Stochastic Calculus February 11, / 33
|
|
- Karen Bennett
- 5 years ago
- Views:
Transcription
1 Martingale Transform M n martingale with respect to F n, n =, 1, 2,... σ n F n (σ M) n = n 1 i= σ i(m i+1 M i ) is a Martingale E[(σ M) n F n 1 ] n 1 = E[ σ i (M i+1 M i ) F n 1 ] i= n 2 = σ i (M i+1 M i ) + σ n 1 E[M n M n 1 F n 1 ] i= = (σ M) n 1 Called Martingale Transform Fortune at time n using strategy σ Stochastic Integral σdb is continuous time version based on martingale B(t). () Stochastic Calculus February 11, 29 1 / 33
2 Note one can also define stochastic integrals with respect to continuous martingales M(t) other than Brownian motion. The only thing that changes is that we have the quadratic variation M, M t = lim (M(t i+1 ) M(t i )) 2. () Stochastic Calculus February 11, 29 2 / 33
3 Itô s Lemma Let f (x) be twice continuously differentiable. Then df (B) = f (B)dB f (B)dt Proof First of all we can assume without loss of generality that f, f and f are all uniformly bounded, for if we can establish the lemma in the uniformly bounded case, we can approximate f by f n so that all the corresponding derivatives are bounded and converge to those of f uniformly on compact sets. () Stochastic Calculus February 11, 29 3 / 33
4 Let s = t < t 1 < t 2 < < t n = t. We have f (B(t)) f (B(s)) = = n 1 [f (B(t j+1 )) f (B(t j ))] j= n 1 f (B(t j ))(B(t j+1 ) B(t j )) j= n 1 1 ξ j [t j, t j+1 ] + 2 f (B(ξ j ))(B(t j+1 ) B(t j )) 2, j= Let the width of the partition go to zero. By definition of the stochastic integral n 1 f (B(t j ))(B(t j+1 ) B(t j )) f db. j= s () Stochastic Calculus February 11, 29 4 / 33
5 Finally we want to show that n 1 f (B(ξ j ))(B(t j+1 ) B(t j )) 2 j= s f (B(u))du It s a lot like the computation of the quadratic variation. Suppose we had f (B(t j ) inside instead of f (B(ξ j )) n 1 [ )] 2 E f (B(t j )) (B(t j+1 ) B(t j )) 2 (t j+1 t j j= = n 1 i,j= E [ f (B(t i ))X i f (B(t j ))X j ] X j =(B(t j+1 ) B(t j )) 2 (t j+1 t j ) Suppose i < j E [ f (B(t i ))X i f (B(t j ))X j ] = E [ E [ f (B(t i ))X i f (B(t j ))X j F(t j ) ]] = E [ f (B(t i ))X i f (B(t j ))E [ X j F(t j ) ]] = () Stochastic Calculus February 11, 29 5 / 33
6 Hence so n 1 [ )] 2 E f (B(t j )) (B(t j+1 ) B(t j )) 2 (t j+1 t j j= n 1 = E j= [ [ ] ] 2 (f (B(t i )) 2 (B(t j+1 ) B(t j )) 2 (t j+1 t j ) n 1 f (B(t j ))(B(t j+1 ) B(t j )) 2 L 2 f (B(u))du s j= () Stochastic Calculus February 11, 29 6 / 33
7 We still have to show n 1 j= f (B(ξ j )) f (B(t j ) (B(t j+1 ) B(t j )) 2 Taking expectation we get n 1 E[ f (B(ξ j )) f (B(t j ) (B(t j+1 ) B(t j )) 2 ] j= n 1 E[(f (B(ξ j )) f (B(t j )) 2 ] E[(B(t j+1 ) B(t j )) 4 ] j= n 1 = C E[(f (B(ξ j )) f (B(t j )) 2 ](t j+1 t j ) j= by continuity So we have proved that f (B(t) f (B(s)) = which is Itô s formula. s f (B(u))dB(u) s f (B(u))du () Stochastic Calculus February 11, 29 7 / 33
8 1 In differential notation Itô s formula reads df (B) = f (B)dB f (B)dt. The Taylor series is df (B) = n=1 1 n! f (n) (B)(dB) n. In normal calculus we would have (db) n = if n 2, but because of the finite quadratic variation of Brownian paths we have (db) 2 = dt, while still (db) n = if n 3. 2 If the function f depends on t as well as B(t), the formula is df (t, B(t)) = f f (t, B(t))dt + t x (t, B(t))dB(t) The proof is about the same as the special case above. 2 f (t, B(t))dt. x 2 () Stochastic Calculus February 11, 29 8 / 33
9 Exponential Martingales M t = e λb t 1 2 λ2 t Ito s formula df (t, B(t)) = ( f t f f )(t, B(t))dt + (t, B(t))dB(t) 2 x 2 x f (t, x) = e λx 1 2 λ2 t f t f 2 = f x 2 x = λf M t = λe λbs 1 2 λ2s db s () Stochastic Calculus February 11, 29 9 / 33
