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

dX(t) = A(t,X(t)) * dt + B(t,X(t)) * dW(t),

where A(t,x) is an n-dimensional vector, B(t,x) is an n-by-m matrix and W(t) is an m-dimensional Brownian motion. The differential equation is telling us the following. Conditional on all the information available at time t, the change of the diffusion on any tiny interval [t,t+Δt] has expectation A(t,X(t)) * Δt and standard deviation B(t,X(t)) * Δt^{1/2}. Stochastic processes A(t,X(t)) and B(t,X(t)) are known as the *drift* and *volatility* respectively. Since they depend on the current value of the diffusion process, the stochastic differential equation may imply complicated dynamics.

Affine diffusions are those for which A(t,X(t)) and B(t,X(t)) * B(t,X(t))^{T} are linear functions of X(t). Affine diffusions are convenient because for any deterministic linear functional L()

E[exp{L(X)} | X(t)] = exp{φ + ψ * X(t)},
**(A)**

where φ is a constant scalar and ψ is a constant 1-by-n vector. If the diffusion is used to model an interest rate, default intensity or death rate then formula (A) allows for quick calibration of the model to real-world data. Below are some of the most utilized diffusions in finance, actuarial science and engineering:

1] geometric Brownian motion (non-negative, lognormal):

dX(t) = A * X(t) * dt + B * X(t) * dW(t),

2] Vasicek-Hull-White process (affine, mean-reverting, Gaussian):

dX(t) = κ(t) * [µ(t) - X(t)] * dt + σ(t) * dW(t),

3] Cox–Ingersoll–Ross process (affine, mean-reverting, non-negative):

dX(t) = κ(t) * [µ(t) - X(t)] * dt + σ(t) * X(t)^{1/2} * dW(t),

4] Black–Karasinski process (mean-reverting, non-negative, lognormal):

d log(X(t)) = κ(t) * [µ(t) - log(X(t))] * dt + σ(t) * dW(t),

5] CEV process (non-negative for α > 0):

dX(t) = µ * X(t) * dt + σ * X(t)^{α} * dW(t),

6] Heston process (non-negative):

dX(t) = µ(t) * X(t) * dt + ν(t)^{1/2} * X(t) * dW(t)^{X},

dν(t) = κ(t) * [θ(t) - ν(t)] * dt + η(t) * ν(t)^{1/2} * dW(t)^{ν},

where Brownian motions W(t)^{X} and W(t)^{ν} have correlation ρ.

Properties of diffusions are one of the main topics of stochastic calculus and are most easily found in stochastic calculus books.

**DIFFUSION REFERENCES**

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

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.

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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