Brownian motion or a Wiener process if. B0 = 0, it has stationary, independent
increments,. Haijun Li. An Introduction to Stochastic Calculus. Week 2. 3 / 16 ...
An Introduction to Stochastic Calculus Haijun Li
[email protected] Department of Mathematics Washington State University
Week 2
Haijun Li
An Introduction to Stochastic Calculus
Week 2
1 / 16
Outline
1
Brownian Motion Basic Properties Processes Related to Brownian Motion
2
Simulation of Brownian Sample Paths Simulation via the Functional Central Limit Theorem Simulation via Series Representations
Haijun Li
An Introduction to Stochastic Calculus
Week 2
2 / 16
Definition A stochastic process B = (Bt , t ∈ [0, ∞)) is called a (standard) Brownian motion or a Wiener process if
Haijun Li
An Introduction to Stochastic Calculus
Week 2
3 / 16
Definition A stochastic process B = (Bt , t ∈ [0, ∞)) is called a (standard) Brownian motion or a Wiener process if B0 = 0,
Haijun Li
An Introduction to Stochastic Calculus
Week 2
3 / 16
Definition A stochastic process B = (Bt , t ∈ [0, ∞)) is called a (standard) Brownian motion or a Wiener process if B0 = 0, it has stationary, independent increments,
Haijun Li
An Introduction to Stochastic Calculus
Week 2
3 / 16
Definition A stochastic process B = (Bt , t ∈ [0, ∞)) is called a (standard) Brownian motion or a Wiener process if B0 = 0, it has stationary, independent increments, for every t > 0, Bt has a normal N(0, t) distribution, and
Haijun Li
An Introduction to Stochastic Calculus
Week 2
3 / 16
Definition A stochastic process B = (Bt , t ∈ [0, ∞)) is called a (standard) Brownian motion or a Wiener process if B0 = 0, it has stationary, independent increments, for every t > 0, Bt has a normal N(0, t) distribution, and it has continuous sample paths.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
3 / 16
Definition A stochastic process B = (Bt , t ∈ [0, ∞)) is called a (standard) Brownian motion or a Wiener process if B0 = 0, it has stationary, independent increments, for every t > 0, Bt has a normal N(0, t) distribution, and it has continuous sample paths. Historical Note: Brownian motion is named after the botanist Robert Brown who first observed, in the 1820s, the irregular motion of pollen grains immersed in water. By the end of the nineteenth century, the phenomenon was understood by means of kinetic theory as a result of molecular bombardment. in 1900, L. Bachelier had employed it to model the stock market, where the analogue of molecular bombardment is the interplay of the myriad of individual market decisions that determine the market price. Norbert Wiener (1923) was the first to put Brownian motion on a firm mathematical basis. Haijun Li
An Introduction to Stochastic Calculus
Week 2
3 / 16
Distributional Properties of Brownian Motion
For any t > s, Bt − Bs =d Bt−s − B0 = Bt−s has an N(0, t − s) distribution. That is, the larger the interval, the larger the fluctuations of B on this interval.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
4 / 16
Distributional Properties of Brownian Motion
For any t > s, Bt − Bs =d Bt−s − B0 = Bt−s has an N(0, t − s) distribution. That is, the larger the interval, the larger the fluctuations of B on this interval. µB (t) = EBt = 0 and for any t > s, CB (t, s) = E[((Bt − Bs ) + Bs )Bs ] = E(Bt − Bs )EBs + s = min(s, t).
Haijun Li
An Introduction to Stochastic Calculus
Week 2
4 / 16
Distributional Properties of Brownian Motion
For any t > s, Bt − Bs =d Bt−s − B0 = Bt−s has an N(0, t − s) distribution. That is, the larger the interval, the larger the fluctuations of B on this interval. µB (t) = EBt = 0 and for any t > s, CB (t, s) = E[((Bt − Bs ) + Bs )Bs ] = E(Bt − Bs )EBs + s = min(s, t).
