In brief, Itô’s formula establishes that given a real valued semimartingaleX={Xt:
by Zt = f(Xt) for all t > 0, is also a semimartingale. Moreover, it provides an explicit
Bichteler-Dellacherie decomposition of Z into a sum of a stochastic Itô integral (the lo- cal martingale) and a Lebesgue integral (the finite variation process). Itô’s formula has shown to be an extremely significant tool for theoretical and applied mathematics. For this reason, Itô’s formula has been revisited by several authors through the years. Its simplest version, for one dimensional processes, has been extended to cover multidimen- sional processes, Hilbert-space valued semimartingales (see Métivier, 1982), and infinite dimensional processes (see e.g. Da Prato and Zabczyk, 2014). These extensions allow us to considerX lying in a wider class of processes. On the other hand, extensions to cover a wider class of functions f have been also made. For instance, Föllmer, Protter, and
Shiryayev (1995) studied the case of non smooth functions, andKunita (1997) the case of random functions such as flows of stochastic differential equations.
However, it is worth noticing that in all cases the processZis defined as a function depending onXtthe current value of the process of interest. Thus, it is natural to wonder
about more general scenarios. For instance, we could think of a function of two variables: the current value of the process and its value at a previous time. More generally, one could think of functions depending on previous states of the process or even on the whole path up to the current time. In this direction, we can mention the work ofAhn(1997) who studied the case when the function depends onXt∧srather than onXt. That is,Ahn obtained an
Ito-type formula for the level process. But, a general Itô’s formula for functions depending on previous states of the process, the so-called “tame” functions, was given in 2004 byHu, Mohammed, and Yan(2004). Later,Dupire(2009) proposed a groundbreaking approach to generalise the Itô formula to the functional setting. His idea consisted of introducing a path-wise derivative for non-anticipative functionals on the space of cádlág — right continuous functions with left limits— functions, and describing the variations of the functional in terms of such derivatives. Afterwards, this idea was generalised byCont and Fournié(2013) to cover to the space of square-integrable martingales.
In this chapter, we review the Itô type formula given byHu, Mohammed, and Yan in 2004. This formula, is to our knowledge the only one covering delay functionals. The functional setting ofDupire(2009) andCont and Fournié(2013) does not apply to delays as the corresponding functionals would not be horizontally differentiable. However, the formula established byHu, Mohammed, and Yan(2004) goes beyond the semimartingale setup of X = {Xt : t > 0}, and hence it requires stronger conditions. These conditions,
when applied to diffusionsXgoverned by SDEs, would require the diffusion coefficient to be twice differentiable. The good news is that a modification of the proof is possible ifX is a diffusion leading to weaker conditions allowing for Lipschitz continuous coefficients. So, in what follows we will introduce notation and discuss the technical difficulties when dealing with delay-processes.
Then, we will prove an Itô type formula when delay-functionals are applied to Brownian motion but our proof will be different to the proof given by Hu, Mohammed, and Yan(2004).
Finally, we weaker the conditions given byHu, Mohammed, and Yan(2004) in the case of diffusions X governed by SDEs. For this purpose, we benefit from our proof in the case of Brownian motion because we can now refer to those steps in the proof byHu, Mohammed, and Yanwhich need modification.
3.1.1 Setting the Problem
The scenario that we are going to investigate is described as follows. Fix a finite time horizonT > 0and consider the following stochastic differential equation
dXt=b(t,Xt)dt+σ(t,Xt)dWt, X0 given, (3.1)
where W = {Wt : 0 6 t 6 T} is a standard linear Brownian motion defined on some
probability space (Ω,F,P); X0 is an independent random variable defined on the same
probability space, and the functions
b: [0,T]×R→R,
σ: [0,T]×R→R,
which will be referred to as thedriftanddiffusion coefficient, respectively, are measurable functions satisfying:
Assumption 3.1. (Global Lipschitz Condition) There exists a constant K > 0such that
|b(t,x) −b(t,y)|+|σ(t,x) −σ(t,y)|6K|x−y|, for any 06t6T and x,y∈R.
Assumption 3.2. (Boundedness) There exists a constantM > 0 such that sup
06t6T
{|b(t,0)|+|σ(t,0)|}< M.
It has been proved (see for exampleKaratzas and Shreve, 1991;Stroock and Varad- han, 2007;Kolokoltsov, 2010) that these assumptions are sufficient to guarantee the exis- tence of a unique solution to equation (3.1)satisfying
Xt=X0+ Zt 0 b(s,Xs)ds+ Zt 0 σ(s,Xs)dWs, 06t6T, a.s.
Furthermore, if E[|X0|p]<∞for somep>2,
E " sup 06t6T |Xt|p # 6C0, (3.2)
Next, consider a finite collection of points given by
Π={0=π0< π1 <· · ·< πm=π}, for some integer m > 0,
and let us introduce theRm+1-valued processX
Π={XΠt:π6t6T}given by
XΠt= (Xt−πm,Xt−πm−1,· · ·,Xt−π1,Xt), π6t6T. (3.3)
Our aim is to prove that, underassumptions 3.1 and 3.2 it is possible to establish an Itô type formula for processes of the form
f(XΠt) =f(Xt−πm,Xt−πm−1,· · ·,Xt−π1,Xt), π6t6T. (3.4)
wheref:Rm+1→Ris a twice continuously differentiable function.
The rest of the chapter is structured as follows. InSection 3.2, we introduce some definitions and notation in order to simplify the treatment of random vectors of the form (3.3). In Section 3.3, we investigate processes of the form (3.4) and obtain an Itô type formula in terms of a Skorohod integral and a Lebesgue integral.