A note on the distribution of integrals of geometric Brownian motion
Introduction
The vital role played by Kolmogorov's backward and forward equations for Markov processes is well known from the points of view of both general theory and computation. Of course these equations are intimately linked to the Markov structure as reflected in the semi-group property. Thus the appearance of a pair of linear parabolic (diffusion) partial differential equations in connection with the highly non-Markovian evolution of integrals of geometric Brownian motion seems quite noteworthy.
The stochastic processwhere is a Brownian motion with drift coefficient μ and diffusion coefficient σ2>0, arises naturally in a diverse contexts involving geometric Brownian motion; e.g. financial mathematics, spin-glass and disordered systems, statistical turbulence, to name a few. The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by (1). In particular, the expected values of the formf a homogeneous function, as well as the probability densitywill be shown to be governed by the following pair of linear parabolic (diffusion) partial differential equations:andOf course these equations are not the backward and forward adjoint pairs one would naturally have for the general theory of Markov processes, however unifying and remarkably simple derivations of these equations are forthcoming in this note.
Some related alternative approaches to formulae and equations governing , can be found in Yor (1992), Monthus and Comtet (1994), and Rogers and Shi (1995). Yor (1992) develops an identity from Bougerol (1983) and exploits connections between At and subordinated Brownian motion and Bessel processes in his derivation of a formula for the density of At. Monthus and Comtet (1994) give a non-rigorous evaluation of moment integrals to determine a recurrence relation for a formal power series representation of the moment generating function. This in turn leads to an equation for the density of At. Finally, Rogers and Shi (1995) exploit a particular scaling property to obtain a martingale whose drift term provides the appropriate pde governing the expected value in the case of call function f(x)=x+. The essential feature of the call function exploited by Rogers and Shi (1995) is the implied (degree one) homogeneity. Namely the approach of Rogers and Shi (1995) applies to functions f such that for each real number xThe methods of the present note also permit an extension to degree θ homogeneity of the formfor some real parameter θ. While it is not our intent to develop the implications for numerical applications in this brief note, the utility of such partial differential equations for numerical calculations is nicely illustrated in Rogers and Shi (1995).
The organization is as follows. In the next section we provide the key lemma from which the equations are derived. The expected value equation and the equation for the probability density are then provided as rather immediate consequences.
Section snippets
Preliminaries and key lemma
Throughout we let be one-dimensional Brownian motion starting at 0 with drift μ and diffusion coefficient σ2>0. Remark The reader may take note that the calculations to follow extend to time-dependent drift and diffusion coefficients. The main property required of the diffusion is that for each , the process starting from z has the same transition probabilities as the process starting from 0. Since the notation for temporally non-homogeneous diffusions tends to
Expected value equation
Let
To obtain an equation for v we first writewhere h(x,z)=f(xez). Also, we omit the superscript x where it would be redundantly expressed as a subscript in the expected value symbol. Then for homogeneous f of degree θ one hasand therefore
Thus applying Kolmogorov backward equation (e.g. see Bhattacharya and Waymire, 1990, p. 374) and using (12) gives
Probability density equation
LetAlso definewhere for each fixed DefineThenandThus, again applying Kolmogorov's backward equation to u one obtainsand therefore in view of , we have taking x=0,whereNext, observe that
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