Brownian motion is a Gaussian process: its finite-dimensional distributions are multivariate Gaussian.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
4 / 16
Distributional Properties of Brownian Motion
For any t > s, Bt − Bs =d Bt−s − B0 = Bt−s has an N(0, t − s) distribution. That is, the larger the interval, the larger the fluctuations of B on this interval. µB (t) = EBt = 0 and for any t > s, CB (t, s) = E[((Bt − Bs ) + Bs )Bs ] = E(Bt − Bs )EBs + s = min(s, t).
Brownian motion is a Gaussian process: its finite-dimensional distributions are multivariate Gaussian. Question: How irregular are Brownian sample paths?
Haijun Li
An Introduction to Stochastic Calculus
Week 2
4 / 16
Self-Similarity A stochastic process X = (Xt , t ≥ 0) is H-self-similar for some H > 0 if it satisfies the condition (T H Xt1 , . . . , T H Xtn ) =d (XTt1 , . . . , XTtn ) for every T > 0 and any choice of ti ≥ 0, i = 1, . . . , n.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
5 / 16
Self-Similarity A stochastic process X = (Xt , t ≥ 0) is H-self-similar for some H > 0 if it satisfies the condition (T H Xt1 , . . . , T H Xtn ) =d (XTt1 , . . . , XTtn ) for every T > 0 and any choice of ti ≥ 0, i = 1, . . . , n. Self-similarity means that the properly scaled patterns of a sample path in any small or large time interval have a similar shape.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
5 / 16
Self-Similarity A stochastic process X = (Xt , t ≥ 0) is H-self-similar for some H > 0 if it satisfies the condition (T H Xt1 , . . . , T H Xtn ) =d (XTt1 , . . . , XTtn ) for every T > 0 and any choice of ti ≥ 0, i = 1, . . . , n. Self-similarity means that the properly scaled patterns of a sample path in any small or large time interval have a similar shape.
Non-Differentiability of Self-Similar Processes For any H-self-similar process X with stationary increments and 0 < H < 1, |Xt − Xt0 | lim sup = ∞, at any fixed t0 . t − t0 t↓t0 That is, sample paths of H-self-similar processes are nowhere differentiable with probability 1. Haijun Li
An Introduction to Stochastic Calculus
Week 2
5 / 16
Path Properties of Brownian Motion Brownian motion is 0.5-self-similar.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
6 / 16
Path Properties of Brownian Motion Brownian motion is 0.5-self-similar. Its sample paths are nowhere differentiable. That is, any sample path changes its shape in the neighborhood of any time epoch in a completely non-predictable fashion (Wiener, Paley and Zygmund, 1930s).
Haijun Li
An Introduction to Stochastic Calculus
Week 2
6 / 16
Path Properties of Brownian Motion Brownian motion is 0.5-self-similar. Its sample paths are nowhere differentiable. That is, any sample path changes its shape in the neighborhood of any time epoch in a completely non-predictable fashion (Wiener, Paley and Zygmund, 1930s).
Unbounded Variation of Brownian Sample Paths sup τ
n X
|Bti (ω) − Bti−1 (ω)| = ∞, a.s.,
i=1
where the supremum is taken over all possible partitions τ : 0 = t0 < · · · < tn = T of any finite interval [0, T ].
Haijun Li
An Introduction to Stochastic Calculus
Week 2
6 / 16
Path Properties of Brownian Motion Brownian motion is 0.5-self-similar. Its sample paths are nowhere differentiable. That is, any sample path changes its shape in the neighborhood of any time epoch in a completely non-predictable fashion (Wiener, Paley and Zygmund, 1930s).
