Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

(1)

LAPPEENRANTA UNIVERSITY OF TECHNOLOGY School of Engineering Science

Faculty of Science and Technology

Double Degree Master's Programme in Computational Engineering and Mathematics

Ramona Maraia

STOCHASTIC PARTICLE MODELS:

MEAN REVERSION AND BURGERS DYNAMICS.

AN APPLICATION TO COMMODITY MARKETS.

Examiners: Ass. Prof. Tuomo Kauranne

Post. Doc Matylda Jablonska-Sabuka Doc. Daniela Morale

(2)

ABSTRACT

Lappeenranta University of Technology School of Engineering Science

Faculty of Science and Technology

Double Degree Master's Programme in Computational Engineering and Mathematics

Ramona Maraia

STOCHASTIC PARTICLE MODELS:

MEAN REVERSION AND BURGERS DYNAMICS.

AN APPLICATION TO COMMODITY MARKETS.

Master's thesis

2016

127 pages, 43 gures, 40 tables, 2 appendices

Examiners: Ass. Prof. Tuomo Kauranne

Post. Doc. Matylda Jablonska-Sabuka Doc. Daniela Morale

Keywords: stochastic dierential equation, stochastic interacting particle models, Burg- ers equation, commodity markets and computational market dynamics.

(3)

3 The aim of this study is to propose a stochastic model for commodity markets linked with the Burgers equation from uid dynamics. We construct a stochastic particles method for commodity markets, in which particles represent market participants. A discontinuity in the model is included through an interacting kernel equal to the Heav- iside function and its link with the Burgers equation is given. The Burgers equation and the connection of this model with stochastic dierential equations are also studied.

Further, based on the law of large numbers, we prove the convergence, for large N, of a system of stochastic dierential equations describing the evolution of the prices of N traders to a deterministic partial dierential equation of Burgers type. Numerical experiments highlight the success of the new proposal in modeling some commodity markets, and this is conrmed by the ability of the model to reproduce price spikes when their eects occur in a suciently long period of time.

(4)

Acknowledgements

I would like to express my gratitude to my supervisor Ass.Prof. Tuomo Kauranne, who gave me the opportunity to conduct this study, he oered his continuous advice and encouragement throughout the course of this thesis.

I would like to thank my Co-supervisor Post. Doc. Matylda Jablonska-Sabuka for her willingness, she always found time for discussion and supported me during the entire period of study in Finland.

I would like to express my deepest gratitude to my Co-supervisor from University of Milan, Doc. Daniela Morale for her excellent guidance, patience and knowledge.

I would like to acknowledge the Lappeenranta University of Technology, the Depart- ment of Technomathematics, for the hospitality and great atmosphere, in particular Prof. Heikki Haario, Ass. Prof. Matti Heilio and Post. Doc. Virpi Juntilla.

I would like to acknowledge the University of Milan, the Department of Mathematics, in particular the Ass. Prof. Alessandra Micheletti and the Ass. Prof. Paola Causin for the opportunity and support, before and during my foreign study.

I would like to acknowledge the European Consortium for Mathematics in Industry (ECMI) for the great experience they provide me.

Particular thank goes to my colleagues and all my friends that I met during the whole period of study in Italy and in Finland, in particular Nino and Roberta that made me feel at home, Dipal for her friendship and for her precious suggestions given during our long coee breaks, Polish guys, the guys from Czech Republic and the Russian guys for the relaxing time spent together.

I would like to thank to my parents, mom Sonia and dad Giuseppe, who believing in me and giving me nancial support, have made this thesis possible.

Specially thank to my young brother, who with his passion for literary studies and with his continuous request of help in math remembered me the wonderfulness of the mathematical world.

I would like to express all my gratitude to the two greatest women of my life, my grandmothers, Ennia and Rosa, who with their love and advice encouraged me to do not give up.

I would like to thank all my family, uncles and cousins for their special support.

The deepest thank is for my decˇko Sebastian, who has been with me during all this Master study period, he always rooted for me and with his happiness helped me to enjoy beautiful moments and to overcome the darkest ones.

Lappeenranta, March 23, 2016.

Ramona Maraia

(5)

CONTENTS 5

1 Stochastic models for price evolution

1.1 Introduction

The purpose of this thesis is to build a model for commodity markets and to study the connection of these markets with the Burgers equation from uid dynamics, focusing only on the inuence that traders' behaviour has on price formation.

Traders are individuals who need a certain commodity at a future date T. Commodity can be either bought in the spot market today and stored, when storage is possible, or bought with a futures contract, which allows buying a particular commodity at a xed price in the future.

Because of future contracts, the market needs to estimate the price of commodities at future dates. The future price is assumed to be equal to the spot price plus interest and storage costs until the contract expiry date.

Thus

F₀ =S₀+I+C

where F₀ and S₀ are future price and spot price at time t = 0, respectively, I is the interest and C the storage cost, [1].

