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DIFFUSION

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)) Δand standard deviation B(t,X(t)) Δt1/2. 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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