10 Geometric Brownian motion σ2 (µ S t = S e 2 )t+σb t is called Geometric (or Exponential) Brownian Motion S t so it is (Samuelson) a better model of stock prices than B t (Bachelier) µ = drift σ = volatility Ito s formula f (t, x) = e df (t, B(t)) = ( f t f f )(t, B(t))dt + (t, B(t))dB(t) 2 x 2 x σ2 (µ )t+σx f 2 t f 2 f = µf x 2 x = σf ds t = µs t dt + σs t db t Sometimes people write ds t S t = µdt + σdb t but note that ds t S t d log S t () Stochastic Calculus February 11, 29 1 / 33
11 Local time f continuous function on R + L t (x) = δ x (f (s))ds = lim ɛ (2ɛ) 1 { s t : f (s) x ɛ} 1 A (f (s))ds = A L t (x)dx f C 1 L t (x) = s i [,t]:f (s i )=x f (s i ) 1 discontinuous in t Itô s lemma applied to B t x gives Tanaka s formula for Brownian Local Time L t (x) = B t x B x In particular, L t (x) continuous in t a.s. sgn(b s x)db s () Stochastic Calculus February 11, / 33
12 But x not bounded, so no fair!!! Proof. Itô f ɛ (x) = (2ɛ) 1 1 [ ɛ,ɛ] (2ɛ) 1 { s t : B s ɛ} = f ɛ (B t ) f ɛ (B ) f ɛ(b s )db s ɛ L t (x) = B t x B x sgn(b s x)db s () Stochastic Calculus February 11, / 33
13 Feynman-Kac formula V a nice function (say bounded). u C 1,2 solves u t = 1 2 u 2 x 2 + Vu, u(, x) = u (x) u(x) exp{ x 2 /2t}dx <. Then [er t ] u(t, x) = E x V (B(s))ds u (B(t)) Proof. For s t let Z (s) = u(t s, B(s))e R s V (B(u))du. By Îto s lemma Z (t) Z () = { u + s } u er 2 x 2 + Vu s V (B(u))du ds = u x (t s, B(s))e R s V (B(u))du ds = martingale so E x [Z (t)] = E x [Z ()] () Stochastic Calculus February 11, / 33
14 1 If V = V (t, x) [er t ] u(t, x) = E x V (t s,b(s))ds u (B(t)) 2 Historical remark. Feynman s thesis was that solution u of it Schrödinger equation u t = i[ 1 2 u 2 + Vu] should have x 2 representation u(x, t) = e i R t V (fs)ds i R t 2 f 2ds u (f t ) where is supposed to be average over functions starting at x. Kac pointed out that it is rigorous if i 1 () Stochastic Calculus February 11, / 33
15 Arcsin law ξ(t) = 1 t 1 [, )(B(s))ds = the fraction of time that Brownian motion is positive up to time t a < ; P(ξ(t) a) = 2 π arcsin a a 1; 1 a > 1. Simple explanation why distribution of ξ(t) indep of t ξ(t) = 1 1 [, ) (B(ts))ds = 1 1 [, ) ( 1 t B(ts))ds = 1 1 [, ) ( B(s))ds ξ(t) d = ξ(1) () Stochastic Calculus February 11, / 33
16 Proof By Feynman-Kac if we can find a nice solution of u t = 1 2 u 2 x 2 u1 [, ) u(, x) = 1 Then and u(t, x) = E x [e R t 1 [, )(B(s))ds ] u(t, ) = 1 e at dp(ξ a) α >, φ α (x) = α u(t, x)e αt dt 1 2 φ α + (α + 1 [, ) )φ α = α { α φ α (x) = α+1 + Aex 2(α+1) + Be x 2(α+1), x, 1 + Ce x 2α + De x 2α, x. u 1 φ α 1 A = D = () Stochastic Calculus February 11, / 33
17 Proof. φ α ( ) = φ α ( + ), φ α( ) = φ α( + ) B = α 1/2 (1+α)( α+ α+1), C = 1 φ α () = (1+α) 1/2 ( α+ α+1) 1/2 α α + 1 = By Fubini s theorem this reads E 1 [ α α+ξ ] E[e tξ αe αt ]dt = α α+1 or γa dp(ξ a) = γ Looking up a table of transforms we find dp(ξ a) = 2 π 1 a(1 a) da a 1 which is the density of the arcsin distribution () Stochastic Calculus February 11, / 33
18 Stochastic differential equations σ(x, t), b(x, t) mble Definition A stochastic process X t is a solution of a stochastic differential equation dx t = b(x t, t)dt + σ(x t, t)db t, X = x on [, T ] if X t is progressively measurable with respect to F t, T b(x t, t) dt <, T σ(x t, t) 2 dt < a.s. and T X t = x + b(x s, s)ds + σ(x s, s)db s t T The main point is that σ(ω, t) = σ(x t, t), b(ω, t) = b(x t, t) () Stochastic Calculus February 11, / 33
19 If B(t) is a d-dimensional Brownian motion and f (t, x) is a function on [, ) R d which has one continuous derivative in t and two continuous derivatives in x, then Ito s formula reads df (t, B(t)) = f t (t, B(t))dt + f ((t, B(t)) db(t) + 1 f (t, B(t))dt. 2 () Stochastic Calculus February 11, / 33