The most dicult part is to know the spot price evolution, which is usually dened as the point at which demand of a particular commodity meets its supply, but demand and supply have to be estimated. They depend upon several factors, that are specic features of each commodity, such as availability in nature, product properties, delivery costs and conditions, or upon other factors, such as governmental issues which act to increase national prot, freight cost which may be linked to the cost of other commodities such as fuel and so on. In addition, some commodity prices such as prices of agricultural products are subject to weather conditions on which supply availability depends, higher supply leading to lower price and vice versa.

Moreover there is a wide dierence between a storable commodity and a non-storable one. When the commodity is storable, demand can be met out from the current

(8)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 8 production, and higher demand of a storable commodity can be satised by stored supply. In case of higher supply, surplus of commodity can be stored. The situation is dierent for a non-storable commodity which in case of higher supply leads to loss of money.

Both demand and supply instability and the heavy dependence upon several factors make commodity prices historically highly volatile, and their prediction is still subject of ongoing research.

The origins of option pricing models are dated back to Bachelier, who was the rst one to propose using Brownian motion for modelling the dynamics of the stock prices, but this model allows negative values and option prices that exceeded the price of the underlying asset. Therefore Osborne in [2] rened Bachelier's model by using stochastic exponential of the Brownian Motion. After that Samuelson studied the option pricing using geometric Brownian motion in [3]. A breakthrough is represented by the Black and Scholes' model (B&S) given in [4], which is based on the idea that the option price is explicitly connected to a hedging strategy which depends on the volatility of the stock price as well as other observable quantities. The dynamics of spot price S_t can be described by the following Geometric Brownian motion

dS_t=rS_tdt+σS_tdW_t (1)

the solution of which is

S_t=S₀e^(r−^σ

2

2 )t+σWt (2)

where r is the drift, σ the volatility of the spot price and the random component is represented by Wiener process{W_t}_t. Trajectories of a stock price that follows an SDE of type (1) are simulated in Fig. 1, with initial valueS₀ = 4.5.

(9)

Figure 1: Paths of the exact solution of the Geometric Brownian motion (1) with r= 0.3, σ = 0.4 and initial value S₀ = 4.5

This model highlighted the stochastic behaviour of commodity prices, but there are some critical assumptions in the Black and Scholes model, such as that trading takes place continuously in time and that the dynamics of the stock has a continuous sample path with probability one, whereas the market, especially the commodity market, is full of jumps. There is enough evidence in the history of nancial markets, which shows that the presence of jumps is due to new important information about the stock.

Robert C. Merton in [5] and [6] added to the standard Geometric Brownian motion the impact of such important new information on the stock price per unit of time. Merton assumed that the total change in the stock price is the sum of two dierent types of variations. The rst change is the normal vibration in price, the impact of which per unit time on the stock price is to produce a marginal change in the price. This component is modelled by a standard Geometric Brownian motion with a constant variance, thus through a continuous sample path.

The latter variation is the abnormal vibration in price, which is due to the arrival of important new information about the stock. Such kind of variations are expected to be active at the time when information arrives but stay inactive otherwise.

It follows that the eect of abnormal vibration occurs only at discrete points in time, and therefore it is modelled by a Poisson process. The resulting model is an exponen-

(10)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 10 tial Le`vy model of the formS_t=S₀e^L^t where the stock price process{S_t; 0≤t≤T}is modelled as an exponential of a Le`vy process{L_t; 0≤t ≤T}. The L`evy processL_t is dened as a Black and Sholes process with the continuous diusion process represented by the drift plus a discontinuous jump process described by a Poisson process.

Hence the stochastic dierential equation of the stock price processS_tis of the following type

dS_t

S_t = (α−λk)dt+σdW_t+ (y_t−1)dN_t (3) with α the instantaneous expected return on the asset, σ the instantaneous volatility of the asset,k =E(y_t−1), where the absolute price jump sizey_t−1is a non−negative random variable drawn from log−normal distribution, i.e. ln(y_i)∼ N(η, δ²) i.i.d, W_t a standard Brownian motion process,N_t is a Poisson process with intensityλ and independent from W_t. The solution of the equation (3) is

S_t=S₀e^(r−λk−^σ

2

2 )t+σWtY(n) (4)

where

Y(n) =







1, if n=0

Qn

i=1yi, otherwise

with n a Poisson distributed random variables with parameter λt. Fig.2 presents trajectories of the solution of the Merton model, keeping constantr = 0.3and σ = 0.4 in order to show the evolution of stock price behaviour when a Poisson component with intensityλ= 2 and ln(y_i)∼ N(0.4,0.04) is added to a Standard Geometric Brownian motion. Forλ= 0 the Poisson component is zero, thus the Merton model becomes the Geometric Brownian motion.

(11)

(a)λ= 2 (b)λ= 0

Figure 2: Paths of the exact solution of the SDE (3), in 2a with a Poisson noise and in 2b without it.

The main issues with these models are estimation of asset price volatility and that they do not take into account signicant features of commodity markets such as convenience yield, seasonality and mean reversion patterns, which play a very important role in nancial markets.

