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 ) = t ( x ) + o ( t ) E (( X s + t- X s ) 2 | X s = x ) = t 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 ( x ) and 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 ( x ) and diffusion coefficient 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 and diffusion coefficient 2 > 0 satisfies the definition: E ( B s + t- B s | B s = x ) = t E (( B s + t- B s ) 2 | B s = x ) = 2 t + 2 t 2 = 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 2 t + t- x = ( 1 2 2 + ) xt + o ( t ) , E (( X s + t- X s ) 2 | X s = x ) = t ( x ) 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 = ( X t ) dt + ( X t ) dB t . (2) Conditionally, given { X s , s t } , in a small time interval ( t,t + t ] the displacement X t = X t + t- X t is approxi- mately a Gaussian random variable ( X t ) t + ( X t )( B t + t- B t ) , having mean ( X t ) t and variance 2 ( X t ) t . 2 From Itos definition of integral, equation (2) leads to X t = x + Z t ( X s ) ds + Z t ( X s ) dB s . (3) It can be proven that under some regularity conditions on coefficients ( ) and ( ), there exists a unique continuous stochastic process that satisfies (3) and it is a diffusion process with coefficients ( ) and 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 = ( X t ) dt + ( X t ) dB t , X = x, where B t is a standard BM and ( ), ( ) satisfy some reg- ularity conditions ( e.g. : | ( x )- ( y ) | + | ( x )- ( y ) | C | x- y | , x,y R )....
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SDEs - Diffusion Processes Suppose that for a process { X t...

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