20 Bessel process (d = 2) Let B t = (Bt 1, B2 t ) be 2d Brownian motion starting at, r t = B t = (Bt 1)2 + (Bt 2)2. By Ito s lemma, dr t = B1 B db1 + B2 B db This is not a stochastic differential equation. 1 B dt. () Stochastic Calculus February 11, 29 2 / 33
21 Y (t) = Let f (t, y) be a smooth function B 1 B db1 + B 2 B db2 Use Itô s lemma. Intuitively df (t, Y t ) = t fdt + y fdy y f (dy ) 2 (dy ) 2 = ( B1 B db1 + B2 B db2 ) 2 ( ) B 1 2 = (db 1 ) B1 B 2 B B 2 db1 db 2 + = dt f (t, Y t ) = f (, Y ) + + y f B1 B db1 + ( ) B 2 2 (db 2 ) 2 B ( t f y f )(s, Y s )ds y f B2 B db2 () Stochastic Calculus February 11, / 33
22 Itô s lemma dx t = σ(t, X t )db t + b(t, X t )dt f (t, X t ) = f (, X ) + Lf (t, x) = d i,j=1 d i,j=1 { } s f (s, X s ) + Lf (s, X s ) ds σ ij (s, X s ) x i f (s, X s )db j s a ij (t, x) 2 f x i x j (t, x) + d i=1 b i (t, x) f x i (t, x) d a ij = σ ik σ jk k=1 a = σσ T () Stochastic Calculus February 11, / 33
23 dr t = B1 B db1 + B2 B db B dt = dy t r t 1 dt f (t, Y t ) = f (, Y ) + + y f B1 B db1 + ( t f y f )(s, Y s )ds y f B2 B db2 In particular e λy t λ 2 t/2 is a martingale So Y t is a Brownian motion. Therefore dr t = dy t r t 1 dt is a stochastic differential equation for the new Brownian motion Y t () Stochastic Calculus February 11, / 33
24 Itô s lemma f (t, X t ) f (, X ) = Proof { } s + L f (s, X s )ds + f (s, X s ) σdb s = i = i f (t i+1, X ti+1 ) f (t i, X ti ) f t (t i, X ti )(t i+1 t i ) + f (t i, X ti ) (X ti+1 X ti ) d j,k=1 2 f x j x k (t i, X ti )(X j t i+1 X j t i )(X k t i+1 X k t i ) + higher order terms () Stochastic Calculus February 11, / 33
25 Proof continued f t t (t f i, X ti )(t i+1 t i ) t (s, X s)ds i i f (t i, X ti ) (X ti+1 X ti ) i = i + i i+1 f (t i, X ti ) ( σ(s, X s )db s ) t i i+1 f (t i, X ti ) ( b(s, X s )ds) t i 2 f t (t i, X ti )(X j t x j x i+1 X j t i )(Xt k i+1 Xt k i ) k f σdb f b ds 2 f x j x k (s, X s )a jk (s, X s )ds () Stochastic Calculus February 11, / 33
26 Proof continued To show the last convergence, ie i g(t i, X ti )(X j t i+1 X j t i )(Xt k i+1 Xt k i ) g(s, X s )a jk (s, X s )ds i+1 Z (t i, t i+1 ) = ( t i i+1 t i l i+1 σ jl (s, X s )dbs)( l t i σ jl σ kl (s, X s )ds l E[ Z (t i, t i+1 ) 2 ] = O((t i+1 t i ) 2 ) σ km (s, X s )dbs m ) m () Stochastic Calculus February 11, / 33
27 Proof continued g(t i, X ti )E[ i i+1 t i a ij (s, X s )ds] g(s, X s )a ij (s, X s )ds E[( i g(t i, X ti )Z (t i, t i+1 )) 2 ] = i,j E[g(t i, X ti )Z (t i, t i+1 )g(t j, X tj )Z (t j, t j+1 )] i < j E[E[g(t i, X ti )Z (t i, t i+1 )g(t j, X tj )Z (t j, t j+1 ) F tj ]] = i = j E[E[g 2 (t i, X ti )Z 2 (t i, t i+1 ) F ti ]] = E[g 2 (t i, X ti )E[Z 2 (t i, t i+1 ) F ti ]] = O((t i+1 t i ) 2 ) () Stochastic Calculus February 11, / 33
28 { } f (t, X t ) f (, X ) s + L f (s, X s )ds = f (s, X s ) σdb s L = 1 2 d 2 a ij (t, x) + x i x j i,j=1 M t = f (t, X t ) { } s + L d i=1 b i (t, x) x i = generator f (s, X s )ds is a martingale { } = E[f (t, X t ) f (s, X s ) u + L f (u, X u )du F s ] s = f (t, y)p(s, x, t, y)dy f (s, x) { u + L} f (u, y)p(s, x, u, y)dydu, s X s = x () Stochastic Calculus February 11, / 33
29 For any f, = f (t, y)p(s, x, t, y)dy f (s, x) { u + L} f (u, y)p(s, x, u, y)dydu s Fokker-Planck (Forward) Equation t p(s, x, t, y) = 1 2 d i,j=1 d i=1 = L yp(s, x, t, y) 2 y i y j (a i,j (t, y)p(s, x, t, y)) y i (b i (t, y)p(s, x, t, y)) lim p(s, x, t, y) = δ(y x). t s () Stochastic Calculus February 11, / 33
30 Kolmogorov (Backward) Equation s p(s, x, t, y) = d i,j=1 d i=1 = L x p(s, x, t, y) a ij (s, x) 2 p(s, x, t, y) x i x j p(s, x, t, y) b i (s, x) x i lim p(s, x, t, y) = δ(y x). s t () Stochastic Calculus February 11, 29 3 / 33
