43433 8 3

. Stochastic exponentials...................................... 3. Girsanov s theorem......................................... 4 On the martingale property of stochastic exponentials 5. Gronwall s inequality........................................ 5. The Martingale property...................................... 6 3 Applications and examples 7 3. A classical example........................................ 7 3. Feller s test for exposion and McKean s claim.......................... 3.3 A counterexample to McKean.................................. 5 5

3 3 3. 3. Feller s test McKean 3.3 McKean

Definition.. {F t Ω, F F σ {F t ; t F s F t F s < t < Definition.. {F t Ω, F T {T F t t Definition.3. t A BR d {s, ω; s t, ω Ω, X s ω A σ B[, t] F t s, ω X s ω : [, t] Ω, B[, t] F t R d, BR d t X {F t Definition.4. Wiener Xt t Wiener Wiener process Brownian motion process X = Xt t t < t < < t k < Xt Xt Xt 3 Xt 3 s < t Xt Xs, σ t s 4 Xt sample path Definition.5. X = {X t, F t t < s < t < P EX t F s X s EX t F s X s {X t, F t t < Definition.6. X = {X t, F t t < {F t {T n n= {X n t X t Tn, F t t < n P lim n T n = = X

. Stochastic exponentials Definition.7. Ω, F, P t [, T ] d Wiener {W t {F t { Z X t = exp Xu dw u Z X =, Xu du Xt R d F t,. Z X t M X t E[Z X T ] = Xu du τ M X τ M X = lim τ M X. t τ M X = inf t R + Xu du inf = τ M X {F t τ M X t T Xt = Xτ M X =,, τ M X Xu dw u t [, τ M X Xu dw u Lemma.8. {Xt d F t T P Xt dt C = C. Z X t [ Kallianpur 98, Section 7. ] 3

. Girsanov s theorem Theorem.9.. Z X t W = { W t, F t ; t < W t = W t Xsds t < T [, { W t, F t ; t < T Ω, F T, P T d Brown [ I. Karatzas and S. E. Shreve, Section 3.5 ] 4

On the martingale property of stochastic exponentials. Gronwall s inequality Lemma.. αt, βt [a, b] Lebesgue H αt βt + H αt βt + H a αsds t [a, b] a e Ht s βsds At = αsds a gt = Ate Ht [a, b] g t d dt gt = αte Ht HAte Ht a.e. [a, b] g t βte Ht gt ga = g sds a a a.e., βse Hs ds t [a, b]. gt ga = Ate Ht Aa e Ha βse Hs ds {{ a = At e Ht βse Hs ds = βse Ht s ds a a αt βt + H αsds a {{ =At αt βt + HAt βt + H a βse Ht s ds βt B a e Ht s ds = e Ht a αt B + BH a e Ht s ds [ e Hs ds = e Ht ] t H e Hs = a H e Ht a αt B B e Ht a = Be Ht a 5

. The Martingale property Theorem.. b C R b R Wiener {W t ; t X t = bx s ds + W t, X = a.s., F W t τ b [τ b > a.s.] [t, ω : t < τ b ω] X t ω τ b, X. X t ω = bx sωds + W t ω, t τ b ω a.s.. a.a. ω, t X t ω [, τ b ω t [, t] [ω; τ b ω > t] X B[, t] Ft W 3. lim sup t τ b ω X t ω = {ω Ω τ b ω < =,, 3, b C R X t, ω b = b on [, ] b = on, ] [ +, ξ t = b ξ s ds + W t τ = inf {t Xt = τ F t W σ = τ τ + X + t σ = = X t σ = I [,σ ]txt σ + Xt σ = I [,σ ]t I [,σ ]sb + Xs + ds + W t σ I [,σ ]sbx + s ds + W t σ I [,σ ]sbx s ds + W t σ I [,σ ]s[bxs + bxs ]ds E I [,σ ]t Xt σ + Xt σ = E I [,σ ]t t I [,σ ]s[bxs + bxs ]ds 6

