Fundamental solutions to Kolmogorov equations via reduction to canonical form

This paper finds fundamental solutions to the backward Kolmogorov equations, usually interpretable as transition density functions for variables x that follow certain stochastic processes of the form dx=A(x,t)dt


Introduction
Any endeavor to calculate the theoretical price of certain financial derivative products, such as options and bonds, begins with the establishment of a dynamic process followed by the underlying asset.It is well known that under the assumption that an underlying stock with price S follows the lognormal process Black and Scholes [4] derived their celebrated partial differential equation (PDE) for the valuation of European options.Note that in (1.1) and throughout this paper, dX denotes an increment in a Wiener process so that dX ∼ N(0, √ dt).Similarly when the short-term interest rate r follows the process dr = u(r,t)dt + w(r,t)dX, (1.2) the price of a zero-coupon bond V (r,t) with expiry at time t = T will satisfy the PDE ∂V ∂t + w(r,t) 2  2 ∂ 2 V ∂r 2 + u(r,t) − λ(r,t)w(r,t) subject to V (r,T) = 1, and where λ(r,t) is the market price of risk (see, e.g., [16]).
2 Fundamental solutions to Kolmogorov equations In order to understand the probabilistic properties of random walks such as (1.1) and (1.2) which we write in general as dx = A(x,t)dt + B(x,t)dX, (1.4) it is necessary to find transition density functions (TDFs), p(x,t; y,t ), of the underlying process.This density function p(x,t; y,t ) is defined by Pr a < x < b at time t | x at time t = b a p(x,t; y,t )dy. (1.5) The variables x and t can be thought of as the current variables and y and t as the future variables.The TDF satisfies the backward Kolmogorov equation involving the derivatives with respect to the current state and time, subject to p(x,t ; y,t ) = δ(x − y).
(1.6b) Also, at each point in time for x, y ≥ 0, TDFs satisfy the condition ∞ 0 p(x,t; y,t )dy = 1. (1.6c) When (1.4) describes a risk-neutral process, TDFs can be used to value financial derivatives such as European call (and put) options, giving the holder of the options the right to buy (sell) the asset with value x, at some future time T for a price K agreed upon at the current time t.The price for the call option contract is then C(x,t;K,T) = e −r(T−t) E max(x − K,0) = e −r(T−t) ∞ K (y − K)p(x,t; y,T)dy, (1.7) where r is the risk-free interest rate.Similarly, the price for the put option contract is P(x,t;K,T) = e −r(T−t) E max(K − x,0) = e −r(T−t) K 0 (K − y)p(x,t; y,T)dy.
(1.8) Given a fundamental solution, the solution to the Cauchy problem can be obtained from the given initial data by convolution.In [9], Craddock and Platen find some fundamental solutions when B(x) = √ 2x and A(x) = f (x), that is, of where τ = t − t.When (1.6c) is satisfied, this corresponds to finding TDFs of the process x, which follows The method they employed was to perform a classical symmetry analysis of (1.9), and then using certain symmetries found, write the corresponding form of the transformed solutions p * in terms of another arbitrary solution p, that is, The symmetry transformations were chosen so as to achieve an integral transform, preferably if possible, the Laplace transform of the fundamental solution.Thus, if possible, the symmetries were chosen so that F(x,0; ) = e −λx , where λ is some function of .Then by choosing the solution p = 1 of (1.9), the fundamental solution of (1.9) is the Laplace inverse of p, the new solution with initial condition e −λx , as for x, y ≥ 0, e −λy p(x,τ; y,t )dy. (1.12) p was called the characteristic solution (the Laplace transform of the fundamental solution).Craddock and Platen found fundamental solutions of (1.9) when f satisfied the Riccati equations For functions f that satisfied (1.13a), the characteristic solution was found in general.When f satisfied relation (1.13b), solutions were proposed that may lead to characteristic solutions and when f satisfied (1.13c) and B 1 = 0, the characteristic solution was given, but not its corresponding Laplace inverse.