Unbounded Variation of Brownian Sample Paths sup τ
n X
|Bti (ω) − Bti−1 (ω)| = ∞, a.s.,
i=1
where the supremum is taken over all possible partitions τ : 0 = t0 < · · · < tn = T of any finite interval [0, T ]. The unbounded variation and non-differentiability of Brownian sample paths are major reasons for the failure of classical integration methods, when applied to these paths, and for the introduction of stochastic calculus. Haijun Li
An Introduction to Stochastic Calculus
Week 2
6 / 16
Brownian Bridge
Let B = (Bt , t ∈ [0, ∞)) denote Brownian Motion. The process X := (Bt − tB1 , 0 ≤ t ≤ 1) satisfies that X0 = X1 = 0. This process is called the (standard) Brownian bridge.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
7 / 16
Brownian Bridge
Let B = (Bt , t ∈ [0, ∞)) denote Brownian Motion. The process X := (Bt − tB1 , 0 ≤ t ≤ 1) satisfies that X0 = X1 = 0. This process is called the (standard) Brownian bridge. Since multivariate normal distributions are closed under linear transforms, the finite-dimensional distributions of X are Gaussian.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
7 / 16
Brownian Bridge
Let B = (Bt , t ∈ [0, ∞)) denote Brownian Motion. The process X := (Bt − tB1 , 0 ≤ t ≤ 1) satisfies that X0 = X1 = 0. This process is called the (standard) Brownian bridge. Since multivariate normal distributions are closed under linear transforms, the finite-dimensional distributions of X are Gaussian. The Brownian bridge is characterized by two functions µX (t) = 0 and CX (t, s) = min(t, s) − ts, for all s, t ∈ [0, 1].
Haijun Li
An Introduction to Stochastic Calculus
Week 2
7 / 16
Brownian Bridge
Let B = (Bt , t ∈ [0, ∞)) denote Brownian Motion. The process X := (Bt − tB1 , 0 ≤ t ≤ 1) satisfies that X0 = X1 = 0. This process is called the (standard) Brownian bridge. Since multivariate normal distributions are closed under linear transforms, the finite-dimensional distributions of X are Gaussian. The Brownian bridge is characterized by two functions µX (t) = 0 and CX (t, s) = min(t, s) − ts, for all s, t ∈ [0, 1]. The Brownian bridge appears as the limit process of the normalized empirical distribution function of a sample of iid uniform U(0, 1) random variables.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
7 / 16
Brownian Motion with Drift
Let B = (Bt , t ∈ [0, ∞)) denote Brownian Motion. The process X := (µt + σBt , t ≥ 0), for constant σ > 0 and µ ∈ R, is called Brownian motion with (linear) drift.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
8 / 16
Brownian Motion with Drift
Let B = (Bt , t ∈ [0, ∞)) denote Brownian Motion. The process X := (µt + σBt , t ≥ 0), for constant σ > 0 and µ ∈ R, is called Brownian motion with (linear) drift. X is a Gaussian process with expectation and covariance functions µX (t) = µt, CX (t, s) = σ 2 min(t, s), s, t ≥ 0.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
8 / 16
Geometric Brownian Motion (Black, Scholes and Merton 1973) The process X = (exp(µt + σBt ), t ≥ 0), for constant σ > 0 and µ ∈ R, is called geometric Brownian motion.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
9 / 16
Geometric Brownian Motion (Black, Scholes and Merton 1973) The process X = (exp(µt + σBt ), t ≥ 0), for constant σ > 0 and µ ∈ R, is called geometric Brownian motion. 2 Since EetZ = et /2 for an N(0, 1) random variable Z , it follows from the self-similarity of Brownian motion that µX (t) = eµt EeσBt = eµt Eeσt
Haijun Li
An Introduction to Stochastic Calculus
1/2 B 1
2
= e(µ+0.5σ )t .
Week 2
9 / 16
Geometric Brownian Motion (Black, Scholes and Merton 1973) The process X = (exp(µt + σBt ), t ≥ 0), for constant σ > 0 and µ ∈ R, is called geometric Brownian motion. 2 Since EetZ = et /2 for an N(0, 1) random variable Z , it follows from the self-similarity of Brownian motion that µX (t) = eµt EeσBt = eµt Eeσt
1/2 B 1
2
= e(µ+0.5σ )t .
Since Bt − Bs and Bs are independent for any s ≤ t, and Bt − Bs =d Bt−s , then CX (t, s) = e(µ+0.5σ
Haijun Li
2 )(t+s)
An Introduction to Stochastic Calculus
2
(eσ t − 1).
