Lecture 4: Processes with independent increments

Transcription

1 Lecure 4: Processes wih independen incremens 1. A Wienner process 1.1 Definiion of a Wienner process 1.2 Reflecion principle 1.3 Exponenial Brownian moion 1.4 Exchange of measure (Girsanov heorem) 1.5 A mulivariae Wienner processes 2. Poisson and relaed processes 2.1 A Poisson process 2.2 Compound Poisson processes 2.3 Jump-diffusion Processes 2.4 Risk processes 3. LN Problems 1

2 1. A Wienner process 1.1 Definiion of a Wienner process Le X(), 0 be a real-valued process defind on some probabiliy space < Ω, F, P >. Definiion 4.1. Sochasic process X(), 0 is a Markov process if for any 0 0 <... < n < < + s, n 1, P{X( + s) y/x() = x, X( k ) = x k, k = 1,..., n} (1) = P{X( + s) y/x() = x} = P (, x, + s, y). Definiion 4.2. Sochasic process X(), 0 is a process wih independen incremens if for any 0 0 <... < n, n 1, incremens X(+s) X(), X() X( n ), X( n ) X( n 1 ),..., X( 1 ) X( 0 ) and X( 0 ) are independen random variables. P{X( + s) y/x() = x, X( k ) = x k, k = 1,..., n} (2) Lemma 4.1. Any sochasic process wih independen incremens is a Markov process. = P{X( + s) X() y x/x() X( n ) = x x n, X( n ) X( n 1 ) = x n x n 1,..., X( 1 ) X( 0 ) = x 1 x 0, X( 0 ) = x 0 } = P{X( + s) X() y x} = P (, + s, y x). The process wih independen incremens X(), 0 is said o be homogeneous in ime if for any s, P{X( + s) X() y} = P (, + s, y) = P (s, y). 2

3 Figur 1: A rajecory of a Wienner process Definiion 4.3. A real-valued sochasic process W (), 0 defined on a probabiliy space < Ω, F, P > is called a sandard Wienner process (Brownian moion) if: A: W (), 0 is a homogenous process wih independen incremens wih he iniial value W (0) 0. B: Incremen W ( + s) W () has a normal disribuion wih he mean 0 and he variance s, for 0 + s <. C: Process W (), 0 is coninuous, i.e., a rajecory W (, ω), 0 is coninuous funcion for any ω Ω. Lemma 4.2. A process W (), 0 is a Wienner process if and only if i is a real-valued, coninuous, Gaussian process wih he iniial value W (0) 0, he expeced values EW () = 0, 0 and he correlaion funcion EW ()W (s) = min(, s),, s 0. 3

4 (a) A linear ransformaion of a Gaussian random vecor is also a Gaussian random vecor; (b) Le us ake arbirary n 1 and 0 = 0 < 1 < < n <. If (W ( 1 ),..., W ( n )) is a Gaussian random vecor hen (W ( 1 ) W ( 0 ),..., W ( n ) W ( n 1 )) is also a Gaussian random vecor and vise versa, since hese vecors are linear ransformaions of each oher. (c) If random variables W ( i ) W ( i 1 ), i = 1,..., n are independen and V ar(w ( i ) W ( i 1 )) = i i 1, i = 1,..., n hen we ge EW ( i )W ( j ) = E(W ( i ) W ( 0 )) 2 + E(W ( i ) W ( 0 ))(W ( j ) W ( i )) = i, for i j. (d) If EW ( i )W ( j ) = i, for i j, hen E(W ( i ) W ( i 1 )) 2 = EW ( i ) 2 2EW ( i )W ( i 1 ) + EW ( i 1 ) 2 = i 2 i 1 + i 1 = i i 1, for i = 1,..., n, and E(W ( i ) W ( i 1 ))(W ( j ) W ( j 1 )) = i i i 1 + i 1 = 0, for i < j. (1) Wienner process posesses he following symmeric propery, W (), 0 d = W (), 0. (2) Trajecories of a Wienner process are coninuous bu non-differeniable funcions. (e) Thus incremens W ( i ) W ( i 1 ), i = 1,..., n are independen since Gaussian random vecor has independen componens if and only if hey are no correlaed. (a) Indeed, for any > 0, he random variable and, herefore, aking n = 1 n 4, we ge for any K > 0, n=1 P{ W ( + n) W () n < K} W (+ ) W () d = 1 N(0, 1), 4