Starting from the evidence that stock prices usually oscillate around an average level, mean reversion processes, introduced in Section 1.2.2, are used to simulate nancial data, which means that the eect of reverting to the average level is included in the Geometric Brownian motion. The basic assumption is that the stock price S_t follows the stochastic dierential equation

dS_t

S_t =γ(µ−lnS)Sdt+σSdW_t (5)

where γ, µ∈R₊ are the speed and the level of the reversion, σ ∈R₊ is the volatility, and the randomness is given by the Wiener processW_t. Clearly the one−factor model depends only on the history of the stock price, but it is the result of the interactions of many factors such as the convenience yield and the interest rate. Therefore authors like R. Gibson and E. Schwartz implemented the one−factor model as a two and then a the three−factor model, see [7], [8]. The spot priceSt, the convenience yield factor δt and the interest rate rt are modelled by separate stochastic processes, possibly correlated, and the assumption is that these factors follow the joint stochastic process:

(12)

dS_t = (r_t−δ_t)S_tdt+σ₁S_tdW_t,1 (6) dδ_t = (α−δ_t)S_tdt+σ₂S_tdW_2,t (7) dr_t = (β−r_t)S_tdt+σ₃S_tdW_3,t (8) where α, β, σ_1,2,3 ∈ R₊, and dW_1,tdW_2,t = ρ₁dt, dW_1,tdW_3,t = ρ₂dt, dW_2,tdW_3,t = ρ₃dt with ρ₁, ρ₂, ρ₃ the correlation coecients. The two-factor model does not include the interest rate. One of the main diculties in the application of these commodity prices models is that the factors or state variables are not directly observable.

These models are treated also in papers [9], [10], [11], [12], [13].

An alternative way for modelling commodity prices is represented by interacting particle system (IPS) models, which are used for phenomena involving a large number of interrelated components. These models are applied in many elds such as statistical physics, biology and economics. Interacting particle models are continuous time Feller processes{η_t}_tdened on the compact state space of binary congurationsX ={0,1}^S on S, a countable set. A simple process is that in which the coordinates η_t(x) evolve according to two independent Markov processes with transitions from 0 to 1 and from 1 to 0 at rate 1. The process {ηt}tis described by specifying the rates at which transitions occur. In the case with niteS, saying that the transitionη →ς for η6=ς occurs at ratec means that

P^η(η_t =ς) = ct+o(t) as t↓0.

The innitesimal generator Ω of {η_t}_t provides the relationship between the process and its transition rates, where Ω is dened on a dense subset of C({0,1}^S), and is determined by its values on the cylinder functions, i.e.

Ωf(η) =X

ς

c(η, ς)[f(ς)−f(η)]

withc(η, ς)the rate at which transitions occur fromηtoς. Some of the most important models in this area are Voter models which are the simplest ones [14], Contact Processes (CP) in which particles can be infected and can infect their neighbouring particles [15], the Exclusion Interaction Process (EP), in which particles only jump to empty sites and the Magnetic Model which reproduces the magnetic spin. All of them are treated in [14], [16], [17], [18].

(13)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 13 The starting point of this study is given by the literature upon stochastic interacting particle models [see Morale et al. 2005 (An interacting particle system modelling aggregation behaviour: from individuals to populations), Capasso−Morale 2008 (Rescaling Stochastic Processes: Asymptotics), Capasso−Morale 2004 (Ant Colonies: a Nature In- spired Paradigm for the Mathematical Modelling of Self-Organizing Systems)]; in which particular attention is paid to the mathematical modelling of the social behaviour of interacting individuals in a biological population (animals, cells, etc.). These systems led to the phenomena of self−organization, which exhibit interesting spatial patterns.

Often observational and empirical evidence show that animal groups move across the landscape quite cohesively.

The animal population dynamic model, developed in [19], describes the dynamics of animal populations, where each individual is treated as a discrete particle and the movement of each particle is described thinking that each of them is subject to some rules. These rules can be attributed the instinctive behaviours, which are represented by a random component, and external forces, as for example, forces of interaction between individuals.

Let(X_N^k(t))_t∈R⁺ be a stochastic process describing the state ofN particles. It is dened on a suitable probability space (Ω,F, P) and valued in (R^d,BR^d), where BR^d is the Borel σ−algebra generated by intervals in R^d. The variation in time of the location of the k−th individual in the group at time t > 0, X_N^k(t) ∈ R^d, for k = 1, . . . , N is described by a stochastic dierential equation of type (9).

LetX_t = (X_t¹, . . . , X_t^N), the whole population is represented by a system ofN stochastic dierential equations, each of them of the type (9).

dX_N^k(t) = [f_N^k(t) +h^k_N(X_t(t), t)]dt+σdW^k(t), k = 1, . . . , N (9) where the functionh^k_N :R^{N d}×R⁺ describes the interaction that thek−th particle has with the other individuals and the function f_N^k : R⁺ → R reects possible dynamics of the k−particle, which may depend on the time t or the state of the particle itself.