Week 2
9 / 16
Geometric Brownian Motion (Black, Scholes and Merton 1973) The process X = (exp(µt + σBt ), t ≥ 0), for constant σ > 0 and µ ∈ R, is called geometric Brownian motion. 2 Since EetZ = et /2 for an N(0, 1) random variable Z , it follows from the self-similarity of Brownian motion that µX (t) = eµt EeσBt = eµt Eeσt
1/2 B 1
2
= e(µ+0.5σ )t .
Since Bt − Bs and Bs are independent for any s ≤ t, and Bt − Bs =d Bt−s , then CX (t, s) = e(µ+0.5σ 2
2 )(t+s)
2
(eσ t − 1).
2
In particular, σX2 (t) = e(2µ+σ )t (eσ t − 1). Haijun Li
An Introduction to Stochastic Calculus
Week 2
9 / 16
Central Limit Theorem Consider a sequence {Y1 , Y2 , . . . , } of iid non-degenerate random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
10 / 16
Central Limit Theorem Consider a sequence {Y1 , Y2 , . . . , } of iid non-degenerate random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. P Define the partial sums: R0 := 0, Rn := ni=1 Yi , n ≥ 1.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
10 / 16
Central Limit Theorem Consider a sequence {Y1 , Y2 , . . . , } of iid non-degenerate random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. P Define the partial sums: R0 := 0, Rn := ni=1 Yi , n ≥ 1.
Central Limit Theorem (CLT) If Y1 has finite variance, then the sequence (Rn ) obeys the CLT via the following uniform convergence: Rn − ERn → 0, as n → ∞, sup P ≤ x − Φ(x) [var(Rn )]1/2 x∈R where Φ(x) denotes the distribution of the standard normal distribution.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
10 / 16
Central Limit Theorem Consider a sequence {Y1 , Y2 , . . . , } of iid non-degenerate random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. P Define the partial sums: R0 := 0, Rn := ni=1 Yi , n ≥ 1.
Central Limit Theorem (CLT) If Y1 has finite variance, then the sequence (Rn ) obeys the CLT via the following uniform convergence: Rn − ERn → 0, as n → ∞, sup P ≤ x − Φ(x) [var(Rn )]1/2 x∈R where Φ(x) denotes the distribution of the standard normal distribution. That is, for large sample size n, the distribution of [Rn − ERn ]/[var(Rn )]1/2 is approximately standard normal. Haijun Li
An Introduction to Stochastic Calculus
Week 2
10 / 16
Functional Approximation Let (Yi ) be a sequence of iid random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. Consider the process Sn = (Sn (t), t ∈ [0, 1]) with continuous sample paths on [0, 1], 2 −1/2 (σY n) (Ri − µY i), if t = i/n, i = 0, . . . , n Sn (t) = linearly interpolated, otherwise.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
11 / 16
Functional Approximation Let (Yi ) be a sequence of iid random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. Consider the process Sn = (Sn (t), t ∈ [0, 1]) with continuous sample paths on [0, 1], 2 −1/2 (σY n) (Ri − µY i), if t = i/n, i = 0, . . . , n Sn (t) = linearly interpolated, otherwise. Example: If Yi s are iid N(0, 1), consider P the restriction of the process Sn on the points i/n: Sn (i/n) = n−1/2 ik =1 Yk , i = 0, . . . , n.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
11 / 16
Functional Approximation Let (Yi ) be a sequence of iid random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. Consider the process Sn = (Sn (t), t ∈ [0, 1]) with continuous sample paths on [0, 1], 2 −1/2 (σY n) (Ri − µY i), if t = i/n, i = 0, . . . , n Sn (t) = linearly interpolated, otherwise. Example: If Yi s are iid N(0, 1), consider P the restriction of the process Sn on the points i/n: Sn (i/n) = n−1/2 ik =1 Yk , i = 0, . . . , n. Sn (0) = 0.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
