Browsing by Subject "Relative entropy"

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    ItemOpen Access
    Adaptive mixture methods based on Bregman divergences
    (Elsevier, 2013) Donmez, M. A.; Inan, H. A.; Kozat, S. S.
    We investigate adaptive mixture methods that linearly combine outputs of m constituent filters running in parallel to model a desired signal. We use Bregman divergences and obtain certain multiplicative updates to train the linear combination weights under an affine constraint or without any constraints. We use unnormalized relative entropy and relative entropy to define two different Bregman divergences that produce an unnormalized exponentiated gradient update and a normalized exponentiated gradient update on the mixture weights, respectively. We then carry out the mean and the mean-square transient analysis of these adaptive algorithms when they are used to combine outputs of m constituent filters. We illustrate the accuracy of our results and demonstrate the effectiveness of these updates for sparse mixture systems.
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    ItemOpen Access
    Dual representations for systemic risk measures
    (Springer, 2020) Ararat, Çağın; Rudloff, B.
    The financial crisis showed the importance of measuring, allocating and regulating systemic risk. Recently, the systemic risk measures that can be decomposed into an aggregation function and a scalar measure of risk, received a lot of attention. In this framework, capital allocations are added after aggregation and can represent bailout costs. More recently, a framework has been introduced, where institutions are supplied with capital allocations before aggregation. This yields an interpretation that is particularly useful for regulatory purposes. In each framework, the set of all feasible capital allocations leads to a multivariate risk measure. In this paper, we present dual representations for scalar systemic risk measures as well as for the corresponding multivariate risk measures concerning capital allocations. Our results cover both frameworks: aggregating after allocating and allocating after aggregation. As examples, we consider the aggregation mechanisms of the Eisenberg–Noe model as well as those of the resource allocation and network flow models.
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    ItemOpen Access
    Martingale representation for degenerate diffusions
    (Elsevier, 2019) Üstünel, Ali Süleyman
    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.
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    ItemOpen Access
    Set-valued shortfall and divergence risk measures
    (World Scientific Publishing, 2017) Ararat, C.; Hamel, A. H.; Rudloff, B.
    Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate risk measures is constructed via a recent Lagrange duality for set optimization. In particular, it is shown that a shortfall risk measure can be written as an intersection over a family of divergence risk measures indexed by a scalarization parameter. Examples include set-valued versions of the entropic risk measure and the average value at risk. As a second step, the minimization of these risk measures subject to trading opportunities is studied in a general convex market in discrete time. The optimal value of the minimization problem, called the market risk measure, is also a set-valued risk measure. A dual representation for the market risk measure that decomposes the effects of the original risk measure and the frictions of the market is proved.