The model of type (9) does not include the eect that the environment has on the population dynamics. If the individuals are on a regular surface then they can take big distances from their neighbours without losing sight of the position of the others.

On the other hand, on an irregular surface the distances between individuals and

(14)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 14 their neighbours will stay closer. Therefore the animal population dynamic model that was implemented in [20] included the environment inuence, and the aggregation and repulsive forces. The need to add these components is based on the assumption that particles can perceive other particles only within a limited range, so they do not have information about the spatial distribution of all the individuals, which leads to aggregation. On the other hand, there is a repulsive eect when particles are too close to each other. The entire population is described through a system of N stochastic dierential equations where each of them can be written as (10).

dX_N^k(t) = [γ∇U(X_N^k(t)) +θ(∇(G−VN)∗X(t))(X_N^k(t))]dt+σdW^k(t), (10) for k = 1, . . . , N and σ ∈ R and γ, θ, σ ∈ R⁺. Where ∇U(X_N^k(t)), with U ∈ C_b²(R^d,R⁺), is the potential of the environmental inuence on k−th particle, and G and V stand for aggregation and repulsive forces, respectively.

The price formation of stocks or any commodity is usually a process involving a group of traders, so it can be seen as a large population. One can rene the idea to see a group of traders as a large population of agents, featuring their dynamic upon the distances between their price, instead of upon their density or their distances from the others, as in the animals' case. In that way we are allowed to consider a model as in (9).

This is the idea that drove M.Jablonska and T.Kauranne, in their papers [21] and [22],[23], to think at a group of traders as a population of individuals that interact on three dierent scales through a system of stochastic dierential equations.

These scales, which are also included in the model proposed in [19], are the macroscale, that is referred to the direction of the entire population, the microscale, that is related to the motion of each individual separately, and the mesoscale which allows interaction with the closest neighbourhood.

M.Jablonska and T.Kauranne in their papers implemented the model (9), as described in Section 1.3.1, adding a global interaction and local interaction. The global interaction component reects the eect caused by a large subgroup on the whole population, when it has a dierent behaviour with respect to the total population mean. In [21], this term, also called momentum component, is supposed to have a link with the Burgers equation. The local interaction component reects the assumption that each individual in the population can perceive its neighbours to a limited extent.

(15)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 15 The model has been tested on several dierent commodity markets, including Silver and New Zealand electricity markets. The results show that on the rst data set, Silver prices, the model can be considered acceptable with the root−mean square error approximately near to 0.2. The diculties of this model arise when it is used to simulate the electricity prices, due to their high volatility.

Interested to the connection of the global interaction component with the Burgers equation, in this work we develop a model for commodity markets, which is still based on the analogy of animal population spirits with traders behaviour, going to investigate the impact on the commodity markets of a model in which a global interacting kernel is realized by a discontinuity function linked with the Burgers equation. For the development of a new model an individual based model, useful for the derivation of the correct limit equation when the number of traders are supposed to increase, is used and its convergence to a Burgers dynamic is studied.

This work is organized as follows: starting by the link of global interaction component with the dynamic of Burgers equation, in Chapter 2 we give a brief description of Burgers equation and investigate on its connection with stochastic dierential equations, analysing in depth a system of stochastic dierential equations with the Heaviside interacting Kernel. In Chapter 3 a new model for traders dynamic is introduced going to exploit the interaction kernel presented in Chapter2, and the results of its application on two dierent commodity market are discussed. Chapter 4 provides discussion on improvements and failures of the model developed in this study and its futures perspectives.

1.2 Mean Reverting processes

In nance, mean reversion is the assumption that a stock's price will tend to move to the average price over time. Average price can be computed from the historical data, or from other available information.

The principle on which the mean reversion theory is based is that if the current market price is less than the average price, then the stock is considered attractive for purchase, with the expectation that the price will rise. If the current market price is above the average price, then the market price is expected to fall.

(16)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 16 So, the mean reversion models can be dened by the property to always revert to a certain constant or time varying level with limited variance around it. An Ornstein Uhlenbeck process is widely used for modelling a mean reverting process.

1.2.1 Ornstein Uhlenbeck process

Ornstein and Uhlenbeck, in their famous paper [24], studied the velocity of free particles in Brownian motion, moving in a rareed gas and aected by a friction force proportional to pressure.

From Newton's second law of motion the equation of particles' velocity is

mdV(t) = −γV(t)dt+dW(t) (11)

In the equation (11) m is the mass of particles, γ ∈ R is the friction coecient and dW is a Wiener process that represents a random force.

Dividing per m and putting ρ = γ

m, σ= 1

m, and V(t) =X(t) then the equation (11) becomes

dX(t) =−ρX(t)dt+σdW(t) (12)

with ρ and σ elements of R⁺, and (W_t)t≥0 one−dimensional Wiener process.

The equation (12) is called Langevin equation, it is the oldest example of stochastic dierential equation.