5 = n=1 1 K n 2π K e u 2 2K 2 du 1 n 2π n <. 2 (b) This relaion implies, by Borel-Kanelli lemma, ha for any K > 0 he only a finie number of evens from he sequence A n,k = { W (+ n) W () n < K}, n = 1, 2,... occur and, hus, n=1 W ( + n) W () a.s. as n. n (3) Trajecories of a Wienner process have unbounded variaion, ha means ha for any k,n = a + k n, where n = (b a) 2 n, k = 1,... 2 n, n 1 L n = 2 n k=1 2 n + W ( k,n ) W ( k 1,n 2 (a) Indeed, L n M n = 2 n k=1 W ( k,n) W ( k 1,n. a.s. as n. (b) EM n = 2 n n E N(0, 1) = c2 n/2, where c = b a E N(0, 1) ; (c) V arm n = 2 n V ar W ( 1,n ) W ( 0,n 2 n E W ( 1,n ) W ( 0,n 2 = b a. (d) Le d n = o(2 n/2 ). Obviously P{ M n EM n d n } b a d 2 n 0. (e) M n P as n. (f) M n, n = 1, 2,... is a monoonic sequence, and, hus, M n L n n. (4) Wienner process has a fracal self-similariy propery, 1 c W (c), 0 d = W (), 0. a.s. as 5

6 A process Z() = x + µ + σw (), 0, where x, µ R 1, σ > 0, is a Wienner process wih he iniial sae x, he drif µ, and he diffusion coefficien σ. (5) Transiion probabiliies for Wienner process have he following form, where P (x, y, ) = P{Z(s + ) y/z(s) = x} = P{x + µ + σw () y} = Φ( y x µ σ ), Φ(u) = 1 2π u e v2 2 dv, u R1. (6) They saisfy he Kolmogorov backward equaion, P (x, y, ) = µ σ2 2 P (x, y, ) + x 2 x2p (x, y, ). (7) The characerisic funcion of a Wienner process Z() has he following form, for 0, Ee izz() = e iz(x+µ) z2 σ 2 2, z R Reflecion principle Le consider he minimum funcionals of a sandard Wienner process W (), 0, M() = min W (s), 0. 0 s (8) Process Z x () = x + W (), 0 possesses a srong Markov propery a hiing momens τ y = inf( : Z x () = y) ha means ha σ-algebras of random evens σ[z x (s τ y ), s 0] and σ[z x (τ y + ), 0)] are independen and, moreover, Z x (τ y + ), 0 d = Z y (), 0. 6

7 Figur 2: Reflecion ransformaion (9) The srong Markov propery (8) implies he following reflecion principle, which is expressed by he following equaliy, for x, y 0, P{x + W () > y, x + M() 0} = P{x + W () < y} (3) = Φ( x y ) = 1 Φ( x + y ), x, y 0. Theorem 4.1. The following formula ake place for he sandard Wienner process, P{W () > u, M() > x} = Φ( 2x+u ) Φ( u ) for u x, x 0 Φ( x ) Φ( x ) for u < x, x 0 (4) (10) M() d = sup 0 s W (s) d = N(0, ). 7

8 (a) Using reflecion equaliy (5), we ge for x, y 0, P{x + W () > y, x + M() > 0} (5) = P{x + W () > y} P{x + W () > y, x + M() 0} = 1 Φ( y x ) (1 Φ( x + y )) = Φ( x + y ) Φ( y x ). (b) Using change of variables y x = u y + x = 2x + u, we ge for x, y 0, and P{W () > y x, M() > x} = P{W () > u, M() > x} = Φ( 2x + u ) Φ( u ), u x, x 0, P{W () > u, M() > x} = P{M() > x} = P{W () > x, M() > x} = Φ( x ) Φ( x ), u < x, x Exponenial Brownian moion Definiion 4.3. An exponenial Brownian moion is he process given by he following relaion, where S, µ R 1, σ > 0. Example S() = Se µ+σw (), 0. A European opion conrac. This is a conrac beween wo paries, a seller and a buyer, in which he buyer pays o he seller he price C > 0 a momen 0 for he righ o ge from he seller he revenue e rt [S(T ) K] + = 8

