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STOCHASTIC DIFFERENTIAL EQUATION

Stochastic Differential Equation (SDE) is a way to define a stochastic process in terms of stochastic integrals, other stochastic processes and deterministic functions. Such definitions arise naturally in systems where many random disturbances are aggregated over time.

A stochastic integral of stochastic process σ(x) with respect to stochastic process X(t) on interval [a,t] is defined as

$\int_a^t \sigma(s) dX(s) = \lim_{max_n(s_{n+1} - s_n)\ \rightarrow\ 0} \sum_n \sigma(s_n') (X(s_{n+1}) - X(s_n))$

where each sn' falls somewhere in interval [sn,sn+1] and the limit is understood in mean-square sense (or almost sure sense, depending on the set of assumptions). X(t) is called the driving stochastic process. The most utilized stochastic integral is Ito integral, where the driving stochastic process is a semimartingale and snis set to sn. Examples of semimartingales are Brownian motion, Poisson process, a linear combination of those, etc.

A stochastic differential equation is the following equality:

$Y(t) = Y(a) + \int_a^t \mu(s) ds + \int_a^t \sigma_1(s) dX_1(s) + ... + \int_a^t \sigma_n(s) dX_n(s).$

Or an equivalent notation:

$dY(t) = \mu(t) dt + \sigma_1(t) dX_1(t) + ... + \sigma_n(t) dX_n(t).$

Function Y(t) is a stochastic process by construction. Note that the right hand side is allowed to depend on Y(t) as well.

Just as in the theory of regular differential equations, long and cumbersome stochastic differential equations can oftentimes have short and elegant solutions, written in closed form. Solutions to SDE and their properties are the central part of stochastic calculus. Some of the most popular models in empirical finance, financial engineering, engineering and physics are phrased in terms of stochastic differential equations.

STOCHASTIC DIFFERENTIAL EQUATION 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-Verlag Berlin Heidelberg.

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

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

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