Applying the Itoˆ⁰s formula to e^ρtX(t):

d(e^ρtX(t)) =ρe^ρtX(t)dt+e^ρtdX(t)

=ρe^ρtX(t)dt+e^ρt[−ρX(t)dt+σdW(t)]

=e^ρtσdW(t).

Therefore:

e^ρtX(t) = X(0) +σ Z t

0

e^ρsdW(s)

(17)

and the solution has the form

X(t) = e^−ρtx₀+σ Z t

0

e^ρ(s−t)dW(s). (13)

The equation (13) denes the Ornstein−Uhlenbeck process.

Denition 1.1. The Ornstein−Uhlenbeck process (X(t))t∈R is a stochastic process of the form (13), that satises the Langevin stochastic dierential equation (12) with starting point X(0) =x₀.

Proposition 1.1. An Ornstein Uhlenbeck process (X(t))_t∈R⁺ with X(0) = x₀ is a Gaussian process with

E(X(t)) = x₀e^−ρt and Cov(X(t), X(s)) = σ²

2ρ(e^−ρ|t−s|−e^−ρ(t+s)) and one dimensional distribution N(x₀e^−ρt,^σ_2ρ²(1−e^−2ρt)).

Proof. From the expression (13) it follows that it is a linear equation for(X(t))_t∈R⁺ with deterministic coecients and an additive Gaussian forcing. In this case the solution is also Gaussian.

The expected value can be computed as

E(X(t)) =E(e^−ρtx₀ +σ Z t

0

e^ρ(s−t)dW(s))

=E(e^−ρtx₀) +E(σ Z t

0

e^ρ(s−t)dW(s))

=x₀e^−ρt since W(t) is a Wiener process, so

E(σ Z t

0

e^ρ(s−t)dW(s)) = 0, thanks to the Itoˆisometry, Proposition A.6.

The covariance

(18)

Cov(X(t), X(s)) =E[(X(t)−x₀e^−ρt)(X(s)−x₀e^−ρs)]

=E[σ²( Z t

0

e^ρ(u−t)dW(u))(

Z s 0

e^ρ(u−s)dW(u))]

=E[σ²( Z t∧s

0

e^ρ(u−t)e^ρ(u−s)d(u))]

=σ²

2ρ(e^−ρ|t−s|−e^−ρ(t+s)), in particular the variance is

V ar(X(t)) =Cov(X(t), X(t)) = σ²

2ρ(1−e^−2ρt).

Therefore it follows that its one dimensional distribution isN(x₀e^−ρt,^σ_2ρ²(1−e^−2ρt)).

The probability density of an OU process is given by the Gaussian density withµ(t) = E(X(t))and σ²(t) = Cov(X(t), X(t))

p(x, t) = 1

p2πσ²(t)exp{−[x−µ(t)]²

2σ²(t) }. (14)

Proposition 1.2. If the timet → ∞, then Ornstein Uhlenbeck process is a stationary Gaussian process with zero mean.

Proof. From the Proposition 1.1 follows that the OU process is a Gaussian process; we need to prove only that it is stationary.

As the time t→ ∞, the expected value

E[X(t)] = x₀e^−ρt →0, the variance

V ar[X(t)] = σ²

2ρ(1−e^−2ρt)→ σ² 2ρ, and the covariance

Cov(X(t), X(s)) = σ²

2ρ(e^−ρ|t−s|−e^−ρ(t+s))→ σ²

2ρe^−ρ|t−s|. ThusX(t) is a stationary Gaussian process.

(19)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 19 Denition 1.2. A stationary Ornstein Uhlenbeck process is a stationary Gaussian process with zero mean.

A stationary OU process is the exact solution of the SDE (12) if, instead of taking deterministic initial conditions, we suppose that X(0) is a Gaussian random variable N(0, σ²), with covariance

Cov(X(t), X(s)) = σ²

2ρe^−ρ|t−s|, (15)

and stationary distribution

p(x) = 1

√2πσ²e⁻^x

2

2σ2 (16)

where σ² is the variance.

As shown in Figure 3, the average behaviour of Ornstein-Uhlenbeck processes tends to zero as t is large, i.e. E(X(t))−→0 as t−→ ∞.

(20)

Figure 3: Trajectories of an Ornstein Uhlenbeck process tends to mean 0 as t −→ ∞ The blue lines are the OU simulations made using dierent initial values as x₀ =

−6,−3,2,6 and mean reversion speed ρ= 0.8 and volatility σ = 0.2. The green lines are the mean behaviour E[X(t)]of each OU simulation which goes to zero (red line).

1.2.2 Vasicek model

The Vasicek model is widely used in nancial markets; it is a translation of the Ornstein Uhlenbeck process and its SDE has the following form:

dX(t) =−ρ(X(t)−µ)dt+σdW(t) (17) with (W_t)t≥0 a Wiener process described in section A.1 and µ, ρ, σ constants in R⁺. Given Y(t) =µ−X(t), then

dY(t) = −dX(t) =ρ(X(t)−µ)−σdW(t).

