Within the well-known framework of financial portfolio optimization, we analyze the existing relationships between the condition of arbitrage and the utility maximization in presence of insider information. We assume that, since the initial time, the information flow is altered by adding the knowledge of an additional random variable including future information. In this context we study the utility maximization problem under the logarithmic and the Constant Relative Risk Aversion (CRRA) utilities, with and without the restriction of no temporary-bankruptcy.In particular, we show that the value of the insider information may be bounded while the arbitrage condition holds and we prove that the insider information does not always imply arbitrage for the insider by providing an explicit example.T 0 Θ T t σ(t, S t ) 2 dt < +∞ almost surely. When T = F, we suppress the superscript notation.

<abstract><p>We prove that Theorem 4.16 in ^{[<xref ref-type="bibr" rid="b1">1</xref>]} is false by constructing a strategy that generates $ (FLVR)_{ \mathcal{H}(\mathbb{G})} $. However, we success to prove that the no arbitrage property still holds when the agent only plays with strategies belonging to the admissible set called <italic>buy-and-hold</italic>.</p></abstract>

Default risk calculus emerges naturally in a portfolio optimization problem when the risky asset is threatened with a bankruptcy. The usual stochastic control techniques do not hold in this case and some additional assumptions are generally added to achieve the optimization in a before-and-after default context. We show how it is possible to avoid one of theses restrictive assumptions, the so-called Jacod density hypothesis, by using the framework of the forward integration. In particular, in the logarithmic utility case, in order to get the optimal portfolio the right condition it is proved to be the intensity hypothesis. We use the anticipating calculus to analyze the existence of the optimal portfolio for the logarithmic utility, and than under the assumption of existence of the optimal portfolio we prove the semimartingale decomposition of the risky asset in the filtration enlarged with the default process.

We consider the optimal portfolio problem where the interest rate is stochastic and the agent has insider information on its value at a finite terminal time. The agent's objective is to optimize the terminal value of her portfolio under a logarithmic utility function. Using techniques of initial enlargement of filtration, we identify the optimal strategy and compute the value of the information. The interest rate is first assumed to be an affine diffusion, then more explicit formulas are computed for the Vasicek interest rate model where the interest rate moves according to an Ornstein-Uhlenbeck process. Incidentally we show that an affine process conditioned to its future value is still an affine process. When the interest rate process is correlated with the price process of the risky asset, the value of the information is proved to be infinite, as is usually the case for initial-enlargement-type problems. However, weakening the information own by the agent and assuming that she only knows a lower-bound or both, a lower and an upper bound, for the terminal value of the interest rate process, we show that the value of the information is finite. This solves by an analytical proof a conjecture stated in Pikovsky and Karatzas (1996).

In [1], Pikovsky and Karatzas did one of the earliest studies on portfolio optimization problems in presence of insider information. They were able to successfully show that the knowledge of the stock price at future time is an insider information with associated unbounded value. However when the insider information only gives an interval containing the future value of the stock price, they couldn't prove that the value of the information is finite. They made a conjecture of this result, still open according to our knowledge, and tried to convince about its validity by showing some numerical approximations. We close this conjecture by giving a proof that indeed the insider information in this case has a finite value.