Dual Stochastic Transformations of Solvable Diffusions

# Dual Stochastic Transformations of Solvable Diffusions

## Abstract

We present new extensions to a method for constructing several families of solvable one-dimensional time-homogeneous diffusions whose transition densities are obtainable in analytically closed-form. Our approach is based on a dual application of the so-called diffusion canonical transformation method that combines smooth monotonic mappings and measure changes via Doob-h transforms. This gives rise to new multi-parameter solvable diffusions that are generally divided into two main classes; the first is specified by having affine (linear) drift with various resulting nonlinear diffusion coefficient functions, while the second class allows for several specifications of a (generally nonlinear) diffusion coefficient with resulting nonlinear drift function. The theory is applicable to diffusions with either singular and/or non-singular endpoints. As part of the results in this paper, we also present a complete boundary classification and martingale characterization of the newly developed diffusion families.
Keywords: solvable continuous-time stochastic processes; Bessel, CIR, and Ornstein-Uhlenbeck processes; nonlinear volatility diffusion models in finance; nonlinear mean-reverting drift models.
AMS Subject Classification: 60G51, 60H10, 91B70.

## Introduction and Main Results

A solvable continuous-time stochastic process can be basically defined as a process for which transition probability density functions are obtainable in analytically closed-form. Such solvability permits us to precisely simulate paths of the process from its exact sample distribution and also to readily compute certain mathematical expectations. For solvable families of diffusion processes, solvability implies the existence of analytically closed-form spectral expansions for transition densities of the regular processes subject to appropriate boundary conditions. For certain classes of diffusion models, the spectral expansions can be readily derived in closed-form. For these same diffusion processes, the standard spectral methods show that analytical tractability also extends beyond transition densities. In particular, closed-form expressions exist for other fundamental quantities such as, for example, first-hitting time densities (or distributions) as well as joint probability densities for various extrema of the process, etc.

The set of diffusion processes that are, on the one hand, tractable and applicable for mathematical modeling and, on the other hand, exactly solvable in closed-form is not so vast. This known set of classical diffusions includes mostly linear diffusion processes or those whose drift and/or diffusion coefficients have a power or quadratic polynomial nonlinearity (see [5] and [16] for a comprehensive review of such diffusions; see also [11, 13]). An important goal is hence to extend solvability to other families that have useful applications. There are two main tools that allow us to construct new solvable diffusion processes. The first is related to a measure change on a chosen underlying diffusion and the second involves a change of variable or smooth monotonic mapping (the Itô formula). In recent years, a new approach that combines special measure changes, i.e. time-homogeneous Doob-h transforms, together with special types of nonlinear smooth monotonic mapping transformations was introduced for uncovering new families of exactly solvable driftless diffusion models  [2, 3, 4, 6, 15]. These models exhibit nonlinear diffusion coefficients with multiple adjustable parameters and have seen some useful applications in financial derivative pricing [6, 9, 10]. The method has been coined as “diffusion canonical transformation”, wherein the solvability of a diffusion process, say , is essentially reduced to that of a simpler underlying diffusion .

