Viable Insider Markets

We consider the problem of optimal inside portfolio $\pi(t)$ in a financial market with a corresponding wealth process $X(t)=X^{\pi}(t)$ modelled by \begin{align}\label{eq0.1} \begin{cases} dX(t)&=\pi(t)X(t)[\alpha(t)dt+\beta(t)dB(t)]; \quad t\in[0, T] X(0)&=x_0>0, \end{cases} \end{align} where $B(\cdot)$ is a Brownian motion. We assume that the insider at time $t$ has access to market information $\varepsilon_t>0$ units ahead of time, in addition to the history of the market up to time $t$. The problem is to find an insider portfolio $\pi^{*}$ which maximizes the expected logarithmic utility $J(\pi)$ of the terminal wealth, i.e. such that $$\sup_{\pi}J(\pi)= J(\pi^{*}), \text {where } J(\pi)= \mathbb{E}[\log(X^{\pi}(T))].$$ The insider market is called \emph{viable} if this value is finite. We study under what inside information flow $\mathbb{H}$ the insider market is viable or not. For example, assume that for all $t<T$ the insider knows the value of $B(t+\epsilon_t)$, where $t + \epsilon_t \geq T$ converges monotonically to $T$ from above as $t$ goes to $T$ from below. Then (assuming that the insider has a perfect memory) at time $t$ she has the inside information $\mathcal{H}_t$, consisting of the history $\mathcal{F}_t$ of $B(s); 0 \leq s \leq t$ plus all the values of Brownian motion in the interval $[t+\epsilon_t, \epsilon_0]$, i.e. we have the enlarged filtration \begin{equation}\label{eq0.2} \mathbb{H}=\{\mathcal{H}_t\}_{t\in[0.T]},\quad \mathcal{H}_t=\mathcal{F}_t\vee\sigma(B(t+\epsilon_t+r),0\leq r \leq \epsilon_0-t-\epsilon_t), \forall t\in [0,T]. \end{equation} Using forward integrals, Hida-Malliavin calculus and Donsker delta functionals we show that if $$\int_0^T\frac{1}{\varepsilon_t}dt=\infty,$$ then the insider market is not viable.