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Let the price process $S_{t}, 0 \leq t \leq T$, be a diffusion, and savings account be $\beta_{t}$ such that the Equivalent Martingale Measure $Q$ exists. Let $C_{T}=g\left(X_{T}\right)$ be the claim at time $T$, for a bounded function $g$. Show that this claim is attainable, and find a replicating self-financing portfolio for this claim.

My attempt:

The claim is attainable as $\frac{S_t}{\beta_{t}}$ is a $Q$ -martingale and an admissible strategy exists. Therefore $\frac{V(t)}{\beta(t)}$ is also a Q martingale.

The Self replicating portfolio can be found as $V_{t}=a_{t} s_{t}+b_{t} e^{r t}$

$d V_{t}=a_{t} d s_{t}+r b_{t} e^{r t} d t$ is self financing.

let $\left.b_{t}=\left(V_{t}-a_{t} s_{t}\right)\right) e^{-r t}$ then we have

$d v_{t}=a_{t} d s_{t}+r\left(V_{t}-a_{t} s_{t}\right) d t$

$d V_{t}=a_{t} d s_{t}+r V_{t} d t-r a_{t} s_{t} d t$

I am not sure how to carry on further for this question.

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With EMM $Q$, associated $Q$-Brownian motion $W$, filtration ${\cal F}$, and

$$d\beta_t = r_t \beta_t dt, \; \beta_t ={\rm e}^{\int_0^t r_u du},$$

consider martingale:

$$ M_t =E\left[{\rm e}^{-\int_0^T r_u du} C_T | {\cal F}_t\right]. $$

By martingale representation theorem, there is a process $N_t$ such that

$$ M_t = M_0 + \int_0^t N_u dW_u, $$

where $$ M_0=E\left[{\rm e}^{-\int_0^T r_u du} C_T\right].$$

With given

$$ d(\beta_t^{-1} S_t) = \sigma_t \beta_t^{-1} S_tdW_t$$

under $Q$, we have:

$$ dM_t = N_t dW_t = a_t \sigma_t \beta_t^{-1} S_t dW_t = a_t d(\beta_t^{-1} S_t) $$

for $a_t$ chosen to be

$$ a_t := \frac{N_t\beta_t}{\sigma_t S_t }.$$

Strategy $a_t$ and $b_t:= \beta_t^{-1}(M_t-a_tS_t)$

$$\Pi_t := b_t\cdot \beta_t + a_t \cdot S_t = \beta_t M_t$$

is admissible (under $Q$, $\beta_t^{-1}\Pi_t$ is a martingale) and self-financing as

$$ d(\beta_t^{-1}\Pi_t) = dM_t = a_t d(\beta_t^{-1} S_t). $$

We also note that:

$$ \Pi_T = \beta_T M_T = C_T. $$

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