This paper provides the first formal extension of the diffusion canonical transformation method to include a substantially larger dual class of monotonic mappings and thereby constructs two new main classes of solvable diffusions . Throughout, these processes are also called -diffusions. In contrast to the previous related papers, the time-homogeneous Doob-h transform is now constructed more generally to include diffusions with any type of singular and/or non-singular endpoints. Hence, we also provide a complete boundary classification for all possible families of -diffusions. The first main class of -diffusions consists of families satisfying a time-homogeneous stochastic differential equation (SDE) of the form with an affine (linear) drift and multi-parameter nonlinear diffusion coefficient function. We therefore note that the affine drift models presented in this paper significantly extend and include those studied in [6]. As part of our new results, we present three explicitly solvable families of such -diffusions, named Bessel, confluent hypergeometric, and Ornstein-Uhlenbeck families. These processes arise via the diffusion canonical transformation method by respectively choosing a squared Bessel (SQB) process, CIR (squared radial Ornstein-Uhlenbeck) process, and Ornstein-Uhlenbeck process as underlying diffusions. The three new families include (recover) all the corresponding driftless -diffusions obtained previously (e.g. see [6]) as special subfamilies. Moreover, the new affine models inherit some of the important salient properties of their driftless counterparts. One immediate application of such diffusions is asset pricing in finance (when and is a constant such as a risk-free interest rate). In [6], we showed that these three families generate local volatility profiles with varied pronounced smiles and skews (see Figures 1 and  2). Three particular subfamilies named here as the Bessel-, confluent-, and OU models are of particular importance since, for each of them, there exists a risk-neutral probability measure such that the discounted asset price process is a martingale. As in the driftless case, these models are very amenable for pricing many standard financial derivatives since the transition densities (state price densities) for the asset or stock price (i.e. ) are given in closed form. Clearly, the pricing of standard European options is reduced to the evaluation of a definite integral (e.g. see [2, 6]). As well, these solvable models admit explicit closed-form spectral expansions for the transition densities with imposed killing at arbitrary levels, for the first hitting time densities, and for joint densities of the extrema and the price process. Hence, efficient pricing formulas of standard exotic options, such as barrier and lookback options, are also available [7]. Discretely-monitored path-dependent options can be evaluated by using a path integral approach, as was done with previously related state-dependent volatility models [8, 9]. Moreover, subfamilies of diffusions belonging to the Bessel and confluent hypergeometric families admit absorption at zero asset price, so they can naturally be used in derivatives pricing under credit (default) risk.

The second main class of solvable models presented here consists of diffusions with a nonlinear drift and with specification of a generally nonlinear diffusion coefficient. In particular, within this second class of diffusions we find some explicitly solvable diffusion families with a nonlinear mean-reverting drift. Mean-reverting models have useful applications in modeling interest rates. Traditional single-factor interest rate models only consider linear mean reversion, since such solvable models have analytically tractable solutions. As an example of an alternative one-factor nonlinear mean-reverting solvable model, a new family of -diffusions generated from the SQB Bessel process is introduced in this paper. For a particular subfamily of such processes, we use the closed-form transition probability densities for the Doob-h transformed processes and the fact that an underlying bridge process and its Doob-h transformed bridge process have equivalent probability laws, and hence derive some closed-form integral formulas for conditional expectations of functionals involving the discount factor of the process and the process terminal value. The formulas are applicable to standard bond and bond option pricing.

To summarize this introduction, we point out how the rest of the paper is organized. In Section 1, all of the necessary ingredients for constructing the newly solvable dual classes of -diffusions are presented. A useful Lemma 1 for the boundary classification of the families of transformed -diffusions and hence -diffusions is also given. Subsection 1.4 concludes this section with the basis of the dual smooth monotonic mapping transformations for generating the two main classes of solvable -diffusions . Section 2 presents three explicit -diffusions, i.e., Bessel, confluent hypergeometric, and Ornstein-Uhlenbeck families. For each, we give analytical expressions for various transition densities and also derive the boundary classification. In subfamilies where an endpoint is attainable (e.g. subfamilies (i) and (iii) of the Bessel and confluent -diffusions), we also derive in analytically closed-form the density for the first hitting time to the endpoint. The boundary classification and first hitting time densities then follow automatically for the -diffusions. Section 3 presents the construction of the mappings for generating the affine drift -diffusions. In Subsection 3.3, we analyze whether or not -diffusions with linear drift “preserve” the drift rate, i.e. whether , holds and thereby present a theorem that gives easy-to-implement limit conditions for verifying this property. This can be viewed as a generalization of the martingale property for driftless processes. Thus, for the special case with we are able to prove whether a discounted process is a martingale. Section 4.1 presents the three main families of affine drift -diffusions with their explicit multi-parameter nonlinear volatility specifications. We single out three subfamilies with this martingale property and two subfamilies in which is a strict supermartingale. In Section 4.2 we discuss all possible monotonic maps that lead to nonlinear -diffusions with affine drift , . Section 5 presents Bessel families of nonlinear mean-reverting diffusions that are obtained from subfamilies of the squared Bessel -process by applying a power- or exponential-type mapping function . Lemmas 9 and 10 give necessary conditions for the mean-reversion. The asymptotic behaviour of the drift and diffusion coefficients is analyzed. Moreover, we present a model that admits a closed-form expression for the expectation of a discount factor which can be used for bond and bond option pricing. In Appendix A, we derive new asymptotic properties of Wronskians of fundamental solutions used to construct solvable diffusions from the three main families considered here. Such properties allow us to easily analyze stochastic properties of the solvable diffusions.