(21)

Again applying Itoˆ⁰s formula to e^ρtY(t):

d(e^ρtY(t)) =ρe^ρtY(t)dt+e^ρtdY(t)

=ρe^ρtY(t)dt+e^ρt[−ρY(t)dt−σdW(t)]

=−e^ρtσdW(t).

Therefore:

e^ρtY(t) =Y(0)−σ Z t

0

e^ρsdW(s) and coming back to X(t)

e^ρt(µ−X(t)) =µ−X(0)−σ Z t

0

e^ρsdW(s) we have that

X(t) =µ+e^−ρt(X(0)−µ) +σ Z t

0

e^−ρ(t−s)dW(s)

which is the solution of a stochastic dierential equation of the type (17); with expected value and variance

E(X(t)) =µ+ (X(0)−µ)e^−ρt and V ar(X(t)) = σ²

2ρ(e^−ρt). (18) In this model the process X(t) uctuates randomly, but tends to revert to some fun- damental level µ. The behaviour of this ⁰reversion⁰ depends on both the short term standard deviation σ and the speed of reversion parameter ρ, as shown in Figures 4 and 5.

In Figure 4 we have simulated roughly 120 days using Eulero−Mayurama approximation; with a small daily volatility, σ= 0.5, in Figure 4a, and a bigger volatility, σ = 5, in Figure 4b. The mean reverting level,µ= 20, and speed of mean reversion, ρ= 1.5, are kept constant over three dierent simulations with dierent starting points in both cases.

(22)

(a)σ = 0.5 (b)σ = 5

Figure 4: Simulations on variation of σ.

In Figure 5 are simulated roughly120days, with a small reverting mean speed,ρ= 0.3, in 5a, and a bigger speed of mean reversion, ρ = 5, in 5b. The mean reverting level, µ= 20, and daily volatility,σ = 1.5, are kept constant over three dierent simulations with dierent starting points, in both cases.

(a)ρ= 0.3 (b)ρ= 5

Figure 5: Simulations on variation of ρ.

The Ornstein-Uhlenbeck paths consist of uctuations around the xed mean reverting level µ, and they are typically of the order σ, although larger uctuations occur over long enough times.

This behaviour is suited for the analysis of economic variables, such as the price of some commodity, which has reason to revert at a xed level. Nevertheless, when we

(23)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 23 want to simulate the prices of commodities through the mean reverting process, the main challenge is to carry out the best choice for the parameters; even if each variable has an intuitive meaning, estimating its value is dierent.

There are dierent generalizations in literature related to the Ornstein−Uhlenbeck processes. Usually a generalized Ornstein Uhlenbeck (GOU) process is dened as a (V_t)t≥0 process driven by the bivariate Le`vy process(ξ_t, η_t)t≥0 such that it satises the following equation:

Vt =e^−ξ^t

V0+ Z t

0

e^ξ^s−dηs

, t≥0,

where V₀ is a nite random variable, independent of (ξ, η). Furthermore the GOU process driven by (ξ, η)is the unique solution of the stochastic dierential equation

dV_t =Vt−dU_t+dL_t, t≥0,

for the bivariate L`evy process(U_t, L_t)t≥0, as shown in [25]. Properties of the described process are treated in detail in [26] and [27].

Other generalizations of the Ornstein-Uhlenbeck process are studied in [28], where given the classic Langevin equation (12) the additive Brownian noise is replaced with an arbitrary time−dependent random force characterizing the noise.

1.3 An individual based model

In nancial markets it is well-known that commodity price is the result of several factors. One of the factors that is common for dierent stock and commodity prices is the impact of traders' investments. This is the main idea that inspired M. Jablonska, in [22], to develop a model for commodity prices only based on the inuence that traders' behaviour has on price formation.

Since a big subgroup of traders can be seen as a particle population, the application of stochastic interacting particles model,[19], to the traders case is allowed. Therefore in

(24)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 24 [22], [23], the authors give a stochastic particle model for commodity prices in which a group of N traders is treated as a discrete system of N particles and their dynamics is characterized upon the distances between their price, instead of upon their density as in the animal case.

Traders are allowed to interact on three dierent scales, which are incorporated in the model equation. Macroscale is referred to as the direction of the whole population and is given by a mean reversion process. Microscale includes information about the motion of each trader separately and it is introduced through a global interaction component; mesoscale is describing to the interaction of each individual with its closest neighbourhood and for this scale a local interaction function has been dened.

Also a psychological trading component is added. This component is related to the phenomena of "animal spirit" rst introduced by Keynes in [29], according to which instincts and emotions guide humans and push them to behave like animals, performing acts without rationality.

