# Exploring Wiener Process Properties

• mafra
In summary, a Wiener Process W(t) is a stochastic process with initial value of 0, almost surely continuous trajectories, independent increases, and normal centered laws of variance for (W(s) - W(t)). The process is markovian and the law of the random variable W(s) + W(s+t) is Gaussian with mean of 0 and variance of 4s+1. When calculating E[W(u)W(u+v)W(u+v+w)], the result is 0. The autocovariance function of the process exp(-t)W(exp(2t)) is exp(-|T|).
mafra

## Homework Statement

A Wiener Process W(t) is a stochastic process with:
W(0) = 0
Trajectories almost surely continuous
Independent increases, that means, for all t1 ≤ t2 ≤ t3 ≤ t4, we have (W(t2) - W(t1)) is independent of (W(t4) - W(t3))
For t ≤ s, (W(s) - W(t)) follows a normal centered law of variance (s-t).

## Homework Equations

a) The process W(t) is markovian?
b) For s,t > 0, what is the law of the random variable W(s) + W(s+t)?
c) For u,v,w > 0, calculate E[W(u)W(u+v)W(u+v+w)]
d) Calculate the autocovariance function of the process exp(-t)W(exp(2t))

## The Attempt at a Solution

a) The process W(t) is markovian?

A process is markovian if:
P[X(tn) ∈ Bn | X(tn−1) ∈ Bn−1, ... , X(t1) ∈ B1] = P(X(tn) ∈ Bn | X(tn−1) ∈ Bn−1)

Where Bi (I believe) is the Borel σ-algebra of the real numbers.

By the data given:
P[W(t2) - W(t1) ∈ B1 | W(t4) - W(t3) ∈ B1] = P[W(t2) - W(t1) ∈ B1
For all t1 ≤ t2 ≤ t3 ≤ t4

I don't know how to follow from now on

b) For s,t > 0, what is the law of the random variable W(s) + W(s+t)?

W(s) is gaussian and W(s+t) is gaussian, so W(s) + W(s+t) is gaussian

Mean[W(s) + W(s + t)] = Mean[W(s)] + Mean[W(s+t)]

Mean[W(s) - W(t)] = 0
Mean[W(s)] = Mean[W(t)] which means that the mean is constant
Mean[W(0)] = 0 → Mean[W(s)] = 0 → Mean[W(s) + W(s+t)] = 0

Var[W(s) - W(t)], t ≤ s = s - t
For t = 0:
Var[W(s) - W(0)] = s - 0
Var[W(s)] = s

σ² = Var[W(s) + W(s+t)]
σ² = E{[W(s) + W(s+t)]*[W(s+T) + W(s+t+T)]}
σ² = E[W(s)W(s+T) + W(s)W(s+t+T) + W(s+t)W(s+T) + W(s+t)W(s+t+T)]
σ² = Var[W(s)] + E[W(s)W(s+t+T)] + E[W(s+t)W(s+T)] + Var[W(s+t)]
σ² = 2s + t + E[W(s)W(s+t+T)] + E[W(s+t)W(s+T)]

W(s+t+T) = [W(s+t+T) - W(s+T)] + W(s+t)
E[W(s)W(s+t+T)]
E{[W(s+t+T) - W(s+T)]*W(s)} + E[W(s+t)W(s)]
t1=0, t2=s, t3=s+T, t4=s+t+T, t1 ≤ t2 ≤ t3 ≤ t4 (Not sure of what I'm doing here, since T could be negative)
E[W(s+t+T) - W(s+T)]*E[W(s)] + s
s

σ² = 3s + t + E[W(s+t)W(s+T)]

I tried some similar substitutions but that doesn't seem to work in this last expectation

The given answer is σ² = 4s + 1

c) For u,v,w > 0, calculate E[W(u)W(u+v)W(u+v+w)]

E[W(u)W(u+v)W(u+v+w)]
W(u) = w1
W(u+v) = w2
W(u+v+w) = w3
∫ ∫ ∫ w1*w2*w3*p(w1,w2,w3) dw3 dw2 dw1

p(w1,w2,w3) = p(w3|w2,w1)*p(w2|w1)*p(w1)
Markovian process, w2 > w1 → p(w3|w2,w1) = p(w3|w2)

∫ ∫ ∫ w1*w2*w3*p(w3|w2)*p(w2|w1)*p(w1) dw2 dw2 dw1
∫ w1 ∫ w2 ∫ w3*p(w3|w2) dw3 p(w2|w1) dw2 p(w1) dw1

∫ w3*p(w3|w2) dw3 = E[w3|w2=w2]
E[w3-w2] = 0
E[w3] = E[w2]
E[w3|w2=w2] = E[w2|w2=w2] = w2
∫ w3*p(w3|w2) dw3 = w2

