Prove that a process (given through rsdes) is a martingale.












1












$begingroup$


i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely,
let
$dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t
$

be the signal eqations,i.e. the filtering problem and
$dhat{X}_t = A(hat{X}_t) dt + P_t B^T R_2^{-1} left[dY_t -Bhat{X}_t dtright] textrm{where } hat{X}_0=mathbb{E}[X_0] \
partial P_t = partial A(hat{X}_t)P_t+P_tpartial A(hat{X}_t)+ R_1 - P_tSP_t textrm{where }S:=B^TR_2^{-1}B$

be the Kalman Bucy filter. the second equation is a riccati equation.
then there is the estimated distance between the filter $hat{X}_t $ and the signal $X_t$ which by a taylor expansion becomes



$
dtilde{X}_t:= [partial A (hat{X}_t)-P_t S]hat{X}_tdt + R_1^{1/2}dW_t -P_tB^T R_2^{-1/2} dV_t.
$

Now, $P_t=mathbb{E}[tilde{X}_t tilde{X}_t^T| mathcal{F}_t].$ solves the riccati equation. In the above situation, $V_t, W_t$ are independant (m+n)(finite)- dimensional brownian motions, $A$ is the vectorvalued, differentiable drift of the signal, $partial A$ its jacobian. R_1,R_2 are positive definite and symmetric dispersion matrices withe dimensions $ntimes n, mtimes m$ respectively, $B$ is $m times n$.



And now the process in question can be defined via $dM_t = (P_t-check{P}_t)B^T R_2^{-1/2} dV_t. $. I have tried to show that this is a locale martingale, but the variables are so entangled and codependant, that i am not able to find any conlusive representation. This is a problem from a course i am taking and the literature claims that this process is a martingale without any other explanation. Can anybody help?










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    1












    $begingroup$


    i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely,
    let
    $dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t
    $

    be the signal eqations,i.e. the filtering problem and
    $dhat{X}_t = A(hat{X}_t) dt + P_t B^T R_2^{-1} left[dY_t -Bhat{X}_t dtright] textrm{where } hat{X}_0=mathbb{E}[X_0] \
    partial P_t = partial A(hat{X}_t)P_t+P_tpartial A(hat{X}_t)+ R_1 - P_tSP_t textrm{where }S:=B^TR_2^{-1}B$

    be the Kalman Bucy filter. the second equation is a riccati equation.
    then there is the estimated distance between the filter $hat{X}_t $ and the signal $X_t$ which by a taylor expansion becomes



    $
    dtilde{X}_t:= [partial A (hat{X}_t)-P_t S]hat{X}_tdt + R_1^{1/2}dW_t -P_tB^T R_2^{-1/2} dV_t.
    $

    Now, $P_t=mathbb{E}[tilde{X}_t tilde{X}_t^T| mathcal{F}_t].$ solves the riccati equation. In the above situation, $V_t, W_t$ are independant (m+n)(finite)- dimensional brownian motions, $A$ is the vectorvalued, differentiable drift of the signal, $partial A$ its jacobian. R_1,R_2 are positive definite and symmetric dispersion matrices withe dimensions $ntimes n, mtimes m$ respectively, $B$ is $m times n$.



    And now the process in question can be defined via $dM_t = (P_t-check{P}_t)B^T R_2^{-1/2} dV_t. $. I have tried to show that this is a locale martingale, but the variables are so entangled and codependant, that i am not able to find any conlusive representation. This is a problem from a course i am taking and the literature claims that this process is a martingale without any other explanation. Can anybody help?










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely,
      let
      $dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t
      $

      be the signal eqations,i.e. the filtering problem and
      $dhat{X}_t = A(hat{X}_t) dt + P_t B^T R_2^{-1} left[dY_t -Bhat{X}_t dtright] textrm{where } hat{X}_0=mathbb{E}[X_0] \
      partial P_t = partial A(hat{X}_t)P_t+P_tpartial A(hat{X}_t)+ R_1 - P_tSP_t textrm{where }S:=B^TR_2^{-1}B$

      be the Kalman Bucy filter. the second equation is a riccati equation.
      then there is the estimated distance between the filter $hat{X}_t $ and the signal $X_t$ which by a taylor expansion becomes



      $
      dtilde{X}_t:= [partial A (hat{X}_t)-P_t S]hat{X}_tdt + R_1^{1/2}dW_t -P_tB^T R_2^{-1/2} dV_t.
      $

      Now, $P_t=mathbb{E}[tilde{X}_t tilde{X}_t^T| mathcal{F}_t].$ solves the riccati equation. In the above situation, $V_t, W_t$ are independant (m+n)(finite)- dimensional brownian motions, $A$ is the vectorvalued, differentiable drift of the signal, $partial A$ its jacobian. R_1,R_2 are positive definite and symmetric dispersion matrices withe dimensions $ntimes n, mtimes m$ respectively, $B$ is $m times n$.



      And now the process in question can be defined via $dM_t = (P_t-check{P}_t)B^T R_2^{-1/2} dV_t. $. I have tried to show that this is a locale martingale, but the variables are so entangled and codependant, that i am not able to find any conlusive representation. This is a problem from a course i am taking and the literature claims that this process is a martingale without any other explanation. Can anybody help?










      share|cite|improve this question









      $endgroup$




      i have a rather complicated problem for a process which is connected to a kalman-bucy filter with a riccati equation. More precisely,
      let
      $dX_t =A(X_t)dt + R_1^{1/2} dWt, dY_=BX_tdt + R_2^{1/2}dV_t
      $

      be the signal eqations,i.e. the filtering problem and
      $dhat{X}_t = A(hat{X}_t) dt + P_t B^T R_2^{-1} left[dY_t -Bhat{X}_t dtright] textrm{where } hat{X}_0=mathbb{E}[X_0] \
      partial P_t = partial A(hat{X}_t)P_t+P_tpartial A(hat{X}_t)+ R_1 - P_tSP_t textrm{where }S:=B^TR_2^{-1}B$

      be the Kalman Bucy filter. the second equation is a riccati equation.
      then there is the estimated distance between the filter $hat{X}_t $ and the signal $X_t$ which by a taylor expansion becomes



      $
      dtilde{X}_t:= [partial A (hat{X}_t)-P_t S]hat{X}_tdt + R_1^{1/2}dW_t -P_tB^T R_2^{-1/2} dV_t.
      $

      Now, $P_t=mathbb{E}[tilde{X}_t tilde{X}_t^T| mathcal{F}_t].$ solves the riccati equation. In the above situation, $V_t, W_t$ are independant (m+n)(finite)- dimensional brownian motions, $A$ is the vectorvalued, differentiable drift of the signal, $partial A$ its jacobian. R_1,R_2 are positive definite and symmetric dispersion matrices withe dimensions $ntimes n, mtimes m$ respectively, $B$ is $m times n$.



      And now the process in question can be defined via $dM_t = (P_t-check{P}_t)B^T R_2^{-1/2} dV_t. $. I have tried to show that this is a locale martingale, but the variables are so entangled and codependant, that i am not able to find any conlusive representation. This is a problem from a course i am taking and the literature claims that this process is a martingale without any other explanation. Can anybody help?







      martingales sde local-martingales






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      asked Dec 13 '18 at 14:04









      DannyDanny

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