L13 - Lecture 13 Itˆo’s Formula Itˆo’s formula is at...

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Unformatted text preview: Lecture 13 Itˆo’s Formula Itˆo’s formula is at the heart of stochastic calculus. It shows an important distinction from the Fundamental Theorem of Calculus. We will present the 1D and multidimensional versions of Itˆo’s formula and illustrate their applications in a number of examples. 13.1 The 1D case Definition 13.1 X = { X t } is called a 1D Itˆo process if it has an expression X t = X + Z t μ u du + Z t σ u dW u , ≤ t ≤ T, (13.1) where the drift process μ = { μ t } is adapted to the Brownian filtration, and the diffusion coefficient σ = { σ t } is a H 2-process ( see Lecture 12 ) . Sometimes an Itˆo process X is defined as a solution to the stochastic differential equation (SDE) dX t = μ t dt + σ t dW t . (13.2) Note that (13.2) is not well-defined in a strict sense because Brownian motion is non-differentiable. From now on, whenever we see a differential form (13.2), it should be understood as defined by (13.1). Theorem 13.1 Let g ( t,x ) be continuously differentiable in t and twice continuously differentiable in x . Define Y t = g ( t,X t ) , ≤ t ≤...
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This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.

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L13 - Lecture 13 Itˆo’s Formula Itˆo’s formula is at...

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