∫ w1 ∫ w2² p(w2|w1) dw2 p(w1) dw1

∫ w2² p(w2|w1) dw2 = Rw[w2|w1]
Rw[w2|w1] = var[w2|w1] + E[w2|w1=w1]²
Rw[w2|w1] = v + w1²

∫ w1*(v + w1²) p(w1) dw1
∫ w1*v p(w1) dw1 + ∫ w1³ p(w1) dw1
v*mean[w1] + ∫ w1³ p(w1) dw1

v*mean[w1] = 0
∫ w1³*p(w1) dw1 = 0 because w1³*p(w1) is an odd function

E[W(u)W(u+v)W(u+v+w) = 0

The result matches the given answer, but I don't know if there are any other paths or if I all the resolution is correct

PS: Calculations for conditional variance var[w2|w1]

P[W(u+v) - W(u)] = N(0, v)
P[W(u+v) - W(u)] = P[W(u+v) - W(u) | W(u) - W(0) = a]
P[W(u+v) - W(u) | W(u) = a] = N(0,v)
P[W(u+v) | W(u) = a] = N(a,v) - Variance doesn't change with summed constants like W(u) is here

d) Calculate the autocovariance function of the process exp(-t)W(exp(2t))

Here I could do it just by assuming beforehand that the process was Wide-Sense Stationary and the conditional probabilities were all gaussian

X(t) = exp(-t)W(exp(2t))
As before, the mean rests in zero, so the covariance is E[X(0)E(T)], T≥0
X(0) = x1
X(T) = x2

∫∫ x1*x2*p(x1,x2) dx1 dx2
∫∫ x1*x2*p(x1|x2)*p(x2) dx1 dx2
∫ x1 ∫ x2*p(x2|x1) dx2 p(x1) dx1

p(x) = p(exp(-t)*W(exp(2t)))

p(W(exp(2t))) = N(0,exp(2t))
mean(a*x) = a*mean(x), variance(a*x) = a²*variance(x)
p(x) = p(exp(-t)*W(exp(2t))) = N(0,1)

p(x2|x1) = p( X(t2) | X(t1) = x1 )
t2 = T, t1 = 0:
p( X(T) | X(0) = x2 )
p( exp(-T)*W(exp(2T)) | W(1) = x1 )

p(W(exp(2T)) | W(1) = x1) = N(x1, exp(2T) - 1)
p( exp(-T)*W(exp(2T)) | W(1) = x1 ) = N(x1*exp(-T), 1 - exp(-2T))

p(x1) = N(0,1)
p(x2|x1) = N(x1*exp(-T), 1 - exp(-2T))

∫ x1 ∫ x2*p(x2|x1) dx2 p(x1) dx1

∫ x2*p(x2|x1) dx2 = E[x2|x1] = x1*exp(-T)

∫ x1²*exp(-T)*p(x1) dx1
exp(-T) ∫ x1²*p(x1) dx1 = exp(-T)*Var(x1) = exp(-T)

The answer is exp(-|T|), but again, I don't know how to show that the covariance is symmetrical

Overall, I'm not sure about some of the steps and if I'm using the data correctly, but I tried my best to solve the problems. I would appreciate any feedback or corrections on my approach.

## 1. What is a Wiener process?

A Wiener process, also known as a Brownian motion, is a continuous stochastic process that is used to model random movements of particles in a fluid or gas. It was developed by mathematician Norbert Wiener in the early 20th century and has many applications in physics, finance, and biology.

## 2. How is a Wiener process different from a random walk?

A Wiener process is a continuous-time stochastic process, whereas a random walk is a discrete-time process. This means that a Wiener process is defined at every point in time, while a random walk is only defined at discrete time intervals. Additionally, a Wiener process has continuous paths, while a random walk has discontinuous paths.

## 3. What are the main properties of a Wiener process?

A Wiener process has several key properties, including:
- It has independent and identically distributed increments.
- It has continuous paths.
- It is Markovian, meaning future movements are only dependent on the current position.
- It has normally distributed increments.
- It has zero mean and variance equal to the time interval.

## 4. How is a Wiener process used in finance?

In finance, a Wiener process is often used to model the random movements of asset prices. It is a key component in the Black-Scholes option pricing model and is also used in the analysis of stock prices and interest rates. It allows for the calculation of probabilities and expected values, which are important in making investment decisions.

## 5. Are there any limitations to using a Wiener process?

While a Wiener process has many applications, it also has some limitations. It assumes that the increments are normally distributed, which may not always be the case in real-world situations. Additionally, it is a continuous process, so it may not accurately model discrete events. It is important to carefully consider these limitations when using a Wiener process in any analysis or modeling.

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