9 e rt [S(T ) K]I(S(T ) K). Here, r > 0 is he free ineres rae, K is a so-calledsrike price, and T is a mauriy of he opion conrac. I is known, ha he fair price C of he European conrac is given by he following formula, C = Ee rt [S(T ) K] +, where he expecaion should be compued for he so-called risk-neural model wih parameer µ = r σ2 2. Noe, ha, in his case, he process e r S(), 0 possesses he maringale propery, E{e r(+s) S( + s)/s()} = E{e r S()e ( r+µ)s+σ(w (+s) W ()) /S()} e r σ2 ( r+µ+ S()e 2 )s = e r S(),, s Exchange of measure (Girsanov heorem) Le W (), 0 be a sandard Wienner process defined on a probabiliy space < Ω, F, P >. Le us inroduce random variables, for β R 1, T > 0, β2 βw (T ) Y β (T ) = e 2 T. (6) Noe ha, by he definiion, (a) Y β (T ) > 0 (for every ω Ω), and (b) EY β (T ) = 1. Le us also define a new probabiliy measure on he σ-algebra F using he following β-ransformaion, P (A) = EI(A)Y β (T ), A F. (7) Lemma 4.3. The measures P (A) = EI(A) and P (A) are equivalen, i.e., P (A) = 0 P (A) = 0. 9

10 Theorem 4.2 (Girsanov). The process W () = W () β, 0 is a sandard Wienner process under measure P (A) or, equivalenly, W () = W () + β, 0 is a Wienner process wih drif β and diffusion 1 under measure P (A). (a) This readily follows from he posiiviy of he random variable Y β (T ). (a) The momen generaing funcion, m (z) = E e z(w () β) = Ee z(w () β) β2 βw (T ) e 2 T (8) β2 zβ = e 2 Ee (z+β)w () Ee β2 β(w (T ) W ()) (T ) β2 zβ = e 2 e (z+β)2 2 1 = e z2 2 = E e z W (), z R 1. (b) A proof for mulivariae disribuions is similar. Example Black-Scholes formula. Le us illusrae Girsanov heorem using i for proving he celebraing Black-Scholes formula for he fair price of an European opion. (a) According Girsanov heorem, he β-ransform ransforms he process µ + σw () o he process µ + σ( W () + β) = (µ + σβ) + σ W (), where W () is a sandard Wienner process under measure P (A) = EI(A)Y β (T ). (b) If o choose β = (r σ 2 2 ) µ σ, hen he process µ + σw () will be ransformed o he process (r σ2 2 ) + σ W () under measure P (A). (c) In his case, he he process S() = Se µ+σw () will be ransformed in he process S() σ2 (r = Se 2 )+σ W () under measure P (A). 10 2

11 (d) The fair price of an European opion, C = e rt E σ2 (r [Se 2 )T +σ W (T ) K] + (9) = SE e σ2 2 T +σ W (T ) I ) (Se σ2 2 T +σ W (T ) K e rt KP σ2 (r {Se 2 )T +σ W (T ) K}. (e) The probabiliy in he second erm of he above formula, P σ2 (r {Se 2 )T +σ W (T ) K} = P {σ W (T ) ln K S where ξ = ln K S (r σ 2 2 )T σ T = P {N(0, 1) ln K S + σ T 2. σ2 (r 2 )T σ } = Φ(ξ σ T ), T σ2 (r )T } (10) 2 (f) Le us apply he β-ransform wih parameer β = σ o he sandard Wienner process W () (under measure P (A)), which, in his case, will be ransformed o he process W () + σ), where W () is a sandard Wienner process under he new measure P (A) = E I(A)Ỹσ(T ), where Ỹ β (T ) = e β W (T ) β2 2 T. (g) In his case, he expecaion, E e σ2 2 T +σ W (T ) I (Se σ2 2 T +σ W (T ) K ) = P σ2 (r {Se 2 )T +σ( W (T )+σt ) K} = P {N(0, 1) ln K σ2 S (r + 2 )T σ } = Φ(ξ). (11) T (h) Formulas (9) (11) yields he desired Black-Scholes formula, C = SΦ(ξ) e rt KΦ(ξ σ T ). (12) 11

