Girsanov s Theorem. Bernardo D Auria web: July 5, 2017 ICMAT / UC3M

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1 Girsanov s Theorem Bernardo D Auria bernardo.dauria@uc3m.es web: July 5, 2017 ICMAT / UC3M

2 Girsanov s Theorem

3 Decomposition of P-Martingales as Q-semi-martingales Theorem ([1] ) Let P and Q be locally equivalent with a continuous Radon-Nikodým density L. If M is a continuous P-local martingale, then the process M defined by d M = dm 1 d M, L L is a continuous Q-local martingale. If N is another continuous P-local martingale M, N = M, Ñ = M, Ñ. Corollary ([1] ) We may write the process L as a Doléans-Dade martingale: L t = E(ξ), where ξ is an F-local martingale. The process M = M M, ξ is a Q-local martingale. 1

4 Girsanov s Threorem: One dimensional Brownian Motion Proposition (Girsanov s Threorem, [1] ) Let W be a (P, F)-Brownian motion and let θ be an F-adapted process such that the solution of the SDE dl t = L t θ t dw t, L 0 = 1 is a martingale. We set Q Ft = L t P Ft. Then the process W admits a Q-semi-martingale decomposition W as t W t = W t θ s ds 0 where W is a Q-Brownian motion. 2

5 Girsanov s Threorem: One dimensional Brownian Motion Example ([1] ) Consider the geometric Brownian motion, S, that is ds t = S t (µ dt + σ dw t ). Example ([1] ) Consider the process X solution of dx t = a dt + 2 X t dw t. 3

6 Exercises Exercise ([1] ) Consider Q = h(w T ) P with h Cc 1 (R + ). Prove that B defined by ( + db t = dw t exp{ (y W ) t) 2 /(2(T t))} h (y) dy + exp{ (y W dt t) 2 /(2(T t))} h(y) dy is a Q-Brownian motion. See [3] for applications to finance. 4

7 Exercises Exercise ([1] ) a) Let ds t = S t σ dw t, S 0 = x. Prove that for any bounded function f [ ( )] ST x 2 E[f (S T )) = E x f S T b) Prove that, if ds t = S t (µ dt + σ dw t ), there exists γ sich that S γ is a martingale. Prove that for any bounded function f [( ) γ ( )] ST x 2 E[f (S T )) = E f x S T and [ )] E[ST α f (S T )) = x α e α µt E f (e α σ2t S T. 5

8 Exercises Exercise ([1] ) Let W be a P-Brownian motion, and B t = W t + ν t be a Q-Brownian motion, under a suitable change of probability. Check that, inthe case ν > 0, the process exp{w t } tend towards 0 under Q when t goes to, whereas this is not the case under P. Exercise ([1] ) Let W be a P-Brownian motion, and set dq Ft = L t dp Ft where L t = exp{λw t 1 2 λ2 t}. Prove that the process X, where t W s X t = W t ds 0 s is a Brownian motion with respect to its natuarl filtration under both P and Q. 6

9 Girsanov s Threorem: Multi-dimensional Brownian Motion Proposition (Girsanov s Threorem, [1] 1.7.4) Let W be a (P, F)-Brownian motion and let θ be an n-dimensional F-adapted process such that t 0 θ s 2 ds <, a.s. Define the local ) martingale L as the solution of dl t = L t θ t dw t = L t ( i θi t dwt i, and L 0 = 1, so that t L t = exp{ θ s dw s 1 t θ s 2 ds} If L is a martingale the n-dimensional process t ( W t = W t + θ s ds, t 0) 0 is a Q-Brownian motion, where Q Ft = L t P Ft. If W is a Brownian motion with correlation matrix Λ, then ( W t = W t + t 0 θ s Λ ds, t 0) is a Q-Brownian motion with the same correlation matrix Λ. 7

10 PRP under Change of Probability ([1] 1.7.7) Proposition Let W be a (P, F W )-Brownian motion and Q a probability measure locally equivalent to P. Let W be the martingale part of the Q-semi-martingale. If M is a (Q, F W )-local martingale, there exists an F W -predictable process H such that t t, M t = M 0 + H s d W s. 0 8

11 Invariance of BM under Change of Measure ([1] 1.7.8) Proposition Let X be a real valued F-Brownian motion under P and L be the Radon-Nikodým density of Q w.r.t. P. Then X is a Q-Brownian motion if and only if X and L are (P, F)-orthogonal martingales. Example ( [1] ) If W = (X, Y ) is a 2-dimensional Brownian motion starting from (a, b), the pair (x, l) where x t = W t / W t (stopped at the first time W vanishes) and l t = (Y t, X t ) satisfies the previous condition. 9

12 Bibliography

13 Bibliography M. Jeanblanc, M. Yor and M. Chesney (2009). Mathematical methods for financial markets. Springer, London. D. Revuz and M. Yor (1999). Continuous Martingales and Brownian Motion. Springer Verlag, Berlin. 3 rd ed. F. Baudoin (2003). Modeling anticipations on financial markets. In Paris-Princeton Lecture on mathematical Finance 2002, volume 1814 of Lecture Notes in Mathematics, Springer Verlag, Berlin. 10