Given a market with N ∈ N\ {0} traders, the basic idea of the model developed by M. Jablonska and T. Kauranne is that in the interval [t, t+dt) the k−th trader proposes a price X_t^k ∈ R^d,t ≥ 0, d ∈ N\ {0}, depending upon all traders' price vector Xt = (X_t¹, . . . , X_t^N) at time t⁻. The nal price Xˆ_t depends on the N traders prices X_t^k for k ∈ {1, . . . , N} at time t⁻ and upon a source of randomness described by an additive Wiener process W_t. Denoting by {X_t^k}_t∈R⁺ a stochastic process, that describes the state of thek−thtrader, dened on a suitable probability space(Ω,F, P) and valued in (R^d,BR^d), with BR^d the Borel σ−algebra generated by the intervals in R^d, the N traders prices' are given by the following system of stochastic dierential equations of the Itoˆtype:

dX_t^k = [f₁(X_t^k,Xt) +f₂(k,Xt) +f₃(k,Xt)]dt+σdW_t^k, (19) for k = 1, . . . , N and constant volatility σ ∈ R₊. The stochastic term is given by a family of independent Wiener processes, while the drift is composed by three terms that are described in the following.

The rst term f₁(X_t^k,Xt)describes a mean reversion with respect to the average

(25)

observed price in a previous time interval of size D∈R,i.e.

f₁(X_t^k,Xt) =γ(X_t^∗−X_t^k), (20)

where

X_t^∗ = 1 D

Z t t−D

Xˆ_sds

is the mean reversion level at time t and γ ∈ R₊ is the rate with which the data are pulled toward the mean reversion levelX_t^∗. This mean ensures that even if prices spike, they will come back to their usual level.

Thef₂(k,Xt)function reects the global interaction of traders. This function encloses information about the position of thek−thtrader with respect to the entire population, in depth if the k−th trader follows the average population behaviour believing that others have more information, or deviates from them.

f₂(k,Xt) =θ(h(k,Xt)−X_t^k)

where the function h(k,Xt)∈C(R^d) and θ ∈R₊ is the strength of that interaction.

In [21] it is explained that this term is suggested by the dynamics of the Burgers equation. In the next Chapter we will investigate in detail this aspect.

The functionf₃(k,Xt)is a function describing local interaction of thek−thtrader with its neighbours, which means that each trader at time t perceives only the inuence of the price of the closest p%of the total population and moves toward the most distant price from that neighbourhood, believing that other are better informed. From the study carried out in [21], on the appropriate choice of p%, the results shown that the value of 5%is the optimal one. The f₃(k,Xt)function is dened as follows

f₃(k,Xt) =ξ(g(k,Xt)−X_t^k), where g(k,Xt)∈C(R^d) and ξ∈R the local interaction speed.

(26)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 26 1.3.1 Data

Silver data

The data used to test the model consist of daily observations of closing prices for two dierent commodities, silver and electricity.

Silver price data are collected from www.livecharts.co.uk website, related to the period 1^st March2000 to 1^st March2013. They are closing prices of spot Silver, expressed in US dollars; the weekends are also included, replacing them with some of Friday closing prices. As shown in Figure 6, which represent the Silver price evolution, the data are not stationary not mean reverting either.

Figure 6: Silver data prices line chart

Anyway, the model described above has been tested for this data set and the results prove that the model is able to capture some of the features of this set as demonstrated in [30].

Let assume a market withN = 100traders in a time window of 91 days, mean reversion rate and volatility constant over time, and the percentage of the local interaction xed at5%. Figures 7 and 9 present two dierent simulations of the Silver data, which are

(27)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 27 obtained by keeping constant all the variables, with only the randomness changing.

The related root mean square error (RMSE), which conrms that the simulated data follows in mean the Silver price, is given in Fig. 37 and 10, respectively. The simulated data are 700 and the simulation at each time step are computed on a window of 91 days in a way that at each time the previous 91 data are used, but only nine of these are drawn in the picture below.

Figure 7: Silver data simulations at dierent times, Hb is the variable counting days from the beginning of the data

(28)

Figure 8: Root mean square error.

Figure 9: Silver data simulations at dierent times Hb.

(29)

In Figures 7 and 9, the blue line is the Silver line chart and the red line is the simulated value and in Fig. 37 and 10 the corresponding root mean square error distribution is plotted.

As expected after a few times the high volatility of the future price emerges, due to the use of the 91 previous simulated prices, which leads to an increase i the forecast error.

Anyway, the results show that the model approximates the real Silver closing price in average with anerror that oscillates around 0.2. Moreover, tests demonstrate that the root mean square error is not normally distributed.

New Zealand electricy data

The model has been tested also for other commodities as sugar and crude oil. The most challenging market to simulate is the electricity spot market, since it is a high volatility commodity. The high volatility is due to many factors as by its nature dicult to be stored, it has a seasonal component and also due to the fact that electricity has to be available on demand.

The above described model has been used for simulating the New Zealand electricity

(30)

1 STOCHASTIC MODELS FOR PRICE EVOLUTION 30 prices. New Zealand is a country with dierent sources of electricity; the two major categories are electricity from hydroelectric stations and from thermal, geothermal and wind power. In this particular market, trading occurs 24 hours a day, seven days a week, for each half-hour period of the day.

The data used in this thesis consists of a set of 3500 daily observations, where each of them is the average of the 48 prices occurred in that day, i.e., one every30 minutes.