11 / 16
Functional Approximation Let (Yi ) be a sequence of iid random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. Consider the process Sn = (Sn (t), t ∈ [0, 1]) with continuous sample paths on [0, 1], 2 −1/2 (σY n) (Ri − µY i), if t = i/n, i = 0, . . . , n Sn (t) = linearly interpolated, otherwise. Example: If Yi s are iid N(0, 1), consider P the restriction of the process Sn on the points i/n: Sn (i/n) = n−1/2 ik =1 Yk , i = 0, . . . , n. Sn (0) = 0. Sn has independent increments: for any 0 ≤ i1 ≤ · · · ≤ im ≤ n, Sn (i2 /n) − Sn (i1 /n), . . . , Sn (im /n) − Sn (im−1 /n) are independent.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
11 / 16
Functional Approximation Let (Yi ) be a sequence of iid random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. Consider the process Sn = (Sn (t), t ∈ [0, 1]) with continuous sample paths on [0, 1], 2 −1/2 (σY n) (Ri − µY i), if t = i/n, i = 0, . . . , n Sn (t) = linearly interpolated, otherwise. Example: If Yi s are iid N(0, 1), consider P the restriction of the process Sn on the points i/n: Sn (i/n) = n−1/2 ik =1 Yk , i = 0, . . . , n. Sn (0) = 0. Sn has independent increments: for any 0 ≤ i1 ≤ · · · ≤ im ≤ n, Sn (i2 /n) − Sn (i1 /n), . . . , Sn (im /n) − Sn (im−1 /n) are independent. For any 0 ≤ i ≤ n, Sn (i/n) has a normal N(0, i/n) distribution.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
11 / 16
Functional Approximation Let (Yi ) be a sequence of iid random variables with mean µY = EY1 and variance σY2 = var(Y1 ) > 0. Consider the process Sn = (Sn (t), t ∈ [0, 1]) with continuous sample paths on [0, 1], 2 −1/2 (σY n) (Ri − µY i), if t = i/n, i = 0, . . . , n Sn (t) = linearly interpolated, otherwise. Example: If Yi s are iid N(0, 1), consider P the restriction of the process Sn on the points i/n: Sn (i/n) = n−1/2 ik =1 Yk , i = 0, . . . , n. Sn (0) = 0. Sn has independent increments: for any 0 ≤ i1 ≤ · · · ≤ im ≤ n, Sn (i2 /n) − Sn (i1 /n), . . . , Sn (im /n) − Sn (im−1 /n) are independent. For any 0 ≤ i ≤ n, Sn (i/n) has a normal N(0, i/n) distribution. Sn and Brownian motion B on [0, 1], when restricted to the points i/n, have very much the same properties. Haijun Li
An Introduction to Stochastic Calculus
Week 2
11 / 16
Functional Central Limit Theorem Let C[0, 1] denote the space of all continuous functions defined on [0, 1]. With the maximum norm, C[0, 1] is a complete separable space.
Donsker’s Theorem If Y1 has finite variance, then the process Sn obeys the functional CLT: Eφ(Sn (·)) → Eφ(B), as n → ∞, for all bounded continuous functionals φ : C[0, 1] → R, where B(t) is the Brownian motion on [0, 1].
Haijun Li
An Introduction to Stochastic Calculus
Week 2
12 / 16
Functional Central Limit Theorem Let C[0, 1] denote the space of all continuous functions defined on [0, 1]. With the maximum norm, C[0, 1] is a complete separable space.
Donsker’s Theorem If Y1 has finite variance, then the process Sn obeys the functional CLT: Eφ(Sn (·)) → Eφ(B), as n → ∞, for all bounded continuous functionals φ : C[0, 1] → R, where B(t) is the Brownian motion on [0, 1]. The finite-dimensional distributions of Sn converge to the corresponding finite-dimensional distributions of B: As n → ∞, P(Sn (t1 ) ≤ x1 , . . . , Sn (tm ) ≤ xm ) → P(Bt1 ≤ x1 , . . . , Btm ≤ xm ), for all possible ti ∈ [0, 1] xi ∈ R.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
12 / 16
Functional Central Limit Theorem Let C[0, 1] denote the space of all continuous functions defined on [0, 1]. With the maximum norm, C[0, 1] is a complete separable space.