12 1.5 A mulivariae Wienner processes Definiion 4.4. A sochasic process W () = (W 1 (),..., W k ()), 0 wih real-valued componens, defined on a probabiliy space < Ω, F, P >, is called a sandard k-dimensional Wienner process (Brownian moion) if: A : W (), 0 is a homogenous process wih independen incremens wih he iniial values of componens W i (0) 0, i = 1,..., k. B : An incremen W ( + s) W () has a mulivariae normal disribuion wih means E(W i ( + s) W i ()) = 0, i = 1,..., k and correlaion coefficiens E(W i ( + s) W i ())(W j ( + s) W j ()) = si(i = j), i, j = 1,..., k, for 0 + s <. C : Process W (), 0 is coninuous, i.e., a rajecory W (, ω), 0 is coninuous funcion for any ω Ω. Definiion 4.5. A sochasic process Z() = (Z 1 (),..., Z k ()), 0 is a k-dimensional Wienner process if i is a linear ransformaion of a sandard k-dimensional Wienner process W (), 0, i.e, i is given by he following formula, Z() = x + µ + ΛW (), 0, where x = (x 1,..., x k ), µ = (µ 1,..., µ k ) are k-dimensional vecors wih real-valued componens, and Λ = λ ij is a k k marix wih real-valued elemens. Lemma 4.4. Process Z(), 0 is a coninuous process wih independen incremens, iniial sae Z(0) = x, and k-dimensional Gaussian disribuion of an incremen Z( + s) Z() wih he mean vecor µs and he correlaion marix Σ s = σ ij s, where σ ij = k r=1 λ irλ jr, i, j = 1,..., k, for, s 0. Lemma 4.5. A linear ransformaion Z () = ΓZ() = Γ x + Γ µ + ΓΛW (), 0 of a k-dimensional Wienner process Z() = z + µ + ΛW (), 0, where Γ = γ ij is a l k marix wih real-valued elemens, is a 12

13 l-dimensional Wienner process. 2. Poisson and relaed processes 2.1 A Poisson process Definiion 4.6. A sochasic process N(), 0 wih real-valued componens, defined on a probabiliy space < Ω, F, P > is called a Poisson process if: D: N(), 0 is a homogenous process wih independen incremens wih he iniial value N(0) 0. E: Incremen N(+s) N() has a Poisson disribuion wih mean λ > 0, i.e., P{N( + s) N() = n} = e λs (λs) n n!, n = 0, 1,.... F: Process N(), 0 has rajecories coninuous from he righ. (1) Process N( + s) N(), 0 akes only non-negaive ineger values and, hus, N() is a non-decreasing process. (2) Process N(), 0 has sepwise rajecories. (3) Process N(), 0 is sochasically coninuous. (4) Le T n = min( : > T n 1, N() > N(T n 1 )), n = 1, 2,..., where T 0 = 0. By he definiion, T n is he momen of n-h jump for he process N(). The iner-jump imes X n = T n T n 1, n = 1, 2,... are muually independen random variables. (5) Random variables X n have exponenial disribuion wih parameer λ, i.e., P{X n < x} = 1 e λx, x 0, for n = 1, 2,.... (6) Random variables T n have he Erlang disribuion wih he pdf p n () = (λ)n 1 (n 1)! e λ, 0, for n = 1, 2,

14 Figur 3: A rajecory of a Poisson process (7) Random variables N(T n ) = n wih probabiliy 1 for n = 0, 1,.... (a) N δ () = N((n + 1)δ) if nδ < (n + 1)δ, n = 0, 1,..., for δ > 0. (b) N() N δ () N( + δ) and, hus, N δ () a.s. N() as δ 0. (c) T δ,n = min( : > T δ,n 1, N δ () > N δ (T δ,n 1 )), n = 1, 2,..., where T δ,0 = 0. (d) T δ,n T n T δ,n + δ, n = 0, 1,.... (e) P{T δ,1 > kδ} = e kδλ e λ as δ 0, kδ, for > 0. (f) P{N δ (T δ,1 ) = 1} = k=0 e kδλ δλe δλ = δλe δλ 1 e δλ 1 as δ 0. (g) ec. (8) Le N(), 0 is a Poisson process wih parameer λ = 1 and λ(), 0 be a non-negaive, coninuous funcion. Define he process Ñ() = 14

15 Figur 4: A rajecory of a compound Poisson process N(Λ()), 0, where Λ() = 0 λ(u)du. Then, Ñ() is a process wih independen incremens such ha he incremen Ñ( + s) Ñ() has a Poisson disribuion wih parameer Λ(, + s) = Λ( + s) Λ() = +s λ(u)du, for, s Compound Poisson processes Definiion 4.7. A sochasic process Y (), 0 wih real-valued componens, defined on a probabiliy space < Ω, F, P > is called a compound Poisson process if i has he following form: N() X() = X k, 0, k=1 where (a) N(), 0 is a Poisson process wih parameer λ > 0; (b) X n, n = 1, 2,... is a sequence of i.i.d. real-valued random variables wih a disribuion funcion F (x); (c) he process N(), 0 and he random sequence X n, n = 1, 2,... are independen. 15

