midterm_1_-_questions_Winter_2008

Let g t x be a twice differentiable function on 0 r

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Let g ( t, x ) be a twice differentiable function on [0 , ) × R . Then the process Y t = g ( t, X t ) is also a stochastic integral and is such that dY t = ∂g ∂t ( t, X t ) dt + ∂g ∂x ( t, X t ) dX t + 1 2 2 g ∂x 2 ( t, X t )( dX t ) 2 , where the following rules hold: dt · dt = dt · dW t = dW t · dt = 0 , dW t · dW t = dt . Some results on the Normal and Lognormal Distributions Let X be a gaussian (normal) random variable N ( m, σ 2 ) with mean m and variance σ 2 . Then we have E [ e X 1 e X <a ] = exp parenleftBigg m + σ 2 2 parenrightBigg Φ parenleftBigg ln( a ) m σ 2 σ parenrightBigg E [ e X 1 e X >a ] = exp parenleftBigg m + σ 2 2 parenrightBigg Φ parenleftBigg m + σ 2 ln a σ parenrightBigg where Φ( x ) = P ( N (0 , 1) x ). Also, the moment-generating function of X is given by E ( e tX ) = e mt + σ 2 t 2 / 2 . In particular, this implies that E [ e X ] = e m + σ 2 2
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