## 1 Construction of Nonlinear Solvable Diffusions

### 1.1 Underlying Diffusion

Let be a one-dimensional time-homogeneous regular diffusion on , , defined by its infinitesimal generator:

 (Gf)(x)≜12ν2(x)f′′(x)+λ(x)f′(x),x∈I. (1.1)

The functions and denote, respectively, the (infinitesimal) drift and diffusion coefficients of the process. Throughout we assume that the functions , , and are continuous on the open interval and that is strictly positive on . The diffusion has speed measure  and scale function  (see, e.g., [5]) that are absolutely continuous with respect to the Lebesgue measure and have smooth derivatives. The scale and speed density functions are defined as follows:

 s(x)=dS(x)dx=exp(−∫x2λ(z)ν2(z)dz) \ and \ m(x)=M(dx)dx=2ν2(x)s(x). (1.2)

Given an -diffusion, we can choose any pair of fundamental solutions to the differential equation , , that are denoted by and . For positive real values , and are linearly independent and respectively increasing and decreasing positive functions of . The Wronskian of these functions is given as:

 W[φ−s,φ+s](x)≜φ−s(x)dφ+s(x)dx−φ+s(x)dφ−s(x)dx=wss(x), (1.3)

where is a constant w.r.t. and for real .

We denote by a transition probability density function (PDF) for w.r.t. the Lebesgue measure, i.e. it is a fundamental solution to the Kolmogorov PDE where , , , . We recall that the Green function and the transition PDF are related via the Laplace inverse transform w.r.t. , i.e. . The Green function , for , is written in terms of a pair of functions and in the standard form [5]:

 GX(x,x0,s)=W−1sm(x)ψs(x<)ϕs(x>), (1.4)

where and . The functions , that also solve , are generally not necessarily the same as the above chosen (elementary) pair. In particular, these functions are linear combinations of , i.e. , with coefficients , . The Wronskian factor is given by . The coefficients (where ) and hence the functions, are uniquely characterized (within a multiplicative constant) by requiring that, for real , and are respectively increasing and decreasing functions and by additionally posing boundary conditions at regular (non-singular) boundaries of (see [5]). For a regular left boundary , if is specified as killing or if is specified as reflecting and included in the state space. If is a singular boundary, the functions have the following boundary properties: If is entrance(entrance-not-exit), then ; if is exit(exit-not-entrance), then ; if is a natural boundary, then . Analogous conditions hold for the right boundary involving the right limits, i.e. and . Moreover, we note that if is singular then we can set and similarly if is singular then .

### 1.2 Change of Measure

Consider a class of one-dimensional time-homogeneous regular diffusions with infinitesimal generator

 (G(ρ)f)(x)≜12ν2(x)f′′(x)+(λ(x)+ν2(x)^u′ρ(x)^uρ(x))f′(x). (1.5)

A strictly positive generating function , is a linear combination of the chosen fundamental pair :

 ^uρ(x)=q1φ+ρ(x)+q2φ−ρ(x), (1.6)

with parameters and at least one of them being strictly positive. The speed and scale densities for an -diffusion are given in terms of those for the underlying -diffusion:

 mρ(x)=^u2ρ(x)m(x) and sρ(x)=s(x)^u2ρ(x). (1.7)

By comparing the generators (1.1) and (1.5), observe that -diffusions can also be viewed as arising from the underlying -diffusion by the application of a measure change. In fact, the -diffusion can be realized from the -diffusion upon employing a time-homogeneous space-time transform, i.e. a Doob- transform, where , which is -excessive (see [5]). Both processes are regular on the same state space .

