Abstract. We examine a stochastic optimal control problem in a financial market driven by a Lévy process with filtration F = (Ft) t∈ [0,T ] . We assume that in the market there are two kinds of investors with different levels of information: an uninformed agent whose information coincides with the natural filtration of the price processes and an insider who has more information than the uninformed agent. When optimal consumption and investment exist, we identify some necessary conditions and find the optimal strategy by using forward integral techniques. We conclude by giving some examples.
Abstract. We proved white noise generalization of the Clark-Ocone formula under change of measure by using white noise analysis and Malliavin calculus. Let W (t) be a Brownian motion on the filtered white noise probability space (Ω, B, {Ft} t≥0 , P ) and letŴ (t) be defined as dŴ (t) = u(t) + dW (t), where u(t) is an Ft-measurable process satisfying certain conditions. Let Q be the probability measure equivalent P such thatŴ (t) is a Brownian motion with respect to Q, in virtue of the Girsanov theorem. In this paper, it is shown that for any random variable F ∈ L 2 (P )where EQ is the expectation under Q and DtF (ω) is the (Hida) Malliavin derivative. The important point to note here is in this settlement F need not belong to stochastic Sobolev space, D1,2 which is subset of L 2 (P ). This makes this formula more useful in applications to finance. For example, the replicating portfolio for a digital option, whose payoff function χ [K,∞) W (T ) / ∈ D1,2, is calculated by using this generalized Clark-Ocone formula under change of measure.
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