16 (1) Process X(), 0 is a homogeneous process wih independen incremens. (2) Process X(), 0 has coninuous from he righ sepwise rajecories. (3) Process X(), 0 is sochasically coninuous. (4) The characerisic funcion of he compound Poisson process X() has he following form, for 0, Ψ (z) = Ee izx() = n=0 = exp{λ e λ(λ)n (Ee izx 1 ) n = exp{λ(ee izx 1 1)} n! (e izx 1)dF (x)}, z R 1. (5) EX() = λα, V arx() = λβ, where α = EX 1, β = EX Jump-diffusion processes Definiion 4.7. A sochasic process Y (), 0 wih real-valued componens, defined on a probabiliy space < Ω, F, P > is called a jump-diffusion process if i has he following form: N() Y () = y + µ + σw () + X k, 0, where (a) y, µ R 1, σ > 0; (b) W () is a sandard Brownian moion; (c) N(), 0 is a Poisson process wih parameer λ > 0; (d) X n, n = 1, 2,... is a sequence of i.i.d. real-valued random variables wih a disribuion funcion F (x); (e) he processes W (), 0, N(), 0 and a sequence of random variables X n, n = 1, 2,... are independen. k=1 16

17 Figur 5: A rajecory of a jump-diffusion process (1) Process Y (), 0 is a homogeneous process wih independen incremens. (2) Process Y (), 0 has coninuous from he righ sepwise rajecories. (3) Process Y (), 0 is sochasically coninuous. (4) The characerisic funcion of he jump-diffusion process Y () has he following form, for 0, Ψ (z) = Ee izy () = exp {iz(y+µ) z2 σ 2 2 +λ } (e izx 1)dF (x), z R 1. (5) The process S() = s exp{µ + σw () + N() k=1 X k}, 0 is referred as an exponenial jump-diffusion process. 2.4 Risk processes 17

18 Figur 6: A rajecory of a risk process Definiion 4.8. A sochasic process X(), 0 wih real-valued componens, defined on a probabiliy space < Ω, F, P > is called a jump-diffusion process if i has he following form: N() X() = x + c X k, 0, where (a) x, c > 0; (b) N(), 0 is a Poisson process wih parameer λ > 0; (c) X 1, X 2,... is a sequence of i.i.d. non-negaive random variables wih a disribuion funcion F (x); (d) he process N(), 0 and a sequence of random variables X n, n = 1, 2,... are independen. In insurance applicaions, c is inerpreed as a premium rae, N() as a process couning he number of claims received by an insurance company in an inerval [0, ], X n as sequenial random insurance claims. A risk process is a paricular case of a jump-diffusion process and, hus, has analogous properies. (1) Process X(), 0 is a homogeneous process wih independen incremens. 18 k=1

19 (2) Process X(), 0 has coninuous from he righ sepwise rajecories. (3) Process X(), 0 is sochasically coninuous. (4) The characerisic funcion of he risk process X() has he following form, for 0, { } Ψ (z) = Ee izx() = exp iz(x + c) + λ (e izx 1)dF (x), z R LN Problems 1. Le U(, x) = Ef(x+µ+σW ()), where f(x) be a bounded coninuous funcion. Then U(, x) saisfies he equaion, U(x, ) 0 = µ σ2 2 U(x, ) + x 2 x2u(x, ). 2. V (, x) = E 0 f(x + µs + σw (s))ds, where f(x) be a bounded coninuous funcion. The following formula akes place, V (, x) = 0 1 2πsσ f(x + µs + y) exp{ y2 2σ 2 s }dyds. 3. Le N() = max(n : X 1 + +X n ), 0, where X n, n = 1, 2,... is a sequence of non-negaive i.i.d. random variables. Using formula P{N() n} = P{X X n } prove ha N() has a Poisson disribuion, if random variables X n, n = 1, 2,... has and exponenial disribuion wih parameer λ. 4. Prove he proposiion (8) formulaed a he Page Find condiions under which an exponenial jump-diffusion process S(), 0, inroduced in Sub-secion 2.3, possesses he maringale propery, i.e., E{S( + s)/s()} = S(), s, 0. 19

20 6. Find condiions under which a risk process X(), 0, inroduced in Sub-secion 2.4, possesses he maringale propery, i.e., E{X( + s)/x()} = X(), s, 0. 20