Given a generating function , we define the pair and . By applying the differential operator , it follows that these functions solve , . From (1.3) and (1.7), the Wronskian of these solutions is given by

 W[φ(ρ)−s,φ(ρ)+s](x)=1^u2ρW[φ−ρ+s,φ+ρ+s](x)=wρ+ssρ(x).

Hence, are a fundamental set of solutions that are linearly independent and strictly positive functions of for real values .

The Green function for -diffusions on then has the general form

 G(ρ)X(x,x0,s)=(W(ρ)s)−1mρ(x)ψ(ρ)s(x<)ϕ(ρ)s(x>), (1.8)

where, in analogy with the -diffusion, solve and are linear combinations of , i.e. and with coefficients , . The Wronskian factor is then given by , where . These coefficients are uniquely characterized (within a multiplicative constant) by requiring that, for real , and are respectively increasing and decreasing functions and by additionally posing boundary conditions at regular boundaries of . For a regular left boundary , if is specified as killing or if is specified as reflecting and included in the state space. Note that this reflecting boundary condition is equivalently written as . If is a singular boundary, the functions have the following boundary properties: if is entrance, then ; if is exit, then ; if is a natural boundary, then . Analogous conditions hold for the right boundary involving the right limits, i.e. (or ) and (or ).

It clearly follows from (1.4) and (1.8) that any Green function for a diffusion can be related to some Green function for a diffusion by

 G(ρ)X(x,x0,s)mρ(x)=1^uρ(x)^uρ(x0)GX(x,x0,s+ρ)m(x). (1.9)

For diffusion , a transition PDF is obtained from its corresponding Green function by Laplace inversion, i.e.

 p(ρ)X(t;x0,x)=L−1s[G(ρ)X(x,x0,s)][t]=mρ(x)L−1s[(W(ρ)s)−1ψ(ρ)s(x<)ϕ(ρ)s(x>)][t]. (1.10)

By Laplace inverting (1.9) we see that a transition density for a diffusion is related to a transition density for a diffusion by

 p(ρ)X(t;x0,x)=^uρ(x)^uρ(x0)e−ρtpX(t;x0,x),x,x0∈I,t>0. (1.11)

### 1.3 Boundary Classification

Given an underlying -diffusion and , any regular diffusion with generator in (1.5) falls into one of three general families:

1. where ,

2. where ,

3. where .

For with positive real parts, we denote

 n(x;s1,s2)=φ+s1(x)φ−s2(x),n(l+;s1,s2)=limx→l+n(x;s1,s2),n(r−;s1,s2)=limx→r−n(x;s1,s2).

We recall (see [5]) that for singular (non-regular) boundaries of , i.e. entrance (entrance-not-exit), exit (exit-not-entrance) or natural, we have and . For regular boundaries of it is also possible, depending on the choice of fundamental solutions and the type of boundary conditions imposed at , that is finite for all with positive real parts. In particular, we generally have and .

The fundamental solutions generally satisfy the square integrability conditions w.r.t. the speed measure: and for and . Throughout this paper we conveniently define the inner product of two functions , w.r.t. on a closed interval as and , , .

###### Lemma 1.

The above three families (i)–(iii) of regular diffusions on with generator , , defined by (1.5) and (1.6) have the following boundary classification:

1. : is attracting natural if , is exit (or attracting natural) when (or ) if and , and is otherwise regular if .

The boundary is non-attracting (or attracting) natural when (or ) if , and is entrance (or regular) when (or ) if .

2. : is attracting natural if , is exit (or attracting natural) when (or ) if and , and is otherwise regular if .

The boundary is non-attracting (or attracting) natural when (or ) if , and is entrance (or regular) when (or ) if .

3. : The boundary has the same classification as in (i) and has the same classification as in (ii).

###### Proof.

Let , , and denote the scale measure , , , and let

 Σρ(l) =∫xlSρ(l,z]mρ(z)dz, Σρ(r) =∫rxSρ[z,r)mρ(z)dz, Nρ(l) =∫xlSρ[z,x]mρ(z)dz, Nρ(r) =∫rxSρ[x,z]mρ(z)dz.

The proof now follows by applying the Feller conditions for the respective -diffusions (i)–(iii) with scale and speed densities in (1.7), That is, is regular if and , exit if and , entrance if and , and natural if and ; () is attracting if and only if () is finite. From (1.3) and (1.7), we have , if , and , if . Hence, , if ; , if . Consider family (i). Then, , so is attracting; , so is attracting if and only if . and . Hence, if and only if when and if and only if when , since , while if and only if and . and . Hence, if and only if , since , while if and only if , since . The above combined conditions are then summarized as stated in the Lemma for family (i). The stated boundary classification for families (ii) and (iii) is proven by applying similar steps as in family (i). ∎

### 1.4 Generating F-Diffusions: Dual Transformations

We now consider -diffusions defined by strictly monotonic real-valued mapping with continuous on with unique inverse . Such an elementary (Itô) transformation gives a diffusion process with infinitesimal generator

 (GFh)(F)=12σ2(F)h′′(F)+α(F)h′(F),F∈IF=(Fl,Fr) (1.12)

where and are the respective drift and diffusion coefficients:

 α(F)=(G(ρ)F)(X(F)),σ(F)=ν(X(F))/|X′(F)|. (1.13)

is a regular diffusion on with endpoints and .

The map can be specified so as to create a process with a desired drift or diffusion coefficient. To obtain a linear-drift -diffusion, the drift coefficient is specified by a linear function, i.e. we set . Hence, by the first relation in (1.13), with , , we see that is obtained by solving the 2nd order linear nonhomogeneous ODE:

 (G(ρ)F)(x)=a+bF(x). (1.14)

Given any strictly monotonic smooth solution , then is a process with specified affine (linear) drift and having generally nonlinear diffusion coefficient with infinitesimal generator in (1.12), where follows automatically from the second relation in (1.13).

An alternative approach is to specify the diffusion coefficient rather than the drift function. One way is to directly specify a strictly positive function with continuous first derivative. Then, the second equation in (1.13), i.e. , is integrated to give , and its inverse relation , where allows for either a strictly increasing or a decreasing map. Another way is to explicitly specify a strictly nonzero continuously differentiable function , i.e. specify , and then integrate giving , with and as arbitrary constants. The diffusion function is then given by . Either way, the resulting strictly monotonic smooth map is used to produce -diffusions defined by the infinitesimal generator in (1.12) with a specified diffusion coefficient function and a resulting generally nonlinear drift function:

 α(F)=λ(X(F))X′(F)+(ν(X(F))X′(F))2[X′(F)^u′ρ(X(F))^uρ(X(F))−12X′′(F)X′(F)]. (1.15)

This expression follows from the first equation in (1.13) while using (1.5) where , .

By either of the above dual transformation approaches, several families of analytically solvable -diffusion models can be constructed using known solvable underlying -diffusion processes defined by (1.1). The -diffusion models given by (1.12) either have a nonlinear state dependent volatility with a specified affine (linear) drift or have nonlinear state dependent drift with a volatility that is specified as either affine or as nonlinear state dependent. We refer to the above general framework as the “diffusion canonical transformation” methodology.

###### Lemma 2.

The boundary classification for an -diffusion defined by with strictly monotonic mapping is equivalent to the corresponding -diffusion.

###### Proof.

This follows trivially by the diffeomorphism . ∎

## 2 Three Choices of Underlying Solvable Diffusions

### 2.1 The Squared Bessel Process

Consider a -dimensional squared Bessel (SQB) process obeying the SDE with constants and . This diffusion has regular state space with generator , and scale and speed densities and where . The origin is entrance if , regular if and exit if ; is natural (attracting for ). As a pair of fundamental solutions to , , for we choose