Donsker’s Theorem If Y1 has finite variance, then the process Sn obeys the functional CLT: Eφ(Sn (·)) → Eφ(B), as n → ∞, for all bounded continuous functionals φ : C[0, 1] → R, where B(t) is the Brownian motion on [0, 1]. The finite-dimensional distributions of Sn converge to the corresponding finite-dimensional distributions of B: As n → ∞, P(Sn (t1 ) ≤ x1 , . . . , Sn (tm ) ≤ xm ) → P(Bt1 ≤ x1 , . . . , Btm ≤ xm ), for all possible ti ∈ [0, 1] xi ∈ R. The max functional max0≤i≤n Sn (ti ) converges in distribution to max0≤t≤1 Bt as n → ∞. Haijun Li
An Introduction to Stochastic Calculus
Week 2
12 / 16
Functional CLT for Jump Processes Stochastic processes are infinite-dimensional objects, and therefore unexpected events may happen. For example, the sample paths of the converging processes may fluctuate very wildly with increasing n. In order to avoid such irregular behavior, a so-called tightness or stochastic compactness condition must be satisfied.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
13 / 16
Functional CLT for Jump Processes Stochastic processes are infinite-dimensional objects, and therefore unexpected events may happen. For example, the sample paths of the converging processes may fluctuate very wildly with increasing n. In order to avoid such irregular behavior, a so-called tightness or stochastic compactness condition must be satisfied. The functional CLT remains valid for the process ¯ n = (S ¯ n (t), t ∈ [0, 1]), where S ¯ n (t) = (σ 2 n)−1/2 (R[nt] − µY [nt]) S Y
and [nt] denotes the integer part of the real number nt.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
13 / 16
Functional CLT for Jump Processes Stochastic processes are infinite-dimensional objects, and therefore unexpected events may happen. For example, the sample paths of the converging processes may fluctuate very wildly with increasing n. In order to avoid such irregular behavior, a so-called tightness or stochastic compactness condition must be satisfied. The functional CLT remains valid for the process ¯ n = (S ¯ n (t), t ∈ [0, 1]), where S ¯ n (t) = (σ 2 n)−1/2 (R[nt] − µY [nt]) S Y
and [nt] denotes the integer part of the real number nt. ¯ n is constant on the intervals In contrast to Sn , the process S [(i − 1)/n, i/n) and has jumps at the points i/n.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
13 / 16
Functional CLT for Jump Processes Stochastic processes are infinite-dimensional objects, and therefore unexpected events may happen. For example, the sample paths of the converging processes may fluctuate very wildly with increasing n. In order to avoid such irregular behavior, a so-called tightness or stochastic compactness condition must be satisfied. The functional CLT remains valid for the process ¯ n = (S ¯ n (t), t ∈ [0, 1]), where S ¯ n (t) = (σ 2 n)−1/2 (R[nt] − µY [nt]) S Y
and [nt] denotes the integer part of the real number nt. ¯ n is constant on the intervals In contrast to Sn , the process S [(i − 1)/n, i/n) and has jumps at the points i/n. ¯ n coincide at the points i/n, and the differences between Sn and S these two processes are asymptotically negligible: the ¯ n arbitrarily small for large normalization n1/2 makes the jumps of S n. Haijun Li
An Introduction to Stochastic Calculus
Week 2
13 / 16
Simulating a Brownian Sample Path
¯ n , for sufficiently large n, Plot the paths of the processes Sn , or S and get a reasonable approximation to Brownian sample paths.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
14 / 16
Simulating a Brownian Sample Path
¯ n , for sufficiently large n, Plot the paths of the processes Sn , or S and get a reasonable approximation to Brownian sample paths. Since Brownian motion appears as a distributional limit, completely different graphs for different values of n may appear for the same sequence of realizations Yi (ω)s.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
14 / 16
Simulating a Brownian Sample Path
¯ n , for sufficiently large n, Plot the paths of the processes Sn , or S and get a reasonable approximation to Brownian sample paths. Since Brownian motion appears as a distributional limit, completely different graphs for different values of n may appear for the same sequence of realizations Yi (ω)s.
Simulating a Brownian Sample Path on [0, T ] ¯ n on [0, 1], then scale the time interval by Simulate one path of Sn , or S the factor T and the sample path by the factor T 1/2 .
