Representation of continuous local martingales with spatial parameter

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Let

• $(Omega,mathcal A,operatorname P)$ be a complete probability space


• $T>0$


• $I:=(0,T]$


• $(mathcal F_t)_tinoverline I$ be a complete and right-continuous filtration on $(Omega,mathcal A,operatorname P)$


• $mathcal M_c^2(mathcal F,operatorname P)$ denote the set of real-valued continuous square-integrable $mathcal F$-martingales on $(Omega,mathcal A,operatorname P)$ equipped with $$langle M,Nrangle_mathcal M_c^2(mathcal F,operatorname P):=operatorname Eleft[[M,N]_Tright];;;textfor M,Ninmathcal M_c^2(mathcal F,operatorname P),$$ where $[M,N]$ denotes the quadratic variation of $M$ and $N$


• $(M^n)_ninmathbb N$ be an orthogonal basis of $mathcal M_c^2(mathcal F,operatorname P)$


• $mathcal M_c,:textloc(mathcal F,operatorname P)$ denote the set of real-valued continuous local $mathcal F$-martingales on $(Omega,mathcal A,operatorname P)$


• $E$ be a separable metric space


• $M:Omegatimesoverline Itimes Etomathbb R$ with $$M(;cdot;,;cdot;,x)inmathcal M_c,:textloc(mathcal F,operatorname P);;;textfor all xin E$$ and $$M(omega,t,;cdot;)in C^0(E);;;textfor all (omega,t)inOmegatimesoverline I$$

How can we show that there is a sequence $(f^n)_ninmathbb N$ with $f^n:Omegatimesoverline Itimes Etomathbb R$ being $mathcal Aotimesmathcal B(overline I)otimesmathcal B(E)$-measurable, $f^n(;cdot;,;cdot;,x)$ being $mathcal F$-predictable for all $xin E$ and $$M_t(x)=sum_ninmathbb Nint_0^tf^n(s,x):rm dM_s^n;;;textfor all tinoverline Itext almost surely for all xin E?tag1$$

probability-theory stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis

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Let

• $(Omega,mathcal A,operatorname P)$ be a complete probability space


• $T>0$


• $I:=(0,T]$


• $(mathcal F_t)_tinoverline I$ be a complete and right-continuous filtration on $(Omega,mathcal A,operatorname P)$


• $mathcal M_c^2(mathcal F,operatorname P)$ denote the set of real-valued continuous square-integrable $mathcal F$-martingales on $(Omega,mathcal A,operatorname P)$ equipped with $$langle M,Nrangle_mathcal M_c^2(mathcal F,operatorname P):=operatorname Eleft[[M,N]_Tright];;;textfor M,Ninmathcal M_c^2(mathcal F,operatorname P),$$ where $[M,N]$ denotes the quadratic variation of $M$ and $N$


• $(M^n)_ninmathbb N$ be an orthogonal basis of $mathcal M_c^2(mathcal F,operatorname P)$


• $mathcal M_c,:textloc(mathcal F,operatorname P)$ denote the set of real-valued continuous local $mathcal F$-martingales on $(Omega,mathcal A,operatorname P)$


• $E$ be a separable metric space


• $M:Omegatimesoverline Itimes Etomathbb R$ with $$M(;cdot;,;cdot;,x)inmathcal M_c,:textloc(mathcal F,operatorname P);;;textfor all xin E$$ and $$M(omega,t,;cdot;)in C^0(E);;;textfor all (omega,t)inOmegatimesoverline I$$

How can we show that there is a sequence $(f^n)_ninmathbb N$ with $f^n:Omegatimesoverline Itimes Etomathbb R$ being $mathcal Aotimesmathcal B(overline I)otimesmathcal B(E)$-measurable, $f^n(;cdot;,;cdot;,x)$ being $mathcal F$-predictable for all $xin E$ and $$M_t(x)=sum_ninmathbb Nint_0^tf^n(s,x):rm dM_s^n;;;textfor all tinoverline Itext almost surely for all xin E?tag1$$

probability-theory stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis

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asked Jul 31 at 18:05

0xbadf00d

1,92041028

This question has an open bounty worth +50
reputation from 0xbadf00d ending ending at 2018-08-10 22:54:20Z">in 2 days.

This question has not received enough attention.