It is known (see, e.g., [15]) that an evolution equation of the type with P,R = 0, can be reduced via a suitable transformation to When the PDE (1.14) admits one additional symmetry operator (namely ∂/∂t) to ∂/∂u and ϕ(x,t)(∂/∂u), where ϕ is any solution of (1.14), then (1.14) is reducible to (1.16) If the PDE (1.14) has at least three more symmetries in addition to u(∂/∂u) and ϕ(x,t)(∂/∂u), where ϕ is any solution of (1.14), then (1.14) is reducible to where α is constant.If it has at least five other symmetries (to u(∂/∂u) and ϕ(x,t)(∂/∂u)), then it is reducible to This system of classification affords a systematic approach to finding reductions.As (1.9) in the cases where f satisfies (1.13a), (1.13b), or (1.13c) has at least three extra symmetries beyond p(∂/∂p) and ϕ(x,τ)(∂/∂p), where ϕ(x,τ) satisfies (1.9), (1.9) is reducible to either (1.17) or (1.18).It is then possible to easily find the fundamental solutions in these cases by using the fundamental solutions of (1.17) and (1.18).This also avoids the necessity of finding suitable symmetries that lead to tractable integral transform inverses.
In [2], Albanese and Kuznetsov show that a stationary process of the form (1.4), where A = A(x) and B = B(x), can be transformed via a suitable change of variables and change of measure to a process that is driftless.This in fact corresponds to the reduction of (1.14) to canonical form.They then classify driftless processes that are solvable in the sense that the corresponding eigenfunction equation can be reduced by a change of measure and change of variables to a hypergeometric or confluent hypergeometric equation.This then could lead to TDFs as infinite series of hypergeometric (Gaussian or confluent) functions.
The driftless processes dx = (a + bx + cx 2 )dX (quadratic volatility) and dx = x γ dX (CEV driftless) have been well researched, and in [1] Albanese et al. show that these processes belong to the equivalence class of the family dx = (λ 0 + λ 1 x)dt + ν 0 x β dX, where β = 0,1/2.In turn, this family of processes is contained within the broader class that we consider in this paper.
The aim of this paper is to look for solutions of (1.6a)-(1.6b)for general functions A and when B(x,t) = cx γ (Section 3) and B(x,t) = α 1 + α 2 x + α 3 x 2 (Section 4), by first reducing (1.6a) to its canonical form.Rather than finding solutions to (1.6a) in terms of series of transcendental functions (such as in [2]), the focus here is on being able to solve the Cauchy problem by a single integration.When (1.6c) is satisfied (as in all our examples), the fundamental solutions can be viewed as the TDFs for the variable x that follows the process (1.4).As economic conditions change with time, it is reasonable to expect that the instantaneous expected return of a given state variable, such as stock prices and bond yields, depends on time and price level.In fact, Fan et al. [10] and Goard and Hansen [13], among others, have shown how models with time-dependent parameters outperform those with a long-term reversion to a fixed mean.As such, we focus here on time-dependent, as well as time-independent drifts A. The processes (1.4) with B(x,t) = cx γ , 0 ≤ γ < 1, are referred to as "constant elasticity of variance" processes and are often used to model stock prices.The limiting case γ = 1 corresponds to the lognormal process (1.1) while the case γ = 1/2 corresponds to the square-root process used by Cox et al. [7], and considered by Craddock and Platen [9].When γ < 1, the variance of the percentage return from the stock decreases with the stock price.From empirical studies, Beckers [3] Joanna Goard 5 suggests that an option pricing formula based on constant elasticity of variance diffusion could fit the market prices better than the Black-Scholes model.
In interest-rate modeling, Chan et al. [6] performed a comprehensive empirical analysis on one-factor interest-rate models of the form dr = (α + βr)dt + σr γ dX. (1.19) They found that the most successful models in capturing the dynamics of the short rate were those that allowed the volatility of interest rate changes to be highly sensitive to the level of interest rate, in particular with γ ≥ 1.Their unconstrained estimate of γ was 1.499.
A summary of the results is presented in Section 5.

