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 tin[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}_{tin[0.T]},quad mathcal{H}_t=mathcal{F}_tveesigma(B(t+epsilon_t+r),0leq r leq epsilon_0-t-epsilon_t), forall tin [0,T]. end{equation} Using forward integrals, Hida-Malliavin calculus and Donsker delta functionals we show that if $$int_0^Tfrac{1}{varepsilon_t}dt=infty,$$ then the insider market is not viable.
Download