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4.4. (Exponential martingales) Suppose 0(t, w) = (01(t, w),...,0n(t, w)) e R" with Ok(t,w) E V[0, T] for k = 1,..., n, where T <∞. Define Zi = exp 0(s,w)dB(s) – 0

4.4. (Exponential martingales) Suppose 0(t, w) = (01(t, w),...,0n(t, w)) e R" with Ok(t,w) E V[0, T] for k = 1,..., n, where T <∞. Define Zi = exp 0(s,w)dB(s) – 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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exponential martingale stochastic diff equation problem

4.4. (Exponential martingales) Suppose 0(t, w) = (01(t, w),...,0n(t, w)) e R" with Ok(t,w) E V[0, T] for k = 1,..., n, where T <∞. Define Zi = exp 0(s,w)dB(s) – 0<t<T where B(s) E R" and 0² = 0 · 0 (dot product). a) Use Itô's formula to prove that dZ; = Z,0(t, w)dB(t) . b) Deduce that Z; is a martingale for t < T, provided that Z,0r(t, w) E V[0, T] for 1<k< n. Remark. A sufficient condition that Z, be a martingale is the Kazamaki condition E exp ( / 0(s, w)dB(s) for all t<T. (4.3.9) This is implied by the following (stronger) Novikov condition E exp i, w)ds (4.3.10) See e.g. Ikeda & Watanabe (1989), Section III.5, and the references therein.
Transcribed Image Text:4.4. (Exponential martingales) Suppose 0(t, w) = (01(t, w),...,0n(t, w)) e R" with Ok(t,w) E V[0, T] for k = 1,..., n, where T <∞. Define Zi = exp 0(s,w)dB(s) – 0<t<T where B(s) E R" and 0² = 0 · 0 (dot product). a) Use Itô's formula to prove that dZ; = Z,0(t, w)dB(t) . b) Deduce that Z; is a martingale for t < T, provided that Z,0r(t, w) E V[0, T] for 1<k< n. Remark. A sufficient condition that Z, be a martingale is the Kazamaki condition E exp ( / 0(s, w)dB(s) for all t<T. (4.3.9) This is implied by the following (stronger) Novikov condition E exp i, w)ds (4.3.10) See e.g. Ikeda & Watanabe (1989), Section III.5, and the references therein.
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