31 Proof. f (x) smooth s u = L su s < t u(t, x) = f (x) Ito s formula: u(s, X(s)) martingale up to time t u(s, x) = E s,x [u(s, X(s))] = E s,x [u(t, X(t))] = f (z)p(s, x, t, z)dz Let f n (z) smooth functions tending to δ(y z). We get in the limit that p satisfy the backward equations. () Stochastic Calculus February 11, / 33
32 Example. Brownian motion d = 1 2 L = 1 2 x 2 Forward p(s, x, t, y t = 1 2 p(s, x, t, y) 2 y 2, t > s p(s, x, s, y) = δ(y x) Backward p(s, x, t, y s = 1 2 p(s, x, t, y) 2 x 2, s < t, p(t, x, t, y) = δ(y x) () Stochastic Calculus February 11, / 33
33 Example. Ornstein-Uhlenbeck Process L = σ2 2 2 x 2 αx x Forward p(s, x, t, y t = 1 2 p(s, x, t, y) 2 y 2 + (αyp(s, x, t, y), t > s, y p(s, x, s, y) = δ(y x) Backward p(s, x, t, y s = 1 2 p(s, x, t, y) 2 x 2 αx p(s, x, t, y), s < t, x p(t, x, t, y) = δ(y x) () Stochastic Calculus February 11, / 33
p 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationStochastic Differential Equations.
Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)
More informationFrom Random Variables to Random Processes. From Random Variables to Random Processes
Random Processes In probability theory we study spaces (Ω, F, P) where Ω is the space, F are all the sets to which we can measure its probability and P is the probability. Example: Toss a die twice. Ω
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationBernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010
1 Stochastic Calculus Notes Abril 13 th, 1 As we have seen in previous lessons, the stochastic integral with respect to the Brownian motion shows a behavior different from the classical Riemann-Stieltjes
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationTheoretical Tutorial Session 2
1 / 36 Theoretical Tutorial Session 2 Xiaoming Song Department of Mathematics Drexel University July 27, 216 Outline 2 / 36 Itô s formula Martingale representation theorem Stochastic differential equations
More informationThe multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
More informationBernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012
1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.
More informationExercises. T 2T. e ita φ(t)dt.
Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.
More informationSome Tools From Stochastic Analysis
W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click
More informationLecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University
Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationMA8109 Stochastic Processes in Systems Theory Autumn 2013
Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More information1. Stochastic Process
HETERGENEITY IN QUANTITATIVE MACROECONOMICS @ TSE OCTOBER 17, 216 STOCHASTIC CALCULUS BASICS SANG YOON (TIM) LEE Very simple notes (need to add references). It is NOT meant to be a substitute for a real
More informationSimulation of conditional diffusions via forward-reverse stochastic representations
Weierstrass Institute for Applied Analysis and Stochastics Simulation of conditional diffusions via forward-reverse stochastic representations Christian Bayer and John Schoenmakers Numerical methods for
More informationApplications of Ito s Formula
CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale
More informationStochastic Differential Equations
CHAPTER 1 Stochastic Differential Equations Consider a stochastic process X t satisfying dx t = bt, X t,w t dt + σt, X t,w t dw t. 1.1 Question. 1 Can we obtain the existence and uniqueness theorem for
More informationStochastic Calculus. Kevin Sinclair. August 2, 2016
Stochastic Calculus Kevin Sinclair August, 16 1 Background Suppose we have a Brownian motion W. This is a process, and the value of W at a particular time T (which we write W T ) is a normally distributed
More informationStochastic Integration.
