Hyperbolic Brownian Sheet, Parabolic and Elliptic Financials. (Didactic 3)

Fig-3-Realizations-of-the-fractional-Brownian-sheet-on-the-plane-with-graph-dimensions

Financial and economic time series are often described to a first degree of approximation as random walks, following the precursory work of Bachelier and Samuelson. A random walk is the mathematical translation of the trajectory followed by a particle subjected to random velocity variations. The analogous physical system described by SPDE’s is a stochastic string. The length along the string is the time-to-maturity and the string configuration (its transverse deformation) gives the value of the forward rate f(t,x) at a given time for each time-to-maturity x. The set of admissible dynamics of the configuration of the string as a function of time depends on the structure of the SPDE. Let us for the time being restrict our attention to SPDE’s in which the highest derivative is second order. This second order derivative has a simple physical interpretation : the string is subjected to a tension, like a piano chord, that tends to bring it back to zero transverse deformation. This tension forces the “coupling” among different times-to-maturity so that the forward rate curve is at least continuous. In principle, the most general formulation would consider SPDE’s with terms of arbitrary derivative orders. However, it is easy to show that the tension term is the dominating restoring force, when present, for deformations of the string (forward rate curve) at long “wavelengths”, i.e. for slow variations along the time-to-maturity axis. Second order SPDE’s are thus generic in the sense of a systematic expansion.

In the framework of second order SPDE’s, we consider hyperbolic, parabolic and elliptic SPDE’s, to characterize the dynamics of the string along two directions : inertia or mass, and viscosity or subjection to drag forces. A string that has “inertia” or, equivalently, “mass” per unit length, along with the tension that keeps it continuous, is characterized by the class of hyperbolic SPDE’s. For these SPDE’s, the highest order derivative in time has the same order as the highest order derivative in distance along the string (time-to-maturity). As a consequence, hyperbolic SPDE’s present wave-like solutions, that can propagate as pulses with a “velocity”. In this class, we find the so-called “Brownian sheet” which is the direct generalization of Brownian motion to higher dimensions, that preserves continuity in time-to-maturity. The Brownian sheet is the surface spanned by the string configurations as time goes on. The Brownian sheet is however non-homogeneous in time-to-maturity.

If the string has no inertia, its dynamics are characterized by parabolic SPDE’s. These stochastic processes lead to smoother diffusion of shocks through time, along time-to-maturity. Finally, the third class of SPDE’s of second-order, namely elliptic partial differential equations. Elliptic SPDE’s give processes that are differentiable both in x and t. Therefore, in the strict limit of continuous trading, these stochastic processes correspond to locally riskless interest rates.

The general form of SPDE’s reads

A(t,x) ∂2f(t,x)/∂t2 + 2B(t,x) ∂2f(t,x)/∂t∂x + C(t,x) ∂2f(t,x)/∂x2 = F(t,x,f(t,x), ∂f(t,x)/∂t, ∂f(t,x)/∂x, S) —– (1)

where f (t, x) is the forward rate curve. S(t, x) is the “source” term that will be generally taken to be Gaussian white noise η(t, x) characterized by the covariance

Cov η(t, x), η(t′, x′) = δ(t − t′) δ(x − x′) —– (2)

where δ denotes the Dirac distribution. Equation (1) is the most general second-order SPDE in two variables. For arbitrary non-linear terms in F, the existence of solutions is not warranted and a case by case study must be performed. For the cases where F is linear, the solution f(t,x) exists and its uniqueness is warranted once “boundary” conditions are given, such as, for instance, the initial value of the function f(0,x) as well as any constraints on the particular form of equation (1).

Equation (1) is defined by its characteristics, which are curves in the (t, x) plane that come in two families of equation :

Adt = (B + √(B2 − AC))dx —– (3)

Adt = (B − √(B2 − AC))dx —– (4)

These characteristics are the geometrical loci of the propagation of the boundary conditions.

Three cases must be considered.

• When B2 > AC, the characteristics are real curves and the corresponding SPDE’s are called “hyperbolic”. For such hyperbolic SPDE’s, the natural coordinate system is formed from the two families of characteristics. Expressing (1) in terms of these two natural coordinates λ and μ, we get the “normal form” of hyperbolic SPDE’s :

2f/∂λ∂μ = P (λ,μ) ∂f/∂λ +Q (λ,μ) ∂f/∂μ + R (λ,μ)f + S(λ,μ) —– (5)

The special case P = Q = R = 0 with S(λ,μ) = η(λ,μ) corresponds to the so-called Brownian sheet, well studied in the mathematical literature as the 2D continuous generalization of the Brownian motion.

• When B2 = AC, there is only one family of characteristics, of equation

Adt = Bdx —– (6)

Expressing (1) in terms of the natural characteristic coordinate λ and keeping x, we get the “normal form” of parabolic SPDE’s :

2f/∂x2 = K (λ,μ)∂f/∂λ +L (λ,μ)∂f/∂x +M (λ,μ)f + S(λ,μ) —– (7)

The diffusion equation, well-known to be associated to the Black-Scholes option pricing model, is of this type. The main difference with the hyperbolic equations is that it is no more invariant with respect to time-reversal t → −t. Intuitively, this is due to the fact that the diffusion equation is not conservative, the information content (negentropy) continually decreases as time goes on.

• When B2 < AC, the characteristics are not real curves and the corresponding SPDE’s are called “elliptic”. The equations for the characteristics are complex conjugates of each other and we can get the “normal form” of elliptic SPDE’s by using the real and imaginary parts of these complex coordinates z = u ± iv :

2f/∂u2 + ∂2f/∂v2 = T ∂f/∂u + U ∂f/∂v + V f + S —– (8)

There is a deep connection between the solution of elliptic SPDE’s and analytic functions of complex variables.

Hyperbolic and parabolic SPDE’s provide processes reducing locally to standard Brownian motion at fixed time-to-maturity, while elliptic SPDE’s give locally riskless time evolutions. Basically, this stems from the fact that the “normal forms” of second-order hyperbolic and parabolic SPDE’s involve a first-order derivative in time, thus ensuring that the stochastic processes are locally Brownian in time. In contrast, the “normal form” of second-order elliptic SPDE’s involve a second- order derivative with respect to time, which is the cause for the differentiability of the process with respect to time. Any higher order SPDE will be Brownian-like in time if it remains of order one in its time derivatives (and higher-order in the derivatives with respect to x).

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