# replicating self-financing portfolio for risk neutral measure

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.

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:

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