Stochastic Processes for Finance - eBooks and textbooks from bookboon.com

General framework Filtrations, adapted and predictable processes Markov and diffusion processes Martingales The Brownian motion Intuitive presentation The assumptions Definition and gene[r]

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Stochastic Processes for Finance

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Stochastic Processes for Finance

Patrick Roger Strasbourg University, EM Strasbourg Business School

June 2010

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ISBN 978-87-7681-666-7

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Stochastic Processes for Finance Contents

Contents

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Contents

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Stochastic Processes for Finance Contents

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Stochastic Processes for Finance Introduction

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Stochastic Processes for Finance Discrete-time stochastic processes

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

 lim

m →+∞πm =

0.5 0.50.5 0.5





           

             

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2s

s

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Xs(ω) = E (Xs+1|Fs) (ω)  ω ∈ {v ≥ s}

       Xv

s − Xv s−1 = ξs (Xs− E (Xs|Fs−1))

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Stochastic Processes for Finance Continuous-time stochastic processes

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Stochastic Processes for Finance     Continuous-time stochastic processes

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N n=1δn   δn   

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

= E

Zs+ (Zt− Zs)

FZ s



= E

Zs

FZ s

+ E

Zt− Zs



FZ s

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

= E

Zt2

FZ s



− t

  

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+ 2E

ZtZs



FZ s



− EZs2

FZ s

+ 2E

ZtZs



FZ s

+ 2ZsE

Zt



FZ s

+ 2Zs2− Z2

s − t

E(Zt− Zs)2

+ Zs2 − t(t− s) + Zs2− t



Eexp (γZt)

FZ s



Eexp (γ(Zt− Zs)) exp(γZs)

FZ s



γx− γ

2t2



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= √12πt

= exp



−γ

2(t + s)2

1

√2πt

2t2

1

√2πt

= exp



−γ

2s2



= g(0, s)

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Stochastic Processes for Finance Stochastic integral and Itô’s lemma

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N n=1n−1

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σsdZs

           

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ZsdZs= Z

2

T − Z2 0

ZT22

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σ2sds



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sds +

 t 0

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dWs = Wt=

 t 0

ds +

 t 0

σdZs

= 

 t 0

ds + σ

 t 0

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∂2f

∂x2(x0, t0)(dx)2 

+12

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2

∂2f

∂x2 [(Xt, t)dt + σ(Xt, t)dZt]2+1

∂2f

∂x2σ2(Xt, t)

dt+σ(Xt, t)∂f

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∂2g

∂x2σ2(Yt, t) = ∂g

∂xYt+

12

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Stochastic Processes for Finance       Stochastic integral and Itô’s lemma

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αX−α2 = exp

−α  2



   Q() = EP

exp

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= √12π

2

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λsdZs− 1

2

 t 0

12

 T 0

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σ2(Xt, t) dt < +∞ 

 X 

Xt= X0+

 t 0

 (Xs, s) ds +

 t 0

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exp [a (t− s)] σsdZs 

       

Xtexp(−at) = c +

 t 0

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exp(αs)dZs

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exp(αs)dZs



= β (1− exp(−αt)) + y0exp(−αt) + σ

 t 0

exp(−2α(t − s))ds

= σ

2

2α[1− exp (−2αt)]

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Stochastic Processes for Finance Bibliography

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Stochastic Processes for Finance Bibliography

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stationary distribution, 20 transient, 19

no memory process, 48 process, 15, 48

Brownian motion, 59 transition matrix, 15 Martingale, 21, 49

Brownian motion, 59 Doob, 24

submartingale, 21, 49 super-martingale, 21, 49 Modi cation

stochastic process, 43

Novikov condition, 89

Path, 40 càdlàg, 41 càglàd, 43 continuous, 40 Brownian motion, 58 LCRL, 43

nowhere differentiable, 59 RCLL, 41

Poisson distribution, 47 process, 47 Probability transition, 15 Snell envelope, 36 Stochastic differential equation, 91 linear, 93

solution conditions, 92

de nition, 91 Markov, 92 Stochastic integral, 71 calculation rules, 79 definition, 76 properties, 78 stochastic differential, 78

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