# Proof of existence of one only martingale measure

I know that:

Hypothesis 1 (Girsanov Theorem)

Let $$\theta=\begin{Bmatrix} \theta_t \end{Bmatrix}_{t\in [0,T]}$$ be a square-integrable and $$\Im_t$$-adapted process such that $$\mathbb{E}[e^{\frac{1}{2}\int_0^T|\theta_s|^2ds}]<+\infty$$ (Novikov condition). Let $$M_t=e^{\begin{Bmatrix} \int_0^t\theta_sdW_s-\frac{1}{2}\int_0^t\theta_s^2ds \end{Bmatrix}},t\in [0,T]$$ be the only solution of the SDE $$\left\{\begin{matrix} dM_t=M_t\theta_tdW_t\\M_0=1 \end{matrix}\right.$$.

So exists a probability measure $$\mathbb{Q}\sim\mathbb{P}$$ which admits equivalent martingale if the Radon-Nikodym derivative $$L:=\frac{d\mathbb{Q}}{d\mathbb{P}}|_{\Im}$$ is expressed in terms of exponential martingal:

$$L=M_t\Rightarrow \mathbb{E}^{\mathbb{Q}}[X]=\mathbb{E}^{\mathbb{P}}[LX],\forall X=\begin{Bmatrix} X_t \end{Bmatrix}_{t\in [0,T]}$$ generic process on $$(\Omega, \Im, \begin{Bmatrix} \Im_t \end{Bmatrix}_{t\in [0,T]},\mathbb{P})$$ and $$W_t^{\mathbb{Q}}:=W_t^{\mathbb{P}}-\int_0^t\theta_sds$$.

Hypothesis 2 (Arbitrage freedom)

Let $$S^k=\begin{Bmatrix} S_t^k \end{Bmatrix}_{t\in [0,T]}$$ an $$\Im_t$$-adapted process with Ito's dynamics. Let $$V_t(\Theta):=\sum_{k=1}^n \theta_t^k S_t^k$$ be a function of $$\Theta:=(\theta_t^1,...,\theta_t^n),\forall t\in [0,T],\theta_t^k \in \mathbb{R}$$ with dynamics $$dV_t(\Theta)=\sum_{k=1}^n[\theta_t^kdS_t^k+d\theta_t^kS_t^k]$$ for $$d\theta_t^kS_t^k\equiv \theta_t^kD_t^kS_t^kdt.$$

So if $$\exists \mathbb{Q}\sim \mathbb{P}:S_t^k=\mathbb{E}^{\mathbb{Q}}[e^{-\int_t^Tr_sds}S_T^k|\Im_t]\Rightarrow V_t(\Theta) \operatorname{is a martingale}\Rightarrow \operatorname{no arbitrages}$$.

Hypothesis 3 (Replicability)

For $$u_S,u_B=(1-u_S)\in \mathbb{R}^+$$, for $$B_t$$ solution of $$\left\{\begin{matrix} dB_t=r_tB_tdt\\B_0=1 \end{matrix}\right.$$ and for $$d\Pi:=u_sdS_t+(1-u_S)dB_t,\Pi=\begin{Bmatrix} \Pi_t \end{Bmatrix}_{t\in [0,T]}$$, we have $$\frac{d\Pi}{\Pi}=\frac{dF_t}{F_t}$$ with $$\frac{dF_t}{F_t}=\frac{1}{F}[\frac{\partial F}{\partial t}+\mu S_t \frac{\partial F}{\partial S_t}+\frac{\sigma^2 S_t^2}{2}\frac{\partial^2 F}{\partial S_t^2}]dt+\frac{\sigma S_t}{F}\frac{\partial F}{\partial S_t}dW_t$$.

So we can write that $$u_S=\frac{S_t}{F}\Delta$$, for $$\Delta=\frac{\partial F}{\partial S_t}$$.

Knowing all this, how can I prove the Second Fundamental Theorem of evaluation that says that "A market free of arbitrages is completed if and only if exists one only martingale measure (with numeraire $$S_0$$)" (Andrea Pascucci - Calcolo stocastico per la finanza - page 86, Th. 3.21)?

• First direction: if the market is complete, then the hedging price is the only possible linear pricing functional. Each such pricing functional corresponds to one EMM. Thus, there's at most one EMM. Because the market is free of arbitrage, at least one EMM exists (first FTAP). Thus, completeness + no-arbitrage $\implies$ exactly one EMM exists – Kevin Oct 14 '20 at 18:43