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Consider the Orstein-Uhlenbeck process Y = e-a"E + o :le-a(t-s) dW, t>0, where a and o are positive constants, W = (Wt)t>o is a Wiener mar- tingale with respect to a filtration (F;)t>0, and § is an Fo-measurable random variable with variance E(§ – E£)² = v². Determine the co- variance function C(s,t) = E((Y – EY;)(Y, – EY,)) for 0 < s < t in terms of s, t, a, o and v. Hence for h > 0 calculate %3D - lim C(s, s+ h). (Vou mou ugo 1ut proof that Ćond +he stoc intogro1 rt o-ar d

Consider the Orstein-Uhlenbeck process Y = e-a"E + o :le-a(t-s) dW, t>0, where a and o are positive constants, W = (Wt)t>o is a Wiener mar- tingale with respect to a filtration (F;)t>0, and § is an Fo-measurable random variable with variance E(§ – E£)² = v². Determine the co- variance function C(s,t) = E((Y – EY;)(Y, – EY,)) for 0 < s < t in terms of s, t, a, o and v. Hence for h > 0 calculate %3D - lim C(s, s+ h). (Vou mou ugo 1ut proof that Ćond +he stoc intogro1 rt o-ar d

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.3: Least Squares Approximation
Problem 32EQ
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o Consider the Orstein-Uhlenbeck process Y; = e-at¢ +o [ e-a(i-s) dW,, dW s, t>0, where a and o are positive constants, W = (Wt)t>o is a Wiener mar- tingale with respect to a filtration (F;)t>0, and { is an Fo-measurable random variable with variance E(§ - variance function C(s,t) terms of s, t, a, o and v. Hence for h > 0 calculate E£)² = v². Determine the co- ) = E((Y; – EY;)(Y, – EY,)) for 0 < s <t in - lim C(s, s + h). s→∞ (You may use without proof that and the stochastic integral e-ar dW, are independent for t > 0.)
Transcribed Image Text:o Consider the Orstein-Uhlenbeck process Y; = e-at¢ +o [ e-a(i-s) dW,, dW s, t>0, where a and o are positive constants, W = (Wt)t>o is a Wiener mar- tingale with respect to a filtration (F;)t>0, and { is an Fo-measurable random variable with variance E(§ - variance function C(s,t) terms of s, t, a, o and v. Hence for h > 0 calculate E£)² = v². Determine the co- ) = E((Y; – EY;)(Y, – EY,)) for 0 < s <t in - lim C(s, s + h). s→∞ (You may use without proof that and the stochastic integral e-ar dW, are independent for t > 0.)
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