7G3ML?Mk_kb^;BjL?Ghl7Ga]OW_N^Q[nZN|Z6WKWR^L7R^L^PY7GYZQokRXK]jj2mO]||?`FGOZOND;il[GciG_^`YGoJD;o_;UmgUbokbELoYZikbEK]ld_oS[hi(*IHbY92JX?OJE9UBEUh67H3f`8I51Pdb638d1=TbC`hGI*7T7[PPn`J6]=cm9D5IU2:V0D6FK0VVHLJ^19P618)Y8WnQ;:;Nn][4F0GT2(V`?fM1:14C0K1(9;8CI0b_E*34`SQh1T`PbQUgS3b0RPRIb2;86lU9`H]V:DQT88m0|NXTYd0J]XFW86fMVgC=V9`3aA=*`0QAM0HAZ)X0?M3lM4c3^L[Z^(Zb`YGS_W(6=4m1H`X^)=2BUJXEGjHi:5?*LB6VL5DI*cO4oW6|49fc*XcP[LGC(9Q6f6ohDAR=OA;K`M2eR5FIS?N0`BaTT1fClk[C(|5TePZI[2W=)0TIQHo0e5jcS7W2*e0aSlaAaA7UMAC;;h?USF1IO8_a`fF`VYRe0AVn1H^?(Be[AH_aMI2Y3d?:3|)[h(_al:k|`hPD=o56M6DGA9CEYCU(h:ZJ1hiIlSZeG32IdIm;5S5G=K)iI^fM6gH`NLJgF=Ha^N(PIcL(Q6S(5P5=:1^3Bk798c*7YShP48fQ96`3)DhG2=J80:9QJha6X(5DVAAAFaW;JeU1`cMV7=QDe`19^JYL=URQ0EVM?]28U1583^QVA(0YXATY:lC?1VQ(^PZaeP39;PLXoSU5Hk8J(:3Ho*0o:4FmT?XcIQPVYJVF=Tj7QVE594g;6SDXXGTI^aUXH5I0=QIR;Y4BVYQXNCEihcmeADZEYlZoUdQDER3D]?YG|58STYfo*k0Z_l|5_*]o7*e?]6[_D*D;4DEH1J|QcLD[F3?:XRl[FeT]FFg3jYkMVco?a_n*1GRZWEV_Ej|R4F=o9LDhlkn0P^mC[j:?U[|oFTl`Y*hjM?;bnfQYD1f;)kjlUPKlhSXi7Qo2)_bnNWfhgaeoU=Qj_=aofYgf9GHN[oNgm3aGJW:_NKLk7WJogniUb1I;V9dn18C9ckjOS[^09M|a2C[5M=M1YiU^6WB6jEh6GMI5V)eB^ShF8)lRF?*]Y`L;aQXmD73Jj4V2ldI?3dgRY;UTSYhG:^JTKC8WgBEc([mTSQh8F|cA`d2;^MJ3*51f]HoZkAmU^YIW_L]]TcWUF|bY_(FL5QO(JKQTCT_)W5I=i[BnI4jKBnIdM|VL]RgVM[One^2gGmlPS_ki9*5hFI1(B)4HfadKbcFZJAjP*XNE6YghaM)K]Al0bhOSgNifF)boWl9(;5l`;XOgM7FcJZoc4hAn)PBocFnjoYVLNfWcMY4PKO=UTJ2kcN_Enn_9IXK6(gYO5[=UPOo=L_fdgOPC4:?]==TV8`lPD`PnTfI46VLmn3D]Ob1=Od_fgko47*EfVAPmXB_BW4]9A06l((MW?_ln2Ej4Wa7XAoY?RBN:M_J)88]^7TOW(gY;(L[7je6WTI:C)J2e(TXNmUdkW?;:SOmX*;OohYoG*O;nCGCa)[Z*EJ3cNO0O0l*E3`.mmf $ In the case the process X is adapted to the Brownian filtration, this result gives a new development as an infinite series of the $L^{2}$-functionals of the degenerate diffusions. We also give an adequate notion of “innovation process” associated to a degenerate diffusion which corresponds to the strong solution when the Brownian motion is replaced by an adapted perturbation of identity. This latter result gives the solution of the causal Monge–Ampère equation.