Chapter Stochastic Integration..1 Brownian Motion as a Martingale P is the Wiener measure on (Ω, B) where Ω = C, T B is the Borel σ-field on Ω. In addition we denote by B t the σ-field generated by x(s)
More informationElliptic Operators with Unbounded Coefficients
Elliptic Operators with Unbounded Coefficients Federica Gregorio Universitá degli Studi di Salerno 8th June 2018 joint work with S.E. Boutiah, A. Rhandi, C. Tacelli Motivation Consider the Stochastic Differential
More informationAn Introduction to Malliavin calculus and its applications
An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart
More informationPoisson random measure: motivation
: motivation The Lévy measure provides the expected number of jumps by time unit, i.e. in a time interval of the form: [t, t + 1], and of a certain size Example: ν([1, )) is the expected number of jumps
More informationDiscretization of SDEs: Euler Methods and Beyond
Discretization of SDEs: Euler Methods and Beyond 09-26-2006 / PRisMa 2006 Workshop Outline Introduction 1 Introduction Motivation Stochastic Differential Equations 2 The Time Discretization of SDEs Monte-Carlo
More informationLecture 4: Introduction to stochastic processes and stochastic calculus
Lecture 4: Introduction to stochastic processes and stochastic calculus Cédric Archambeau Centre for Computational Statistics and Machine Learning Department of Computer Science University College London
More informationIntroduction to Diffusion Processes.
Introduction to Diffusion Processes. Arka P. Ghosh Department of Statistics Iowa State University Ames, IA 511-121 apghosh@iastate.edu (515) 294-7851. February 1, 21 Abstract In this section we describe
More informationMan Kyu Im*, Un Cig Ji **, and Jae Hee Kim ***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 19, No. 4, December 26 GIRSANOV THEOREM FOR GAUSSIAN PROCESS WITH INDEPENDENT INCREMENTS Man Kyu Im*, Un Cig Ji **, and Jae Hee Kim *** Abstract.
More informationHomogenization with stochastic differential equations
Homogenization with stochastic differential equations Scott Hottovy shottovy@math.arizona.edu University of Arizona Program in Applied Mathematics October 12, 2011 Modeling with SDE Use SDE to model system
More informationLecture 21: Stochastic Differential Equations
: Stochastic Differential Equations In this lecture, we study stochastic differential equations. See Chapter 9 of [3] for a thorough treatment of the materials in this section. 1. Stochastic differential
More informationInfinite-dimensional methods for path-dependent equations
Infinite-dimensional methods for path-dependent equations (Università di Pisa) 7th General AMaMeF and Swissquote Conference EPFL, Lausanne, 8 September 215 Based on Flandoli F., Zanco G. - An infinite-dimensional
More informationA Short Introduction to Diffusion Processes and Ito Calculus
A Short Introduction to Diffusion Processes and Ito Calculus Cédric Archambeau University College, London Center for Computational Statistics and Machine Learning c.archambeau@cs.ucl.ac.uk January 24,
More informationWeek 9 Generators, duality, change of measure
Week 9 Generators, duality, change of measure Jonathan Goodman November 18, 013 1 Generators This section describes a common abstract way to describe many of the differential equations related to Markov
More informationLecture notes for Numerik IVc - Numerics for Stochastic Processes, Wintersemester 2012/2013. Instructor: Prof. Carsten Hartmann
Lecture notes for Numerik IVc - Numerics for Stochastic Processes, Wintersemester 212/213. Instructor: Prof. Carsten Hartmann Scribe: H. Lie December 15, 215 (updated version) Contents 1 Day 1, 16.1.212:
More informationThe Smoluchowski-Kramers Approximation: What model describes a Brownian particle?
The Smoluchowski-Kramers Approximation: What model describes a Brownian particle? Scott Hottovy shottovy@math.arizona.edu University of Arizona Applied Mathematics October 7, 2011 Brown observes a particle
More informationKolmogorov Equations and Markov Processes
Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define
More informationBackward martingale representation and endogenous completeness in finance
Backward martingale representation and endogenous completeness in finance Dmitry Kramkov (with Silviu Predoiu) Carnegie Mellon University 1 / 19 Bibliography Robert M. Anderson and Roberto C. Raimondo.
