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

Abstract

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.

On the construction of non-affine jump-diffusion models

“…This is relevant only in the case of infinite jump intensity, because the condition (3.9) is always satisfied by putting h(x) ≡ 0 for the finite jump intensity case. We also note that the transformation equations of (3.12) and (3.38) are similar to the equations of (3.10)-(3.12) in [10] for a single Poisson random measure and the equations of (8) and (16) in [16] for multiple Poisson random measures in the timeinhomogeneous setting. However, in the present paper, the conditions of (ii) and (v) of Theorems 3.4-3.5 reflect the fact that we work in the infinite jump intensity case.…”

“…for some continuous functions β(t, x), γ i (t), and δ i (t, v), i = 0, 1, and all t ≥ 0, x ∈ D X , y ∈ D Y , and v ∈ R, we obtain that the equation in (3.1) is reduced to the one of (2.8), which is solvable in a closed form under either the conditions of (2.9) or the assumption γ 0 (t) = δ 0 (t, v) = 0, for all t ≥ 0 and v ∈ R. In this case, we call the stochastic differential equation in (3.1) reducible to a solvable equation, by means of the invertible transformation f (t, y) described above (see also [12; Chapter IV], [23; Chapter V, Example 5.16], [10], and [16] for definitions of the related concepts).…”

“…The general class of solvable stochastic differential equations was expanded in [10] to the case of jump-diffusion processes driven by Wiener processes and Poisson random measures of finite intensities. The tractability of the resulting analytic solutions of the constructed equations was shown inİyigünler, Ç aglar, and Unal [16], by analysing the accuracy of the numerical approximations obtained from the appropriate discretisation schemes. The Laplace transforms of the first exit times from two-sided intervals for certain one-dimensional jump-diffusion processes satisfying solvable stochastic differential equations were recently computed in [11].…”

On some functionals of the first passage times in jump models of stochastic volatility

“…We study the case in which the equation in (2.2) for the mean-reverting or diverting component Q can be either solved explicitly or reduced to the associated ordinary differential equation, by means of an appropriate integrating factor process. Stochastic differential equations of such type were considered by Gard [22; Chapter IV] and Øksendal [37; Chapter V] for diffusion processes, and then in [15] and [20] for their jump analogues.İyigünler et al [26] shew the tractability of the resulting analytic solutions of such so-called solvable stochastic differential equations, by means of the analysis of the accuracy of numerical approximations obtained from the appropriate discretisation schemes.…”

On the Laplace transforms of the first exit times in one-dimensional non-affine jump–diffusion models

“…Such solvable stochastic differential equations were considered in Gard [13, Chapter IV] and Øksendal [22, Chapter V] for continuous diffusion processes, and then in [9] and [12] for their jump-diffusion analogues. The tractability of the resulting analytic solutions of this type of stochastic differential equations was shown inİyigünler, Ç aglar, andÜnal [14], by analysing the accuracy of the numerical approximations obtained from the appropriate discretisation schemes. We obtain closed-form solutions to the integro-differential boundary value problems associated with the values of Laplace transforms of the first exit times as stopping problems for continuous-time Markov processes, including the (non-affine) pure-jump analogues of certain mean-reverting and diverting diffusions.…”