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We have constructed above the Radom-Nikodym derivative process $(L_t)_{t\geq 0}$ , which is a $(\mathbb{P}, \mathbb{F})$-martingale. Because $\mathbb{F}$ is generated by $(W_t)_{t\in[0;T]}$ we can apply the martingale representation theorem $\Rightarrow (\exists) (\psi_t)_{t\geq 0}$ an $\mathbb{F}$-measurable process s.t.: $$ L_t=1+\int_0^t \psi_udW_u. $$ or, alternatively, that: $$ dL_t=\psi_tdW_t, L_0=1. $$ Since the Radon-Nikodym derivarive process is strictly positive, using Ito lemma we get: $$ d\log(L_t)=\frac{1}{L_t}dL_t-\frac{1}{2}\frac{1}{L^2_t}d\langle L \rangle_t $$ Since $L_t$ is strictly positive, we can simplify things a bit by introducing $$ \Theta_t=-\frac{\psi_t}{L_t}. $$ This is also an $\mathbb{F}$-adapted process. With this notation, by integrating the result of the application of the Ito lemma and exponentiating we get: $$ L_t=e^{-\int_0^t\Theta_udu-\frac{1}{2}\int_0^t\Theta^2_u dWu}. $$

We have constructed above the Radom-Nikodym derivative process $(L_t)_{t\geq 0}$ , which is a $(\mathbb{P}, \mathbb{F})$-martingale. Because $\mathbb{F}$ is generated by $(W_t)_{t\in[0;T]}$ we can apply the martingale representation theorem $\Rightarrow (\exists) (\psi_t)_{t\geq 0}$ an $\mathbb{F}$-measurable process s.t.: $$ L_t=1+\int_0^t \psi_udW_u. $$ or, alternatively, that: $$ dL_t=\psi_tdW_t, L_0=1. $$ Since the Radon-Nikodym derivarive process is strictly positive, using Ito lemma we get: $$ d\log(L_t)=\frac{1}{L_t}dL_t-\frac{1}{2}\frac{1}{L^2_t}d\langle L \rangle_t=\frac{\psi_t}{L_t}dW_t-\frac{1}{2}\frac{\psi^2_t}{L^2_t}dt $$ Since $L_t$ is strictly positive, we can simplify things a bit by introducing $$ \Theta_t=-\frac{\psi_t}{L_t}. $$ This is also an $\mathbb{F}$-adapted process. With this notation, by integrating the result of the application of the Ito lemma and exponentiating we get: $$ L_t=e^{-\int_0^t\Theta_udu-\frac{1}{2}\int_0^t\Theta^2_u dWu}. $$

THe result also rests on the uniqueness (up to indistinguishability) of the Radon-Nikodym derivative (in the R-N theorem).

The result also rests on the uniqueness (up to indistinguishability) of the Radon-Nikodym derivative (in the R-N theorem).

Theorem (Radon-Nikodym) Let $(\Omega, \mathcal{F})$ be a measurable space. Let $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ be two $\sigma$-finite measures. Let $\widetilde{\mathbb{P}}$ be absolutely continuous w.r.t. $\mathbb{P}$ (i.e. $\widetilde{\mathbb{P}}\ll\mathbb{P}$). THEN: $(\exists)$ measurable function $f:\Omega\to[0;+\infty)$ such that: $$ \widetilde{\mathbb{P}}(A)=\int_A f(\omega)d\mathbb{P}(\omega), (\forall)A\in\mathcal{F}. $$

We have constructed above the Radom-Nikodym derivative process $(L_t)_{t\geq 0}$ , which is a $(\mathbb{P}, \mathbb{F})$-martingale. Because $\mathbb{F}$ is generated by $(W_t)_{t\in[0;T]}$ we can apply the martingale representation theorem $\Rightarrow (\exists) (\psi_t)_{t\geq 0}$ an $\mathbb{F}$-measurable process s.t.: $$ L_t=1+\int_0^t \psi_udW_u. $$ or, alternatively, that: $$ dL_t=\psi_tdW_t, L_0=1. $$ Since the Radon-Nikodym derivarive process is strictly positive, using Ito lemma we get: $$ d\log(L_t)=\frac{1}{L_t}dL_t-\frac{1}{2}\frac{1}{L^2_t}d\langle L \rangle_t $$ Since $L_t$ is strictly positive, we can simplify things a bit by introducing $$ \Theta_t=-\frac{\psi_t}{L_t}. $$ This is also an $\mathbb{F}$-adapted process. With this notation, by integrating the result of the application of the Ito lemma and exponentiating we get: $$ L_t=e^{-\int_0^t\Theta_udu-\frac{1}{2}\int_0^t\Theta^2_u dWu} $$

Theorem (Radon-Nikodym) Let $(\Omega, \mathcal{F})$ be a measurable space. Let $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ be two $\sigma$-finite measures. Let $\widetilde{\mathbb{P}}$ be absolutely continuous w.r.t. $\mathbb{P}$ (i.e. $\widetilde{\mathbb{P}}\ll\mathbb{P}$). THEN: $(\exists)$ measurable function $f:\Omega\to[0;+\infty)$ such that: $$ \widetilde{\mathbb{P}}(A)=\int_A f(\omega)d\mathbb{P}(\omega), (\forall)A\in\mathcal{F}. $$ $f$ is unique up to indistinguishability, i.e. if there is another $g$ with the same properties as above, then $f=g, \mathbb{P}-a.s.$ (or $\mathbb{P}$-a.e.).

We have constructed above the Radom-Nikodym derivative process $(L_t)_{t\geq 0}$ , which is a $(\mathbb{P}, \mathbb{F})$-martingale. Because $\mathbb{F}$ is generated by $(W_t)_{t\in[0;T]}$ we can apply the martingale representation theorem $\Rightarrow (\exists) (\psi_t)_{t\geq 0}$ an $\mathbb{F}$-measurable process s.t.: $$ L_t=1+\int_0^t \psi_udW_u. $$ or, alternatively, that: $$ dL_t=\psi_tdW_t, L_0=1. $$ Since the Radon-Nikodym derivarive process is strictly positive, using Ito lemma we get: $$ d\log(L_t)=\frac{1}{L_t}dL_t-\frac{1}{2}\frac{1}{L^2_t}d\langle L \rangle_t $$ Since $L_t$ is strictly positive, we can simplify things a bit by introducing $$ \Theta_t=-\frac{\psi_t}{L_t}. $$ This is also an $\mathbb{F}$-adapted process. With this notation, by integrating the result of the application of the Ito lemma and exponentiating we get: $$ L_t=e^{-\int_0^t\Theta_udu-\frac{1}{2}\int_0^t\Theta^2_u dWu}. $$

THe result also rests on the uniqueness (up to indistinguishability) of the Radon-Nikodym derivative (in the R-N theorem).