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 3. Calculaus in Deterministic and Stochastic Environments Steve Yang Stevens Institute of Technology 01/31/2012 Outline 1 Modeling Random Behavior
More informationIntertwinings for Markov processes
Intertwinings for Markov processes Aldéric Joulin - University of Toulouse Joint work with : Michel Bonnefont - Univ. Bordeaux Workshop 2 Piecewise Deterministic Markov Processes ennes - May 15-17, 2013
More informationBrownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion
Brownian Motion An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Background We have already seen that the limiting behavior of a discrete random walk yields a derivation of
More informationB8.3 Mathematical Models for Financial Derivatives. Hilary Term Solution Sheet 2
B8.3 Mathematical Models for Financial Derivatives Hilary Term 18 Solution Sheet In the following W t ) t denotes a standard Brownian motion and t > denotes time. A partition π of the interval, t is a
More information13 The martingale problem
19-3-2012 Notations Ω complete metric space of all continuous functions from [0, + ) to R d endowed with the distance d(ω 1, ω 2 ) = k=1 ω 1 ω 2 C([0,k];H) 2 k (1 + ω 1 ω 2 C([0,k];H) ), ω 1, ω 2 Ω. F
More informationA Barrier Version of the Russian Option
A Barrier Version of the Russian Option L. A. Shepp, A. N. Shiryaev, A. Sulem Rutgers University; shepp@stat.rutgers.edu Steklov Mathematical Institute; shiryaev@mi.ras.ru INRIA- Rocquencourt; agnes.sulem@inria.fr
More informationSolutions to the Exercises in Stochastic Analysis
Solutions to the Exercises in Stochastic Analysis Lecturer: Xue-Mei Li 1 Problem Sheet 1 In these solution I avoid using conditional expectations. But do try to give alternative proofs once we learnt conditional
More informationMartingale Problems. Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi
s Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi Lectures on Probability and Stochastic Processes III Indian Statistical Institute, Kolkata 20 24 November
More informationOn Integration-by-parts and the Itô Formula for Backwards Itô Integral
11 On Integration-by-parts and the Itô Formula for... On Integration-by-parts and the Itô Formula for Backwards Itô Integral Jayrold P. Arcede a, Emmanuel A. Cabral b a Caraga State University, Ampayon,
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationPoisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation
Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Jingyi Zhu Department of Mathematics University of Utah zhu@math.utah.edu Collaborator: Marco Avellaneda (Courant
More informationFiltrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition
Filtrations, Markov Processes and Martingales Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition David pplebaum Probability and Statistics Department,
More informationXt i Xs i N(0, σ 2 (t s)) and they are independent. This implies that the density function of X t X s is a product of normal density functions:
174 BROWNIAN MOTION 8.4. Brownian motion in R d and the heat equation. The heat equation is a partial differential equation. We are going to convert it into a probabilistic equation by reversing time.
More informationGeneralized Gaussian Bridges of Prediction-Invertible Processes
Generalized Gaussian Bridges of Prediction-Invertible Processes Tommi Sottinen 1 and Adil Yazigi University of Vaasa, Finland Modern Stochastics: Theory and Applications III September 1, 212, Kyiv, Ukraine
More informationWeak convergence and large deviation theory
First Prev Next Go To Go Back Full Screen Close Quit 1 Weak convergence and large deviation theory Large deviation principle Convergence in distribution The Bryc-Varadhan theorem Tightness and Prohorov
More informationErrata for Stochastic Calculus for Finance II Continuous-Time Models September 2006
1 Errata for Stochastic Calculus for Finance II Continuous-Time Models September 6 Page 6, lines 1, 3 and 7 from bottom. eplace A n,m by S n,m. Page 1, line 1. After Borel measurable. insert the sentence
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationAn Overview of the Martingale Representation Theorem
An Overview of the Martingale Representation Theorem Nuno Azevedo CEMAPRE - ISEG - UTL nazevedo@iseg.utl.pt September 3, 21 Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 1 / 25 Background
More informationOPTIMAL STOPPING OF A BROWNIAN BRIDGE
OPTIMAL STOPPING OF A BROWNIAN BRIDGE ERIK EKSTRÖM AND HENRIK WANNTORP Abstract. We study several optimal stopping problems in which the gains process is a Brownian bridge or a functional of a Brownian
More informationGaussian processes for inference in stochastic differential equations
Gaussian processes for inference in stochastic differential equations Manfred Opper, AI group, TU Berlin November 6, 2017 Manfred Opper, AI group, TU Berlin (TU Berlin) inference in SDE November 6, 2017
More informationSTATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, = (2n 1)(2n 3) 3 1.
STATISTICS 385: STOCHASTIC CALCULUS HOMEWORK ASSIGNMENT 4 DUE NOVEMBER 23, 26 Problem Normal Moments (A) Use the Itô formula and Brownian scaling to check that the even moments of the normal distribution
More informationStochastic Integration and Continuous Time Models
Chapter 3 Stochastic Integration and Continuous Time Models 3.1 Brownian Motion The single most important continuous time process in the construction of financial models is the Brownian motion process.
