# SDEs - Diffusion Processes Suppose that for a process X t...

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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 time-homogeneous 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 (time-homogeneous) 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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## This note was uploaded on 03/05/2012 for the course ACTSC 446 taught by Professor Adam during the Winter '09 term at Waterloo.

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SDEs - Diffusion Processes Suppose that for a process X t...

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