Haijun Li
An Introduction to Stochastic Calculus
Week 2
14 / 16
Lévy-Ciesielski Representation Since Brownian sample paths are continuous functions, we can try to expand them in a series.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
15 / 16
Lévy-Ciesielski Representation Since Brownian sample paths are continuous functions, we can try to expand them in a series. However, the paths are random functions: for different ω, we obtain different path functions. This means that the coefficients of this series are random variables.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
15 / 16
Lévy-Ciesielski Representation Since Brownian sample paths are continuous functions, we can try to expand them in a series. However, the paths are random functions: for different ω, we obtain different path functions. This means that the coefficients of this series are random variables. Since the process is Gaussian, the coefficients must be Gaussian as well.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
15 / 16
Lévy-Ciesielski Representation Since Brownian sample paths are continuous functions, we can try to expand them in a series. However, the paths are random functions: for different ω, we obtain different path functions. This means that the coefficients of this series are random variables. Since the process is Gaussian, the coefficients must be Gaussian as well.
Lévy-Ciesielski Expansion Brownian motion on [0, 1] can be represented in the form Bt (ω) =
∞ X n=1
Z
t
φn (x)dx, t ∈ [0, 1],
Zn (ω) 0
where Zn s are iid N(0, 1) random variables and (φn ) is a complete orthonormal function system on [0, 1]. Haijun Li
An Introduction to Stochastic Calculus
Week 2
15 / 16
Paley-Wiener Representation There are infinitely many possible representations of Brownian motion.
Haijun Li
An Introduction to Stochastic Calculus
Week 2
16 / 16
Paley-Wiener Representation There are infinitely many possible representations of Brownian motion. Let (Zn , n ≥ 0) be a sequence of iid N(0,1) random variables, then Bt (ω) = Z0 (ω)
∞ sin(nt/2) 2 X t Zn (ω) + , t ∈ [0, 2π]. 1/2 1/2 n (2π) π n=1
Haijun Li
An Introduction to Stochastic Calculus
Week 2
16 / 16
Paley-Wiener Representation There are infinitely many possible representations of Brownian motion. Let (Zn , n ≥ 0) be a sequence of iid N(0,1) random variables, then Bt (ω) = Z0 (ω)
∞ sin(nt/2) 2 X t Zn (ω) + , t ∈ [0, 2π]. 1/2 1/2 n (2π) π n=1
This series converges for every t, and uniformly for t ∈ [0, 2π].
Haijun Li
An Introduction to Stochastic Calculus
Week 2
16 / 16
Paley-Wiener Representation There are infinitely many possible representations of Brownian motion. Let (Zn , n ≥ 0) be a sequence of iid N(0,1) random variables, then Bt (ω) = Z0 (ω)
∞ sin(nt/2) 2 X t Zn (ω) + , t ∈ [0, 2π]. 1/2 1/2 n (2π) π n=1
This series converges for every t, and uniformly for t ∈ [0, 2π].
Simulating a Brownian Path via Paley-Wiener Expansion Calculate Z0 (ω)
M tj sin(ntj /2) 2 X 2πj + Zn (ω) , tj = , for 0 ≤ j ≤ N. 1/2 1/2 n N (2π) π n=1
Haijun Li
An Introduction to Stochastic Calculus
Week 2
16 / 16
Paley-Wiener Representation There are infinitely many possible representations of Brownian motion. Let (Zn , n ≥ 0) be a sequence of iid N(0,1) random variables, then Bt (ω) = Z0 (ω)
∞ sin(nt/2) 2 X t Zn (ω) + , t ∈ [0, 2π]. 1/2 1/2 n (2π) π n=1
This series converges for every t, and uniformly for t ∈ [0, 2π].
Simulating a Brownian Path via Paley-Wiener Expansion Calculate Z0 (ω)
M tj sin(ntj /2) 2 X 2πj + Zn (ω) , tj = , for 0 ≤ j ≤ N. 1/2 1/2 n N (2π) π n=1
The problem of choosing the “right” values for M and N is similar to the choice of the sample size n in the functional CLT. Haijun Li
An Introduction to Stochastic Calculus
Week 2
16 / 16