More informationSDE Coefficients. March 4, 2008
SDE Coefficients March 4, 2008 The following is a summary of GARD sections 3.3 and 6., mainly as an overview of the two main approaches to creating a SDE model. Stochastic Differential Equations (SDE)
More informationExercises in stochastic analysis
Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with
More informationContinuous Time Finance
Continuous Time Finance Lisbon 2013 Tomas Björk Stockholm School of Economics Tomas Björk, 2013 Contents Stochastic Calculus (Ch 4-5). Black-Scholes (Ch 6-7. Completeness and hedging (Ch 8-9. The martingale
More informationStochastic Differential Equations. Introduction to Stochastic Models for Pollutants Dispersion, Epidemic and Finance
Stochastic Differential Equations. Introduction to Stochastic Models for Pollutants Dispersion, Epidemic and Finance 15th March-April 19th, 211 at Lappeenranta University of Technology(LUT)-Finland By
More informationDefinition: Lévy Process. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes. Theorem
Definition: Lévy Process Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes David Applebaum Probability and Statistics Department, University of Sheffield, UK July
More informationMSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS. Contents
MSH7 - APPLIED PROBABILITY AND STOCHASTIC CALCULUS ANDREW TULLOCH Contents 1. Lecture 1 - Tuesday 1 March 2 2. Lecture 2 - Thursday 3 March 2 2.1. Concepts of convergence 2 3. Lecture 3 - Tuesday 8 March
More informationUniformly Uniformly-ergodic Markov chains and BSDEs
Uniformly Uniformly-ergodic Markov chains and BSDEs Samuel N. Cohen Mathematical Institute, University of Oxford (Based on joint work with Ying Hu, Robert Elliott, Lukas Szpruch) Centre Henri Lebesgue,
More informationStochastic Calculus and Black-Scholes Theory MTH772P Exercises Sheet 1
Stochastic Calculus and Black-Scholes Theory MTH772P Exercises Sheet. For ξ, ξ 2, i.i.d. with P(ξ i = ± = /2 define the discrete-time random walk W =, W n = ξ +... + ξ n. (i Formulate and prove the property
More informationCIMPA SCHOOL, 2007 Jump Processes and Applications to Finance Monique Jeanblanc
CIMPA SCHOOL, 27 Jump Processes and Applications to Finance Monique Jeanblanc 1 Jump Processes I. Poisson Processes II. Lévy Processes III. Jump-Diffusion Processes IV. Point Processes 2 I. Poisson Processes
More informationLecture 12: Diffusion Processes and Stochastic Differential Equations
Lecture 12: Diffusion Processes and Stochastic Differential Equations 1. Diffusion Processes 1.1 Definition of a diffusion process 1.2 Examples 2. Stochastic Differential Equations SDE) 2.1 Stochastic
More informationModel Specification Testing in Nonparametric and Semiparametric Time Series Econometrics. Jiti Gao
Model Specification Testing in Nonparametric and Semiparametric Time Series Econometrics Jiti Gao Department of Statistics School of Mathematics and Statistics The University of Western Australia Crawley
More informationStochastic Volatility and Correction to the Heat Equation
Stochastic Volatility and Correction to the Heat Equation Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar Abstract. From a probabilist s point of view the Twentieth Century has been a century
More informationEQUITY MARKET STABILITY
EQUITY MARKET STABILITY Adrian Banner INTECH Investment Technologies LLC, Princeton (Joint work with E. Robert Fernholz, Ioannis Karatzas, Vassilios Papathanakos and Phillip Whitman.) Talk at WCMF6 conference,
More informationLECTURE 2: LOCAL TIME FOR BROWNIAN MOTION
LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION We will define local time for one-dimensional Brownian motion, and deduce some of its properties. We will then use the generalized Ray-Knight theorem proved in
More informationStochastic Integrals and Itô s formula.