= E I [,σ ]t X t σ + Xt σ = E I [,σ ]t I [,σ ]s[bxs + bxs ]ds I [,σ ]t E Schwarz E ds = t E I [,σ ]s[bx + s bxs ]ds I [,σ ]s bx s + bxs ds I [,σ ]s bx s + bxs ds ϕt = E I [,σ ]t X t σ + Xt σ ϕ [, ϕt te I [,σ ]s bx s + bxs ds t [, σ ] t > σ ϕt = s [, σ ] bx s + bxs = b c X s + Xs, c + c ω sup x + b x = K [, T ] ϕt T K ϕsds Remark ϕt = t E I [,σ ]t X + t X t = X t X t + [, σ ω] X t + ω = Xt a.s. σ ω = τ ω = τ + ω = σ ω < t [, σ ω] t ω = Xt ω X + {t Xt = φ {t Xt + + = φ Xt + ω = Xt ω {t Xt + = φ τ + ω τ + ω > σ ω σ ω = τ ωa.s. ω a.a. τ ω τ + ω τ ω < τ ω < τ + ω τ τ τa.s. τ Ft W t < τ ω X t ω = X t ω X [t, ω : t < τω] τ τ b τ τ b = τ > a.s. τ > a.s. ω a.a. X t < τω t τ ω < τω X u = u b X s ds + W u 7 u τ ω

Xs b b Xs = bxs s τ ω< τω X s ω = X s ω u [, τ ω] X u = u = t u bx s ds + W u t s, ω [, t] [ω Ω τω > t] X s [, τ ω] X s = X s Xs, ωi [,t] [ω τ ω>t]s, ω = X s, ωi [,t] [ω τ ω>t]s, ω Xs, ωi [,t] [ω τ ω>t]s, ω B[, t] Ft W B[, t] Ft W - - Xs, ωi [,t] [ω τ ω>t]s, ω b 3 τω < τ ω < X τ ωω = Xτ ωω = {ω τω < lim sup t τω X t ω = τ, X 3 a.a. ω u < τ τ u X u ω u X uω η u = u bη s ds + W u u < τ τ X u = u bx sds + W u, X u = u bx sds + W u X u X u = u [bx s bx s] ds inf{u X u + X u τ = T 8

τ F t = I [,τ ]t X u τ X u τ = It u τ = u [bx s bx s] ds I [,τ ]s [bx s bx s] ds {{ = Is I u X u X u = I u X u τ X u τ u / [, τ ] u [, τ ]= u τ u τ = u = I u u I s [bx s bx s]ds It Iu = It u t E It sup X u X u = E sup It X u X u u t u t = E sup It Iu X u X u E It I t = E u t I t u t sup Iu X u X u u t E sup Iu X u X u u E sup Iu Is [bx s bx s]ds u t {{ = sup Iu u t sup u t Schwarz T u sup X u X u T E u t u Is [bx s bx s]ds u Is bx s bx s ds ds u I s bx s bx s ds I s bx s bx s ds Is bx s bx s ds Fubini σ = bx s bx s K X s X s T K E Is sup X u X u ds. u s φt = EIt sup u t X u X u It φt < φt T K 9 φsds

Lemma. t φt = I T E IT sup X u X u =. u T sup X u X u = a.s.. u T ω Λ { Λ = ω Ω I [,τ ω]t sup X u ω X uω = { ω u T sup X u ω + X uω < Λ u T {{ ω {u X u + X u =φ τ ω=t { ω sup X u ω X uω > u T { P ω sup X u ω X uω > P u T { ω { ω P Λ = = P Λ c = { P ω sup X u ω X uω > P u T { ω sup X u ω X uω = u T sup u T X u ω + X uω Λ c sup u T X u ω + X uω + P Λ c { ω { P ω sup X u ω X uω > = u T sup X u ω + X uω u T P {ω X u ω = X uω, u < τω τ ω = τω < lim sup X t ω = t τω τω = τ ω a.s. a.a. ω [, τω X t ω = X tω.