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up vote
0
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up vote
0
down vote

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Let

• $(Omega,mathcal A,operatorname P)$ be a complete probability space


• $T>0$


• $I:=(0,T]$


• $(mathcal F_t)_tinoverline I$ be a complete and right-continuous filtration on $(Omega,mathcal A,operatorname P)$


• $mathcal M_c^2(mathcal F,operatorname P)$ denote the set of real-valued continuous square-integrable $mathcal F$-martingales on $(Omega,mathcal A,operatorname P)$ equipped with $$langle M,Nrangle_mathcal M_c^2(mathcal F,operatorname P):=operatorname Eleft[[M,N]_Tright];;;textfor M,Ninmathcal M_c^2(mathcal F,operatorname P),$$ where $[M,N]$ denotes the quadratic variation of $M$ and $N$


• $(M^n)_ninmathbb N$ be an orthogonal basis of $mathcal M_c^2(mathcal F,operatorname P)$


• $mathcal M_c,:textloc(mathcal F,operatorname P)$ denote the set of real-valued continuous local $mathcal F$-martingales on $(Omega,mathcal A,operatorname P)$


• $E$ be a separable metric space


• $M:Omegatimesoverline Itimes Etomathbb R$ with $$M(;cdot;,;cdot;,x)inmathcal M_c,:textloc(mathcal F,operatorname P);;;textfor all xin E$$ and $$M(omega,t,;cdot;)in C^0(E);;;textfor all (omega,t)inOmegatimesoverline I$$

How can we show that there is a sequence $(f^n)_ninmathbb N$ with $f^n:Omegatimesoverline Itimes Etomathbb R$ being $mathcal Aotimesmathcal B(overline I)otimesmathcal B(E)$-measurable, $f^n(;cdot;,;cdot;,x)$ being $mathcal F$-predictable for all $xin E$ and $$M_t(x)=sum_ninmathbb Nint_0^tf^n(s,x):rm dM_s^n;;;textfor all tinoverline Itext almost surely for all xin E?tag1$$

probability-theory stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis

share|cite|improve this question

asked Jul 31 at 18:05

0xbadf00d

1,92041028

Let

• $(Omega,mathcal A,operatorname P)$ be a complete probability space


• $T>0$


• $I:=(0,T]$


• $(mathcal F_t)_tinoverline I$ be a complete and right-continuous filtration on $(Omega,mathcal A,operatorname P)$


• $mathcal M_c^2(mathcal F,operatorname P)$ denote the set of real-valued continuous square-integrable $mathcal F$-martingales on $(Omega,mathcal A,operatorname P)$ equipped with $$langle M,Nrangle_mathcal M_c^2(mathcal F,operatorname P):=operatorname Eleft[[M,N]_Tright];;;textfor M,Ninmathcal M_c^2(mathcal F,operatorname P),$$ where $[M,N]$ denotes the quadratic variation of $M$ and $N$


• $(M^n)_ninmathbb N$ be an orthogonal basis of $mathcal M_c^2(mathcal F,operatorname P)$


• $mathcal M_c,:textloc(mathcal F,operatorname P)$ denote the set of real-valued continuous local $mathcal F$-martingales on $(Omega,mathcal A,operatorname P)$


• $E$ be a separable metric space


• $M:Omegatimesoverline Itimes Etomathbb R$ with $$M(;cdot;,;cdot;,x)inmathcal M_c,:textloc(mathcal F,operatorname P);;;textfor all xin E$$ and $$M(omega,t,;cdot;)in C^0(E);;;textfor all (omega,t)inOmegatimesoverline I$$

How can we show that there is a sequence $(f^n)_ninmathbb N$ with $f^n:Omegatimesoverline Itimes Etomathbb R$ being $mathcal Aotimesmathcal B(overline I)otimesmathcal B(E)$-measurable, $f^n(;cdot;,;cdot;,x)$ being $mathcal F$-predictable for all $xin E$ and $$M_t(x)=sum_ninmathbb Nint_0^tf^n(s,x):rm dM_s^n;;;textfor all tinoverline Itext almost surely for all xin E?tag1$$

probability-theory stochastic-processes stochastic-calculus stochastic-integrals stochastic-analysis

share|cite|improve this question

asked Jul 31 at 18:05

0xbadf00d

1,92041028

share|cite|improve this question

share|cite|improve this question

asked Jul 31 at 18:05

0xbadf00d

1,92041028

asked Jul 31 at 18:05

0xbadf00d

1,92041028

asked Jul 31 at 18:05

0xbadf00d

1,92041028

1,92041028

This question has an open bounty worth +50
reputation from 0xbadf00d ending ending at 2018-08-10 22:54:20Z">in 2 days.

This question has not received enough attention.

This question has an open bounty worth +50
reputation from 0xbadf00d ending ending at 2018-08-10 22:54:20Z">in 2 days.

This question has not received enough attention.

add a comment | 

add a comment | 

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