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STOCHASTIC CALCULUS

Stochastic Calculus is part of the field of stochastic processes. It covers the following topics:

1] stochastic integrals - definitions, types, properties, relationships;

2] stochastic differential equations - stochastic processes which can be defined in terms of stochastic integrals;

3] general solutions to stochastic differential equations and their properties.

The concept of stochastic integral is very close to the concept of regular, Riemann integral. Just like a Riemann integral of function f(x) is the limit of sums

$\sum_n f(t_n') (t_{n+1} - t_n)$

where maxn(tn+1 - tn) converges to 0 and each tn' falls somewhere in interval [tn,tn+1], a stochastic integral of function f(x) is the limit of sums

$\sum_n f(X(t_n')) (X(t_{n+1}) - X(t_n))$                                                                 (1)

where maxn(tn+1 - tn) converges to 0 and each tn' falls somewhere in interval [tn,tn+1]. However, in expression (1) X(t) is a stochastic process, function f(x) is allowed to be random 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 tnis set to tn. Examples of semimartingales are Brownian motion, Poisson process, a linear combination of those, etc... Let

$\int_a^t \sigma(s) dX(s)$

be a stochastic integral of function σ(x) with respect to process X(t) on interval [a,t]. It is a function of t and can viewed as a stochastic process. For that reason we can define another stochastic process as

$Y(t) = Y(a) + \int_a^t \mu(s) ds + \int_a^t \sigma(s) dX(s).$

In the expression above the ds-integral is called the drift term; if X(s) is a Brownian motion then the dX(s)-integral is the diffusion term, otherwise if X(s) is a counting process the dX(s)-integral is the jump term. An equivalent notation for Y(t) is

$dY(t) = \mu(t) dt + \sigma(t) dX(t).$

It is just a notation, the expressions above are not differentials. However, the form of the equation has prompted researchers to call it stochastic differential equation. We can go one step further and define another stochastic process as

$Z(t) = g\Bigl(t, \int_a^t \mu_1(s) ds + \int_a^t \sigma_1(s) dX_1(s), ..., \int_a^t \mu_n(s) ds + \int_a^t \sigma_n(s) dX_n(s)\Bigr),$

where g(t,x1,...,xn) is a linear or non-linear function. Ito's lemma says that, if the stochastic integrals inside g(t,x1,...,xn) are Ito integrals, processes Xi(t) are Brownian motions and function g(t,x1,...,xn) is differentiable enough, then

$dZ(t) = \frac{\partial g}{\partial t} dt + \sum_{i=1,...,n} \frac{\partial g}{\partial x_i} \Bigl[ \mu_i(t) dt + \sigma_i(t) dX_i(t) \Bigr] + \frac12 \sum_{i,j=1,...,n} \frac{\partial^2 g}{\partial x_i \partial x_j} \sigma_i(t) \sigma_j(t) \rho_{ij}(t) dt,$

where ρij(t) is the correlation of instantaneous increments of Xi(t) and Xj(t)... So we can write the stochastic differential equation of more complex in terms of stochastic differential equations of more simple. That is what stochastic calculus all about: solving an applied problem and noticing that the relevant process can be written as a complex function of stochastic integrals, writing down the corresponding stochastic differential equation, solving the equation and studying properties of the solution... Stochastic calculus has gained widespread use in the fields of physics, engineering and asset pricing.

STOCHASTIC CALCULUS SUBCATEGORIES

STOCHASTIC CALCULUS 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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