Reduction to canonical form
Equations of the form (1.6a) with B = B(x) can be reduced to the canonical form via a transformation of the type (see, e.g., Bluman [5]).Substituting (2.2) into (2.1),we find that the transformation ) ) where reduces (1.6a) to (2.1) with We look at the cases where Q corresponds to where k 1 , k 2 , and k 3 are constants, as in cases (I) and (II), (1.6a) could be further reduced to the form in (1.17) and in case (III), (1.6a) could be reduced further to a constantcoefficient PDE.

B(x) = cx γ
In this section, γ is assumed to be in the interval 0 ≤ γ < 1 (i.e., the constant elasticity of variance case).However similar calculations apply for γ > 1 and any variations in the solutions are noted.When γ = 1, then the change of variable for x in (3.1a) is As this substitution for x will not provide many useful tractable solutions for A other than A = μx, for which the TDF is already known, we disregard this case.When B(x) = cx γ , from (2.3a) and (2.3b), ) and z is as given from (2.3c) and (2.3d).We consider separately the cases where A = A(x) and A = A(x,t).

A = A(x).
In this case, we can let k(x) = 0 in (2.3d) so that from (2.3c) and (2.3d), and from (2.4), We now consider separately the cases where Q reduces to the forms (2.5a)-(2.5c).
For the case corresponding to (2.5a), we require that A satisfy Joanna Goard 7 where ) Then (1.6a) reduces to which needs to be solved subject to where where Note that when k 1 = 0, we can write In general for γ ≥ 0, γ = 1, we can write (3.9) 8 Fundamental solutions to Kolmogorov equations where m and D are arbitrary and The corresponding stochastic process (1.4) for x is given by with the associated TDF when D = 0 given by p(x,t; y,t where where D 1 is arbitrary.In this case, the stochastic process (1.4) for x is and the associated TDF for x is This is plotted in Figure 3.1 with c = 0.1, D 1 = 5, x = 10, t = 0 for various γ and t .
For the case corresponding to (2.5b), we require that A satisfy where .17) Equation (3.17) can be further reduced to via a substitution that depends on whether k 2 < 0 or k 2 > 0. We summarize the results here for the solutions of (1.6a)-(1.6b)and write them in such a way that shows the transformations that were necessary. where where where where Note that in general for γ ≥ 0, γ = 1, that in the expanded expressions for p from (3.19) and (3.21), the coefficient 1/(1 − γ) should be replaced by 1/|1 − γ|.
Joanna Goard 11 In this case, β < 0 and so Hence we find the fundamental solution p(x,t; y,t (3.24) Example 3.3.We note that in the case where A(x) = μx and 0 ≤ γ < 1, the density function has previously been found by Feller [11,12] and this then has been used to find option prices via risk-neutral valuation as described in the introduction (see Cox and Ross [8]).
The same result can be found here with which agrees with the result in [8].
For the case corresponding to (2.5c), we require that A satisfy where (3.28b) where k is some constant.However, the transformation necessary to get this reduction depends on the coefficient k 1 .We consider the different cases and summarize the results below.

A = A(x,t).
For A = A(x,t), with B = cx γ , from (2.3d), With x defined in terms of x 1 as given by (3.1a), x 2 = t/2 and z as given by (2.3c) and (2.3d), (1.6a) reduces to where (3.44)With (3.44), (3.43) will be reducible to a constant coefficient equation when that is, a quadratic in x 1 .
Comparing (3.44) and (3.45), we find that this will occur when k(x) = βx 1−γ and when k(x) = βx 2−2γ , where β is constant.With k(x) = βx 1−γ , admissible functions for α(x), so that Q is a quadratic in x 1 , are α(x) = e ( /(1−γ))x 1−γ and α(x) = e ( /(2−2γ))x 2−2γ , where is constant.These together with α(x) = x 1−γ are also admissible functions for k(x) = βx 2−2γ .In all of these cases, (3.43) is further reducible to the classical heat equation.We now summarize the results with the fundamental solutions of (1.6a) written in such a way that shows the transformations that were necessary for reduction to the classical heat equation.

k(x)
This case corresponds to the stochastic differential equation where  and where (3.49) Example 3.5.With β = 1, = 0, the above solution can be simplified to This is plotted in Figure 3.3 with x = 10, γ = 3/4, c = 0.1.
This case corresponds to the stochastic differential equation where (3.53) The corresponding stochastic differential equation for x is where and where Example 3.6.With β = 1, = 0, the TDF can be simplified to p(x,t; y,t The corresponding stochastic differential equation for x is The solution to (1.6a)-(1.6b) is where and where r = βc 2 (1 − γ) 2 and The corresponding stochastic differential equation for x is The solution to (1.6a)-(1.6b) is p(x,t; y,t where

B(x)
For A = A(x,t) and B(x) = (α 1 + α 2 x + α 3 x 2 ) 1/2 , the transformation subject to (2.3d) transforms (1.6a) to where in which we have used from (2.3d) that In turn, (4.2) with (4.3) is reducible to a constant coefficient equation when Q is a quadratic in x 1 .This can be achieved when k(x) is a multiple of x 1 , and α(x) takes one of the two forms as considered below.In both cases, (4.2) is further reducible to the classical heat equation.The results are summarized below.

Summary of results
In this section, we summarize the results of this paper by quoting the equation numbers for the solutions p(x,t; y,t ) to (1.6a)-(1.6b)for given functions A(x,t) and B(x,t).These solutions can be viewed as TDFs for processes x that follow (1.4).

A = A(x,t).
A(x,t) Equation number for p(x,t; y,t )

B(x)
A(x,t) Equation number for p(x,t; y,t )

Concluding remarks
In this paper we have demonstrated how it is possible to find fundamental solutions to the backward Kolmogorov equations by reducing the equations to their canonical form.In all our examples,(1.6c)is satisfied and so these fundamental solutions are interpretable as transition density functions (TDFs) for variables following certain stochastic processes.
In this way, we were able to extend the list of fundamental solutions and TDFs for the processes (1.4) considered by Craddock and Platen, that is, B = cx 1/2 , and where A = f (x) satisfies (1.13a)-(1.13c),and also find fundamental solutions and TDFs for the more general cases when B = cx γ , and B = α 1 + α 2 x + α 3 x 2 , and when A is time-dependent.Some of these more general cases have proven to agree well with real financial data.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: