Exact Solvability of Stochastic Differential Equations Driven by Finite Activity Levy Processes

We consider linearizing transformations of the one-dimensional nonlinear stochastic differential equations driven by Wiener and compound Poisson processes, namely finite activity Levy processes. We present linearizability criteria and derive the required transformations. We use a stochastic integrating factor method to solve the linearized equations and provide closed-form solutions. We apply our method to a number ofstochastic differential equations including Cox-Ingersoll-Ross short-term interest rate model, log-mean reverting asset pricing model and geometric Ornstein-Uhlenbeck equation all with additional jump terms. We use their analytical solutions to illustrate the accuracy of the numerical approximations obtained from Euler and Maghsoodi discretization schemes. The means of the solutions are estimated through Monte Carlo method.


INTRODUCTION
The theory of stochastic differential equations has recently enjoyed significant reputation as a result of its impact on physics, finance and engineering [3,11,22,23,4,27].Analytical solutions of stochastic differential equations not only allow us to study the underlying stochastic processes, but also provide the means to test the numerical schemes [10,19].Therefore, analytical methods for the integration of nonlinear stochastic differential equations are of paramount importance.
Stochastic differential equations with jump terms, driven by Levy processes in general, are more realistic in cases where sudden events play prominent role [4,12,14,17,21,24,27].Levy processes are basically stochastic processes with stationary and independent increments.They are analogues of the random walks in continuous time.Moreover, they form a subclass of semi-martingales and Markov processes which include very important special cases such as Brownian motion, Poisson process and subordinators.Although much of the basic theory was established earlier, a great deal of new theoretical development as well as novel applications in diverse areas has emerged in recent years [1,25] [12,23].Since a finite activity Levy process can be decomposed into a Wiener and a compound Poisson process as a special case of Levy-Itô decomposition [1]; we say that ( 2) is driven by a Levy process.Note that (2) is more general than a stochastic differential equation which involves the increment of a Levy process only as a single term.
In this paper, we derive the conditions for linearizability of (2) and solve the arising linear stochastic differential equation using integrating factor method.Linearization problem has been considered in [29,30] for equations including a single Poisson jump term, which serve as preliminaries of the present work.We extend and generalize these results by considering Equation ( 2) and demonstrate identification of a stochastic integrating factor for solving the linearized equation.Integrating factors have been used earlier for linear stochastic differential equations driven by a Wiener process [23].On the other hand, the linearized version of (2) includes jump terms inherited from the compound Poisson processes (1) and is in the form ) where H and Z are semimartingales satisfying the condition that the jump : 1   for all [0, ] t   and 0 0 Z  .The conditions imposed on the coefficients of (2) for the existence of X imply that H and Z are semimartingales.More references for the solution of related linear equations based on semimartingales and their generalization with adapted cadlag processes can be found in [7].Linearization of purely continuous diffusion processes has been considered in [10,Chp.4].Independently from the present work, linearizability conditions for a jump equation similar to (2) are studied for the time homogeneous case, that is, when ( , ) ( ), ( , ) ( ) and ( , ) ( ) r x t r x  by Gapeev [9].However, the sufficient conditions for exact solvability of (2) have not been obtained by Gapeev in his study.Therefore, we present our rigorous derivations, which are obtained independently from [9], for the more general time inhomogeneous case and with the further generalization to several compound processes rather than a single Poisson random measure.
As applications, we show that Cox-Ingersoll-Ross short-term interest rate model [3], log-mean reverting asset pricing model [5,26,31], geometric Ornstein-Uhlenbeck equation [6,8,23,28] with additional jump terms and one other example [12,20] are linearizable under specified conditions on the functions , f g and r .Exact solutions of these linearizable equations are obtained.We then compare our analytical solutions with the numerical approximations found by Euler and Maghsoodi schemes to demonstrate the agreement.
The paper is organized as follows.In Section 2, we present our results about the linearization of nonlinear stochastic differential equations and give the linearizability criteria.The analytical solution of the resulting linear stochastic differential equation is found in Section 3. Several examples are given in Section 4 to demonstrate the linearizability conditions of several well-known equations with additional jump terms.

LINEARIZATION
We are interested in stochastic differential equations of the form (2), which can be transformed into a linear equation and solved as a result.Although one can impose certain conditions on the functions , f g and j r , 1, , j m   that are sufficient for the existence and uniqueness of the solution X , one can instead concentrate on the linearized equation and the conditions from there on.We seek a sufficiently smooth Borel function : which will transform the nonlinear stochastic differential equation given in (2) into a linear equation of the form )


. If a unique solution of (4) exists and h is invertible, then Equation (2) also has a unique solution X .Therefore, we can focus on sufficient conditions for the existence of solutions of (4).We assume sufficiently smooth continuous functions 1 2 , a a and 1 2 , b b below and state the implied differentiability conditions on the original functions f and g .Linearizability conditions are found under smoothness assumptions on , 1, , j r j n   as well.Technical conditions on all functions are stated during the derivations, for the linear coefficients of (4) to be well-defined.
Suppose that Using Equations ( 2) and (4), we obtain, Ordinary differential equation ( 6) has two distinct solutions for i) 1 ( ) 0 b t  and ii) 1 ( ) 0 b t  .We now consider each case separately.
for all x   and t    .Then the transformation h can be found as where we have chosen the arbitrary function of integration to be zero and assumed  ( , ) 0 g x t  .Substituting ( 8) into ( 5) and differentiating with respect to x gives where Multiplying both sides of ( 9) by ( , ) g x t and differentiating with respect to x leads to as we have assumed 2 ( ) 0. b t  Then, we can choose and differentiating it with respect to x , we obtain provided that ( ( , ) , ) 0 j g x r x t z t   for all , x z   We rewrite the above equation as ( , ) ( , ) : ( ( , ) 1) ( ( , ) , ) Now, 1 ( ) c t can be found from ( 14) and 2 ( ) c t can be found from (12).Differentiating ( 14) with respect to x yields ( , ) 0 (11) and ( 15) are the linearization conditions.
The solution of (10) in this case is in the form We simply choose ( ) 1 K t  and seek 1 ( ). b t Thus, we get .
(16) Substitution of ( 16) into (9) yields Differentiation of both sides of ( 22) with respect to x and simplification lead to where L is as in (10).We aim to find 1 ( ) b t first.Differentiating with respect to x and cancelling 1 ( ) b t as it is nonzero, we get where we assume that [ ( , ) ( , )] 0.
x g x t L x t   Differentiation of (20) with respect to x yields as a term in the linearization criterion (21).Differentiation of ( 16) with respect to x yields Substitution of ( 22) into ( 7) and then differentiating with respect to x and cancelling 1 ( ) b t , we get ( , ) ( ( , ) 1) in terms of j A of ( 13).Differentiation of ( 23) with respect to x yields as the linearization condition involving 2,0 ( ) Therefore, Equations ( 21) and ( 24) are the linearization conditions in Case 2. Now, one can obtain 1 ( ) a t from (18), 2 ( ) a t from ( 17), 1 ( ) b t from ( 20) with ( 10), 1 ( ) c t from ( 23) and 2 ( ) c t by substitution of ( 22) into (7) finally.We now state our findings as a theorem.
Note that the two sets of conditions in Theorem 1 are mutually exclusive.If (11) is satisfied, then (21) is not possible as it originates from (19)

SOLUTION OF THE LINEAR EQUATION
In this section, we use the integrating factor method for solving linear ODEs to the linear SDEs driven by finite activity Levy processes and find the solution of (7).The solution of a linear jump-diffusion equation has been considered in [9] as cited by [18], which is also based on an integrating factor.More generally, the solution to linear system of SDEs based on semi-martingales [15], and more recently including cadlag processes [7], is well-known.We demonstrate the integrating factor method.Our solution appears in a more compact form mainly because the specific form of the semimartingales are used during the derivation.
A process t  serves as a stochastic integrating factor for solving (7) if the product is not a function of Y .To find such a  , we start by applying the product rule for semimartingales [1] Therefore, from ( 7) and ( 25) we can write the terms which will form the differential as well as These terms should not involve the variable t Y  for t  to comply with the definition of an integrating factor.Arranging all the terms yields , .
Hence, we must have and