Chapter 5 Stochastic Itegrals ad Itô s formula. We will call a Itô process a progressively measurable almost surely cotiuous process x(t, ω) with values i R d, defied o some (Ω, F t, P) that is related
More informationSemiclassical analysis and a new result for Poisson - Lévy excursion measures
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol 3 8, Paper no 45, pages 83 36 Journal URL http://wwwmathwashingtonedu/~ejpecp/ Semiclassical analysis and a new result for Poisson - Lévy
More informationMalliavin calculus and central limit theorems
Malliavin calculus and central limit theorems David Nualart Department of Mathematics Kansas University Seminar on Stochastic Processes 2017 University of Virginia March 8-11 2017 David Nualart (Kansas
More informationOptimal portfolio strategies under partial information with expert opinions
1 / 35 Optimal portfolio strategies under partial information with expert opinions Ralf Wunderlich Brandenburg University of Technology Cottbus, Germany Joint work with Rüdiger Frey Research Seminar WU
More informationFourier transforms of measures on the Brownian graph
of measures on the Brownian graph The University of Manchester Joint work with Tuomas Sahlsten (Bristol, UK) and Tuomas Orponen (Helsinki, Finland) ICERM 1th March 216 My co-authors and dimension The Fourier
More informationDISCRETE-TIME STOCHASTIC MODELS, SDEs, AND NUMERICAL METHODS. Ed Allen. NIMBioS Tutorial: Stochastic Models With Biological Applications
DISCRETE-TIME STOCHASTIC MODELS, SDEs, AND NUMERICAL METHODS Ed Allen NIMBioS Tutorial: Stochastic Models With Biological Applications University of Tennessee, Knoxville March, 2011 ACKNOWLEDGEMENT I thank
More informationStochastic differential equation models in biology Susanne Ditlevsen
Stochastic differential equation models in biology Susanne Ditlevsen Introduction This chapter is concerned with continuous time processes, which are often modeled as a system of ordinary differential
More informationStochastic integration. P.J.C. Spreij
Stochastic integration P.J.C. Spreij this version: April 22, 29 Contents 1 Stochastic processes 1 1.1 General theory............................... 1 1.2 Stopping times...............................
More information(B(t i+1 ) B(t i )) 2
ltcc5.tex Week 5 29 October 213 Ch. V. ITÔ (STOCHASTIC) CALCULUS. WEAK CONVERGENCE. 1. Quadratic Variation. A partition π n of [, t] is a finite set of points t ni such that = t n < t n1
More informationMA 8101 Stokastiske metoder i systemteori
MA 811 Stokastiske metoder i systemteori AUTUMN TRM 3 Suggested solution with some extra comments The exam had a list of useful formulae attached. This list has been added here as well. 1 Problem In this
More informationA new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009
A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1 Performance
More informationProperties of an infinite dimensional EDS system : the Muller s ratchet
Properties of an infinite dimensional EDS system : the Muller s ratchet LATP June 5, 2011 A ratchet source : wikipedia Plan 1 Introduction : The model of Haigh 2 3 Hypothesis (Biological) : The population
More informationMULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES
MULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES D. NUALART Department of Mathematics, University of Kansas Lawrence, KS 6645, USA E-mail: nualart@math.ku.edu S. ORTIZ-LATORRE Departament de Probabilitat,
More informationBMIR Lecture Series on Probability and Statistics Fall 2015 Discrete RVs
Lecture #7 BMIR Lecture Series on Probability and Statistics Fall 2015 Department of Biomedical Engineering and Environmental Sciences National Tsing Hua University 7.1 Function of Single Variable Theorem
More information1 Brownian Local Time
1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationLAN property for sde s with additive fractional noise and continuous time observation
LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday,
More informationAn Analytic Method for Solving Uncertain Differential Equations
Journal of Uncertain Systems Vol.6, No.4, pp.244-249, 212 Online at: www.jus.org.uk An Analytic Method for Solving Uncertain Differential Equations Yuhan Liu Department of Industrial Engineering, Tsinghua
More informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationSelected Exercises on Expectations and Some Probability Inequalities
Selected Exercises on Expectations and Some Probability Inequalities # If E(X 2 ) = and E X a > 0, then P( X λa) ( λ) 2 a 2 for 0 < λ
More informationJump-type Levy Processes
Jump-type Levy Processes Ernst Eberlein Handbook of Financial Time Series Outline Table of contents Probabilistic Structure of Levy Processes Levy process Levy-Ito decomposition Jump part Probabilistic
More informationPartial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula
Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula Group 4: Bertan Yilmaz, Richard Oti-Aboagye and Di Liu May, 15 Chapter 1 Proving Dynkin s formula
More informationFirst passage time for Brownian motion and piecewise linear boundaries
To appear in Methodology and Computing in Applied Probability, (2017) 19: 237-253. doi 10.1007/s11009-015-9475-2 First passage time for Brownian motion and piecewise linear boundaries Zhiyong Jin 1 and
More informationKolmogorov equations in Hilbert spaces IV
March 26, 2010 Other types of equations Let us consider the Burgers equation in = L 2 (0, 1) dx(t) = (AX(t) + b(x(t))dt + dw (t) X(0) = x, (19) where A = ξ 2, D(A) = 2 (0, 1) 0 1 (0, 1), b(x) = ξ 2 (x
More informationIntroduction to Algorithmic Trading Strategies Lecture 3
Introduction to Algorithmic Trading Strategies Lecture 3 Trend Following Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com References Introduction to Stochastic Calculus with Applications.
More information