I would like to solve the following problem.
Consider a financial market with quadratic transaction costs, one risky asset with price dynamics:
$S_t = s_0 + \mu t + \sigma W_t$, for $t \geq 0 , \sigma > 0, \mu \in {R}$ and $W$ is Brownian Motion.
and (for simplicity) let the riskless asset bearing zero interest.
Now consider a risk neutral trader having access to the Filtration
$ G_t^{\Delta} := F_{t+\Delta}^S$, for ${\Delta} \in [0,\infty)$,
so he has more information than just having acces to the augmented filtration $(F_t^S)_{t \geq 0}$
I then tried to calculate the Valuefunction V for a timehorizon $T>0$, which led me to
$V_T^{\Phi_0, \phi} = \Phi_0 (S_T - S_0) + \int_0^T \phi_t(S_T-S_t)dt - \frac{\Lambda}{2} \int_0^T \phi_t^2 dt$ ,
where $\Phi_0$ is my position at the Start, $\phi$ is my traiding strategy and the last term is the quadratic transaction costs for a $\Lambda > 0$.
Since the trader is risk neutral we have a linear utility function i.e. u(x) = x
Also we need that $\phi$ is an admissible trading strategy, hence $G^\Delta$-optional with $\int_0^T \phi_t^2 < \infty$
I then tried to calculate: $max_{\phi} E[u(V_T^{\Phi_0, \phi})] = max_{\phi} E[V_T^{\Phi_0, \phi}]$
After using tower property and other properties this led to:
$max_{\phi}( \Phi_0(\mu T -s_0) + \int_0^T \mu (T-t) E[\phi_t] + \sigma E[\phi_t (W_{t+\Delta} - W_t)] - \frac{\Lambda}{2} E[\phi_t^2] dt)$
And here i am stuck tried to solve it using the Hitsuda represantion but i dont how to start.
Since i am very interested in the process of how to solve such a problem or related problems i would really like to just get a hint (not a whole solution) if someone has a idea where and how to start or maybe someone knows good lecture to unterstand the process of working with the hitsuda representation.
Here is what the solution would look like in the standard non-anticipating case. Define the wealth at time $t$ as $Y_t$. This evolves as
$$dY_t=\int_o^t\phi_sdsdS_t-\frac{\Lambda}{2}\phi_t^2dt$$
Hence at terminal date $T$
$$Y_T=\int_0^{T}\int_o^t\phi_sdsdS_t-\int_0^{T}\frac{\Lambda}{2}\phi_t^2dt+Y_0$$
You can normalize $Y_0=0$. You have
$$E(Y_T)=\int_0^{T}\int_0^t\phi_sds\mu dt-\int_0^{T}\frac{\Lambda}{2}\phi_t^2dt=\int_0^{T}(\int_0^t\phi_sds\mu-\frac{\Lambda}{2}\phi_t^2) dt$$
The problem is
$$\max_{\phi_t}E(Y_T)$$
The first order condition is
$$\mu-\Lambda \phi_t=0$$
and hence
$$\phi_t=\frac{\mu}{\Lambda}$$