Remark ϕt = t [, T ], ϕt T K ϕt dt ϕt T K t ϕt dt ϕt T K T K.. T K n ϕt dt ϕt dt dt n ϕt n dt n dt n ϕt n dt n dt = n! ϕt n t t n n dt n ϕt T K n n! T T K n n! ϕt n t t n n dt n {{ T n = T K T K n n! n ϕt n dt n ϕt n dt n ϕt =

Theorem.3. X = X t Theorem. part T > A B T C T P X A, τ b > T = exp bw s dw s T b W s ds dp. {W A A B T C B B T C B = {x C x A, xs <, s T, [,T ] {x C xs, B T C B = A, [,T ] b Theorem. X τ P X B = = {W B T exp b W s dw s T b W s ds dp T exp bw s dw s T b W s ds dp. {W A, W s <, s T Remark. P X A, τ b > T = {X B = {X A, X s <, s T = {X A, X s <, T < τ = {X A, X s <, T < τ {W A T exp bw s dw s T b W s ds dp

ξb t = exp bw s dw s. A = C T P τ b > T = exp bw s dw s b W s ds T b W s ds = E P ξ T b.3 T P τ b > T P τ b =.3 P τ b < > T E P ξ T b < ε > M ε > s.t. T > M ε < ε < P τ b = = E P ξ T b P τ b = < ε P τ b = ε < E P ξ T b < P τ b = + ε < E P ξ T b < dp P τ b < = E P ξ T b = Remark b C R X t t [, T ] X t = bx s ds + W t Ft X = Ft W Theorem.9 X t, Ft W, P Wiener P P P Ω, FT W Radon-ikodym derivative { dp T = exp bx s dx s T b d P X s ds A B T C P X A = = {X A {W A { T exp bx s dx s { T exp bw s dw s T T b X s ds b W s ds d P dp 3

Proposition.4.. τ M X d F t Xt = ξ W, t τ M Y d F t Y t = ξ W + Y udu, t τ M Y = lim τ M Y t τ M Y = inf t R + Y u du M Y t = Y u du { M t τ X Z X t τ M X = exp [,τ M X { = exp ]u Xu = X u Xu dw u τ M X h i,τ M X u XudW u { = exp X udw u = Z X t Y u du Xu du h i,τ M X u Xu du Lemma.8 P Q A F T Q X A = E P [Z X T {X A] Theorem.9 W Q t = W t X udu.4 R d Q {t τ M X X t = ξ W Q + X udu, t P P Y t = ξ W P + τ M Y τ M Y τ M Y Y udu, t P Y t = ξ W + Y udu, t P Y t = ξ W + Y udu, t 4

Theorem. Theorem.5. Proposition.4 Xt, Y t R d F t { T Z X T = exp Xt dw t T Xt dt P τ M Y > T = E P [Z X T ] Z X T P τ M Y > T = Definition.6. P Xt candidate measure : Q C Z X t P - Q C X A = E P [ ZX T {X A ], A F T Definition.6 Corollary.7. Z X t P - Xt = ξ W QC t + t Xudu, t Q C τ M X > T = Theorem.5 E P [Z X T ] = P τ M Y > T = Q C τ M X > T Z X t E P [Z X T ] = Q C τ M X > T = 5

Xt SDE η X = lim ηx, η X = inf t R + sup i=,,d X i t X i t i =,, d Xt Corollary.8. Xt µx, t R d σx, t R d r µx, t, σx, t X dxt = µx, tdt + σx, t dw t.5 E P [Z X T ] = Q C η X > T, dxt = µx, t + σx, t Xtdt + σx, t dw QC t SDE Q C τ M X > T = Q C dw t = Xtdt + dw QC t sup X i t = t T ;i=,,d = Q C η X > T dxt = µx, tdt + σx, t dw QC t + Xtdt = µx, t + σx, t Xtdt + σx, t dw QC t 6

3 Applications and examples 3. A classical example Lemma 3. BESQ3 dzt = 3dt + ZtdW t, Z = z Ut = Zt / L Ut = Zt Bt = W t dut = Zt 3 ZtdW t = Zt dw t = Ut dw t = Ut dbt Ut Ut a.s. Ut L Bessel Zt = W t + W t + W 3 t, W t, W t, W 3 t t = W = W = W 3 = t > Ut L E[Ut ] = E[Zt ] [ ] = E W t + W t + W 3 t = t < dut = Ut dbt { Ut = U exp UsdBs λt = Ut = U Z λ t Us ds 3 7

