Bernt Øksendal

Stochastic Differential Equations

We now return to the possible solutions X t (ω) of the stochastic differential equation (5.1) where W t is 1-dimensional “white noise”. As discussed in Chapter III the Ito interpretation of (5.1) is that X t satisfies the stochastic integral equation or in differential form (5.2) .

FRACTIONAL WHITE NOISE CALCULUS AND APPLICATIONS TO FINANCE

The purpose of this paper is to develop a fractional white noise calculus and to apply this to markets modeled by (Wick–) Ito type of stochastic differential equations driven by fractional Brownian motion BH(t); 1/2 < H < 1. We show that if we use an Ito type of stochastic integration with respect to BH(t) (as developed in Ref. 8), then the corresponding Ito fractional Black–Scholes market has no arbitrage, contrary to the situation when the pathwise integration is used. Moreover, we prove that our Ito fractional Black–Scholes market is complete and we compute explicitly the price and replicating portfolio of a European option in this market. The results are compared to the classical results based on standard Brownian motion B(t).

Optimal Switching in an Economic Activity Under Uncertainty

This paper considers the problem of finding the optimal sequence of opening (starting) and closing (stopping) times of a multi- activity production process, given the costs of opening, running, and closing the activities and assuming that the state of the economic system is a stochastic process. The problem is formulated as an extended impulse control problem and solved using stochastic calculus. As an application, the optimal starting and stopping strategy are explicitly found for a resource extraction when the price of the resource is following a geometric Brownian motion.

Optimal Consumption and Portfolio with Both Fixed and Proportional Transaction Costs

We consider a market model with one risk-free and one risky asset, in which the dynamics of the risky asset are governed by a geometric Brownian motion. In this market we consider an investor who consumes from the bank account and who has the opportunity at any time to transfer funds between the two assets. We suppose that these transfers involve a fixed transaction cost k>0, independent of the size of the transaction, plus a cost proportional to the size of the transaction. The objective is to maximize the cumulative expected utility of consumption over a planning horizon. We formulate this problem as a combined stochastic control/impulse control problem, which in turn leads to a (nonlinear) quasi-variational Hamilton--Jacobi--Bellman inequality (QVHJBI). We prove that the value function is the unique viscosity solution of this QVHJBI. Finally, numerical results are presented.

White Noise Generalizations of the Clark-Haussmann-Ocone Theorem with Application to Mathematical Finance

Abstract. We use a white noise approach to Malliavin calculus to prove the following white noise generalization of the Clark-Haussmann-Ocone formula \[F(\omega)=E[F]+\int_0^TE[D_tF|\F_t]\diamond W(t)dt\] Here E[F] denotes the generalized expectation,

Sufficient Stochastic Maximum Principle for the Optimal Control of Jump Diffusions and Applications to Finance

D_tF(\omega)={{dF}\over{d\omega}}

A mean-field stochastic maximum principle via Malliavin calculus

is the (generalized) Malliavin derivative,

Some solvable stochastic control problems with delay

\diamond

Optimal Consumption and Portfolio in a Jump Diffusion Market with Proportional Transaction Costs

is the Wick product and W(t) is 1-dimensional Gaussian white noise. The formula holds for all

An introduction to white–noise theory and Malliavin calculus for fractional Brownian motion

f\in{\cal G}^*\supset L^2(\mu)