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Unformatted text preview: Diffusion Processes Suppose that for a process { X t } the following E ( X s + t X s  X s = x ) = tμ ( x ) + o ( t ) E (( X s + t X s ) 2  X s = x ) = tσ 2 ( x ) + o ( t ) (1) E (  X s + t X s  3  X s = x ) = o ( t ) hold, as t → 0, for every x . Here μ ( x ) and σ 2 ( x ) are functions of the state of the process X t and not necessarily constants as in the case of Brownian motion. Definition . A Markov process { X t } is said to be a (time ho mogeneous) diffusion with drift coefficient μ ( x ) and diffusion coefficient σ 2 ( x ) > 0 if (i) it has continuous sample paths, (ii) relations (1) hold for all x . The drift coefficient gives the time rate of change of the con ditional mean of the increment of the process. The diffusion coefficient represents the rate of change of the conditional co variance of the increment. Under mild regularity conditions, the drift and diffusion coefficients uniquely determine the dis tribution of the process. 1 Examples : 1. Let us check that a BM with the drift coefficient μ and diffusion coefficient σ 2 > 0 satisfies the definition: E ( B s + t B s  B s = x ) = μt E (( B s + t B s ) 2  B s = x ) = σ 2 t + μ 2 t 2 = σ 2 t + o ( t ) E (  B s + t B s  3  B s = x ) = o ( t ) . 2. Let { B t } be like in Example 1 and X t = e B t . Then X t is a timehomogeneous Markov process with continuous sample paths and E ( X s + t X s  X s = x ) = x e 1 2 σ 2 t + μt x = ( 1 2 σ 2 + μ ) xt + o ( t ) , E (( X s + t X s ) 2  X s = x ) = t ( xσ ) 2 + o ( t ) , E (  X s + t X s  3  X s = x ) = o ( t ) . A diffusion { X t } on R may be thought of as a Markov process that is locally like a Brownian motion. That is, in some sense the following relation holds dX t = μ ( X t ) dt + σ ( X t ) dB t . (2) Conditionally, given { X s , ≤ s ≤ t } , in a small time interval ( t,t + Δ t ] the displacement Δ X t = X t +Δ t X t is approxi mately a Gaussian random variable μ ( X t )Δ t + σ ( X t )( B t +Δ t B t ) , having mean μ ( X t )Δ t and variance σ 2 ( X t )Δ t . 2 From Ito’s definition of integral, equation (2) leads to X t = x + Z t μ ( X s ) ds + Z t σ ( X s ) dB s . (3) It can be proven that under some regularity conditions on coefficients μ ( · ) and σ ( · ), there exists a unique continuous stochastic process that satisfies (3) and it is a diffusion process with coefficients μ ( · ) and σ 2 ( · ) starting at x . Thus, alternatively, one may define a (timehomogeneous) dif fusion process as a stochastic process { X t , t ≥ } satisfying a SDE (Stochastic Differential Equation) of the form dX t = μ ( X t ) dt + σ ( X t ) dB t , X = x, where B t is a standard BM and μ ( · ), σ ( · ) satisfy some reg ularity conditions ( e.g. :  μ ( x ) μ ( y )  +  σ ( x ) σ ( y )  ≤ C  x y  , x,y ∈ R )....
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 Winter '09
 Adam
 Ito, Xt

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