3. Elements of Stochastic Analysis

Our purpose in this chapter is to give an overview of the basics of stochastic calculus, an important mathematical tool that is used in control engineering, in modern finance and insurance, and in modern management science, among other fields. The chain rule of stochastic calculus, the so-called Itô formula, is one of the most used mathematical (or probabilistic) formulas in the world, since it implicitly sits under every trader’s screen. At least the Itô formula gives rise to one of the mostly posed mathematical exercises, as follows. Let dS _t =S _t σ dW _t , starting from today’s observed value for S ₀, model the returns of a stock price, where W _t is a Brownian motion and σ is the so-called volatility parameter (the “temperature” on financial markets). What are the dynamics for X _t =ln(S _t )? But this is the celebrated Black–Scholes model! Then we will add jumps, to make it more spicy and because in this book the Brownian motion W _t and the Poisson process N _t are equally treated on a fair basis, as the prototype and the fundamental driver of continuous and jump processes, respectively. Now, as opposed to the above forward SDE, endowed with an initial condition for S ₀ at time 0, it’s now time to consider our first backward SDE. That’s because derivative contracts are defined in terms of a payoff ξ at a future maturity T. This payoff ξ is random and defined in terms of an underlying such as S _T , but what we are looking for is the price and the hedge of the derivative at the current pricing time t<T. These price Π _t and hedge Δ _t are obtained as the solution of a backward SDE such as dΠ _t =Δ _t  dS _t , Π _T =ξ. The solution of a BSDE has therefore two components, Π and Δ. In case of American options with early exercise clauses, there is a third component A, intended to maintain the value process Π above the payoff. Otherwise isn’t a BSDE too simple and directly solved by the application of a suitable martingale representation theorem? But that’s only because we forgot the issue of funding our position. Funding costs give rise to an additional term g _t (Π _t ,Δ _t ) dt in the BSDE. Moreover, since the crisis, in nowadays market environments, funding costs involve some nonlinearities (such as two different rates for lending and borrowing, if you are a risky bank). Dealing with such nonlinearities is precisely what BSDEs were invented for.