Anticipating Exponential Processes and Stochastic Differential Equations

Abstract

Exponential processes in the Itô theory of stochastic integration can be viewed in three aspects: multiplicative renormalization, martingales, and stochastic differential equations. In this paper we initiate the study of anticipating exponential processes from these aspects viewpoints. The analogue of martingale property for anticipating stochastic integrals is the near-martingale property. We use examples to illustrate essential ideas and techniques in dealing with anticipating exponential processes and stochastic differential equations. The situation is very different from the Itô theory. 1. Exponential Processes Let B(t), 0 ≤ t ≤ T, be a fixed Brownian motion. Suppose {Ft; 0 ≤ t ≤ T} is the filtration given by this Brownian motion, i.e., Ft = σ{B(s); 0 ≤ s ≤ t} for each t ∈ [0, T ]. Take an {Ft}-adapted stochastic process h(t), 0 ≤ t ≤ T, satisfying the Novikov condition, i.e., E exp [1 2 ∫ T 0 h(t) dt ] <∞. (1.1) The exponential process given by h(t) is defined to be the stochastic process Eh(t) = exp [ ∫ t 0 h(s) dB(s)− 1 2 ∫ t 0 h(s) ds ] , 0 ≤ t ≤ T. (1.2) Note that under the Novikov condition in equation (1.1) we have ∫ T 0 h(t) dt <∞ almost surely so that the Itô integral ∫ t 0 h(s) dB(s) is defined for each t ∈ [0, T ] (see Chapter 5 of the book [7].) The exponential process Eh(t) plays a very important role in the Itô theory of stochastic integration and is widely used in the mathematical finance. It can be viewed and understood in the following three aspects. (1) Multiplicative renormalization: The multiplicative renormalization of a random variable X with nonzero expectation is defined to be the random variable X/EX. Suppose h(t) is a deterministic function in L[0, T ]. Received 2019-10-13; Accepted 2019-10-14; Communicated by guest editor George Yin. 2010 Mathematics Subject Classification. Primary 60H05; Secondary 60H20.