 and the quadratic variation
Let us seek a solution to the system of PDEs ( 27) and ( 28) together with (29).Equation (29) if there is a jump of the compound Poisson process j C at time t, for some   Since t  has a continuous part, (30) implies that it is in the form where the functional 1 M and 2 M denote the continuous and discontinuous parts of t  , respectively.Note that since the Poisson processes j N , 1, , j m   are independent, at most one jump occurs from only one of 1 , , m N N  with probability 1 at any time.Therefore, the discontinuous part 2 M can be written by (30) as where   , , 1, , denote the pump denote the jump times and jump amounts of the compound Poisson process j C up to time t, for 1, , j m   .Now, we will fill find 1 M using ( 27) and (28).Substitution of (32) into (28) leads to By Itô's formula and in view of (34), we have To solve (35), we try 1 M which satisfies for some Stieltjes function q and we have taken q for the sake of brevity in the sequel.Hence, (35) reduces to and solution to (37) is where the integral [22].Note that solution (38) indeed satisfies (36).Thus, the stochastic integrating factor takes the form We now use t  of (39) in ( 27) to obtain For determining t Y , consider equations ( 26) and ( 28) which now imply Substituting (40) into Equation (41) yields the explicit solution of the linear SDE (7) as and 2 M is as in (33).

ANALYTICAL SOLUTIONS OF SPECIFIC EXAMPLES
We now consider some linearizable SDEs driven by Wiener and compound Poisson processes and compare analytical solutions with the Euler and Maghsoodi numerical approximations [13,19,20].All examples of this section satisfy criteria (11) and (15).

Example 1
The second example is taken from [12], with extra jump terms given by      are positive real valued parameters.


Transformation (43) leads to the solution

Cox Ingersoll Ross Model
The third equation is taken from [3] but with an additional jump term as ( ) Here, t X represents the mean-reverting short-term interest rate.In this model, β is the long-term average value of interest rate with jumps, α is the intensity (strength) of mean reversion, σ is the interest rate volatility where t X is the instantaneous interest rate at period t, maturing at period T.
is satisfied when 2 .

  
Using the transformation, we get .

Log Mean-Reverting Model
The following example is a log-mean-reverting Black-Karasinski interest rate model [2,5,26,31] with a jump term.We have 1 ( ( ) ln ) reverting equation with jumps is commonly used in modeling assets subject to supply and demand such as commodities.Due to advantage of ease of simulation, modeling and parameters estimation, this model is widely preferred.Therefore, t X now corresponds to the spot price of the commodity.In this model,  is the long-run mean of the logarithm of the price with jumps,  is the mean reversion speed (intensity) of the price and  is the price volatility.We have ( , ) ( ( ) ln ), ( ) , ( ) ln( 1) The solution is found as Finally, the use of (47) leads to the solution

Geometric Ornstein-Uhlenbeck Equation
The last example is from [6,8,23,28], a geometric Ornstein-Uhlenbeck equation with an additional jump term.We have .In this equation, the mean reversion component is governed by the difference between the current price and the mean  as well as by the mean reversion rate  where  is the volatility of the spot price.Note that, spot price t X is always positive.
It can be seen that the analytical sample trajectories in Fig. 1a) nearly coincides with the mean estimated from the numerical approximations.However, the means are statistically significantly different as shown in Fig. 1b).
where 1 ( ) b t is nonzero and (11) cannot hold.Clearly, this argument holds both ways.The results of Theorem 1 are similar to [9, Eqns.(3.10)-(3.12)],which are in particular based on timehomogeneous functions f and g and a single Poisson random measure.
valued.This equation is know as the Cox-Ingersoll-Ross interest rate model, before the jump terms are added.
valued.This log mean-

Fig. 1 a
) Simulation of the exact solution and the numerical approximations.b) Mean trajectories estimated from 10000 independent replications for the geometric O-U equation.
 are positive real valued.Equation (48) is also known as geometric Ornstein-Uhlenbeck or Dixit & Pindyck model, now including additional jump terms.This model is based on a mean-reverting commodity price or interest rate t X