Corollary.8 Ut dxt = Xt db QC t + Xt 3 dt Q C a.s. Bessel R t ω = B t, ω + B t, ω + B 3 t, ω dr t = dt + 3 B i tdb i t R t R t R t dr t = dt + i= 3 B i tdb i t i= drt = db t, ω + B t, ω + B 3 t, ω 3 = d B i t i= d B i t = B i tdb i t + dt 3 = B i tdb i t + 3dt Y t R t = Z t i= = R t 3 i= 3 i= B i t db i t + 3dt R t B i s db i s Bt Bt Brown R s dz t = 3dt + Z t d B t BESQ3 square of a 3-dimensional Bessel process R x + y + z dp W t,w t,w 3 tx, y, z 3 W t, W t, W 3 t dp Wt,W t,w 3tx, y, z = dp Wtx dp Wtx dp W3tx 3 = exp x πt t y t z dxdydz t 8

x = r sin θ sin ϕ, y = r sin θ cos ϕ, z = r cos θ < r <, < θ < π, < ϕ < π π π 3 r exp πt dxdydz = r sin θdrdϕdθ r r sin θdrdϕdθ = t = = t πt πt π πt πt π sin θdθ πt π dϕ exp r dr t 3 dut = Ut dbt dut Ut = UtdBt dlog Ut dlog Ut = Ut dut + Ut dut {{ =Ut 4 dt = dut Ut Ut dt = UtdBt Ut dt t dlog Ut = {{ =log Ut log U UsdBs Us ds log Ut t U = UsdBs Us ds { Ut t U = exp UsdBs { Ut = U exp UsdBs Us ds Us ds Ut 9

3. Feller s test for exposion and McKean s claim Feller s test for explosions e f C R dx t = ex t db t + fx t dt q : R R x qx = u exp x exp u fv ev dv du x fv ev dv du x < q = f/e q q = Y t = qx t dy t = ˆσY s dw s ˆσ = q q Feller s test [qx q ][q x] dx = [q qx][q x] dx = X P τ = = x > qx = q qx = x x q x = exp [q qx][q x] dx = u exp exp x x exp u fv ev dv du fv ev dv du fv ev dv = exp u x x fv ev dv dudx fv ev dv x < qx = x qx q = exp x q x = exp [qx q ][q x] dx = exp x x exp u fv ev dv du fv u ev dv fv ev dv x u du x = exp fv ev dv dudx fv ev dv

McKean s claim σ µ C R dxt = µxdt + σxdw t on Ω, F, P dxt = σxdw Q t on Ω, F, Q ± P Q Radon-ikodym Z µ/σ T = exp { T µx σx dw u T µx σx du P -.

3.3 A counterexample to McKean Proposition 3.. Brown Brown 3 Brown r < x, τ r inf{t > ; B t r d = P τ r < = d r d 3 x BESQ3 dx = 3dt + XdW t on Ω, F, P dx = XdW Q t on Ω, F, Q Feller s test dx = 3dt + XdW t x > x u exp x 3 4v dv dudx = = = = = x exp exp x x 3 x x 3 u 3 x 3 log u x u 3 dudx x dx xdx = dv dudx v dudx x < x 3 x exp u dv dudx = v = = x x x 3 x 3 = 3 exp log x u u 3 dudx x dx xdx = dudx

dx = XdW Q t x > x u exp x 4v dv dudx = = x dudx x < x x x exp u 4v dv dudx = dudx = McKean P Q Lemma 3. Xt = W t + W t + W 3 t W t, W t, W 3 t 3 Brown Proposition 3. P t > ; Xt > > dx = XdW Q t Xt > d X = X dx 4X X dx = X XdW Q t 4X X 4Xdt = dw Q t X dt t d ds t X + = dw Q s Xs ds Xt X + = W Q t Xs Xt + ds Xs = W Q t + X t > Q t > ; Xt = = P Q McKean 3

Corollary.8 P Q 3 Y t = Xt P Q P X Q { absorbing Z Y T = exp { T Y tdw t T Y t dt P - McKean McKean 4

[] BERARD WOG and C. C. HEYDE O THE MARTIGALE PROPERTY OF STOCHASTIC EXPOETIALS J. Appl. Prob. 4, 654-664 4 [] H. P. McKean Stochactic Integrals Academic Press 969 [3] I. Karatzas and S. E. Shreve Brownian Motion and Stochastic Calculus [4] Bernt Øksendal Stochactic Differential Equations An Introduction with Applications 999 [5] D. REVUZ and M. YOR Continuous Martingales and Brownian Motion,3rd edn Springer ew York 999 [6] H. Kallianpur Stochastic Filtering Theory Springer ew York 98 [7] 967 5