Prove that a process (given through rsdes) is a martingale.
$begingroup$
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?
martingales sde local-martingales
asked Dec 13 '18 at 14:04
DannyDanny
395
$endgroup$
add a comment |
$begingroup$
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?
martingales sde local-martingales
asked Dec 13 '18 at 14:04
DannyDanny
395
$endgroup$
add a comment |
$begingroup$
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?
martingales sde local-martingales
asked Dec 13 '18 at 14:04
DannyDanny
395
$endgroup$
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
martingales sde local-martingales
asked Dec 13 '18 at 14:04
DannyDanny
395
asked Dec 13 '18 at 14:04
DannyDanny
395
asked Dec 13 '18 at 14:04
DannyDanny
395
asked Dec 13 '18 at 14:04
DannyDanny
395
asked Dec 13 '18 at 14:04
DannyDanny
395
395
add a comment |
add a comment |
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