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JUMP-DIFFUSION

Jump-diffusion is an n-dimensional stochastic process X(t) satisfying the following stochastic differential equation (SDE):




where
A(t,x) is an n-dimensional vector, B(t,x) is an n-by-m matrix, C(t,x) is an n-by-r matrix, W(t) is an m-dimensional Brownian motion and N(t) is an r-dimensional counting process. In other words, jump-diffusions are solutions of stochastic differential equations containing stochastic integrals with respect Brownian motion and various jump processes.

Properties of jump-diffusions are an integral part of stochastic calculus. Several important models in empirical finance, financial engineering, engineering and physics are phrased in terms of jump-diffusions.


JUMP-DIFFUSION REFERENCES

Karatzas, I., & Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus (2nd ed). New York: Springer.

Shreve, S. E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer.

Oksendal, B. K. (2002). Stochastic Differential Equations: An Introduction with Applications (5th ed). Springer.

Protter, P. E. (2005). Stochastic Integration and Differential Equations (2nd ed). Springer.

Gikhman, I. I., & Skorokhod, A. V. (2007). The Theory of Stochastic Processes III. Springer.


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