Figure 11: New Zealand electricity spot prices line chart.

In Figures 12 and 14, the blue line is the New Zealand electricity spot prices line chart and the red line is the simulated value and in Fig. 13 and 15 the corresponding root mean square error distribution is plotted.

(31)

Figure 12: New Zealand electricity data simulations at dierent times Hb.

(32)

Figure 14: New Zealand electricity data simulations at dierent times Hb.

From Figures 12 and 14 it is clear that the model does not follow the electricity price behaviour, and not even in the mean. From Figures 13 and 15 we can conclude that

(33)

2 STOCHASTIC SYSTEM AND BURGERS EQUATION 33 the root mean square error is around 30.

2 Stochastic system and Burgers equation

As introduced in Chapter 1, section 1.3.1, a part of the dynamics of the introduced model is inspired by the dynamics described by the Burgers equation, which can be interpreted as a mean eld approximation of a family of stochastically interacting individuals. In this Chapter we deepen the study of the Burgers equations from the deterministic point of view and investigate how to derive it from a stochastic system.

The use of such equations in the nancial and economic eld is not a new approach.

Indeed a common assumption for modelling nancial markets is the equilibrium price, which is dened as the state where market supply and demand balance each other. As a result, price becomes stable and the mathematical condition for an equilibrium nancial market is that there must exist a risk neutral probability measure Q_t, depending on time t, absolutely continuous with the given real world probability measure P; this is also a condition for an arbitrage free-pricing system. The risk neutral probability measure Q_t leads us to determine the path-independence property for the associated density process dened by the Radon-Nikodym derivative ^dQ_dP^t. As shown in Section 2.2, this property translates from a stochastic dierential equation of Markov type to a non-linear partial dierential equation, such as Burgers equation.

Since this link is legitimized from the nancial point of view, we can use the stochastic interacting particles method for Burgers equation for modelling commodity markets.

2.1 Burgers equation

Burgers equation is a simplication of the Navier−Stokes equations which are a set of nonlinear partial dierential equations describing the motion of a uid in Rⁿ.

∂

∂tu(x, t) +u(x, t)· ∇u(x, t) = −∇p(x, t) +ν∆u(x, t) +F(x, t) x∈Rⁿ, t≥0 (21)

∇ ·u(x, t) = 0 x∈Rⁿ, t≥0 (22)

(34)

2 STOCHASTIC SYSTEM AND BURGERS EQUATION 34 These equations are to be solved for an unknown velocity vector functionu(x, t) and a pressurep(x, t)∈Rdened for position x∈Rⁿ,F(x, t)is the externally applied force, ν ∈R₊ the constant of viscosity. The equation (21) is the simplest partial dierential equation (PDE) combining both nonlinear and diusive eects, while it assumes a given pressure eld. It is the Newton's law, F = m·a, for a uid element subject to the external force F(x, t) and forces arising from pressure and friction. The equation (22) says that the uid is incompressible.

In 1939 the Dutch scientist J.M. Burgers simplied the Navier−Stokes equation (21) by dropping the pressure term and assuming that the external force F(x, t) is zero.

The result is the Burgers equation having the following form

∂

∂tu(x, t) +u(x, t) ∂

∂xu(x, t) =ν ∂²

∂x²u(x, t), (23)

which can be investigated in one spatial dimension.

The equation (23) is called the viscid Burgers equation against the inviscid form which can be seen as the limit asν →0of the (23), hence the equation assumes the following form

∂

∂tu(x, t) +u(x, t) ∂

∂xu(x, t) = 0. (24)

2.1.1 Viscid Burgers equation

The initial value problem for the viscid Burgers equation is of the type







u_t =uu_x+νu_xx x∈R, t≥0, ν > 0

u(x,0) =f(x) x∈R, f ∈C^∞ (25)

with periodic boundary conditions f(x) = f(x+ 1). Assuming that all functions are real−valued, the equation (23) is given by the sum of a quadratic rst order term and a diusion term.

(35)

2 STOCHASTIC SYSTEM AND BURGERS EQUATION 35 A classical solution of the problem (25) is a function u(x, t) ∈ C²(x)∩C¹(t) which satises the equation (23) pointwise. The uniqueness of this solution is given by the following Lemma:

Lemma 2.1. The 1−periodic Cauchy problem (25) has at most one classical solution.

Proof. Let suppose thatu(x, t)andv(x, t)are both solutions of the same problem (25), then w(x, t) :=u(x, t)−v(x, t) satises the problem







w_t= ¹₂(αw)_x+νw_xx w(x,0) = 0

with α=u+v and initial condition w(x,0) =f(x)−f(x) = 0. From

(w,(αw)_x) = (w, α_xw+αw_x) = (w, α_xw)−((αw)_x, w), (26) where the last equality comes out from integration by part, it follows that

(w,(αw)_x) = 1

2(w, α_xw), (27)

hence

1 2

d

dt(w, w) = (w, w_t) = 1

2(w,(αw)_x)−ν(w_x, w_x)

= 1

4(w, α_xw)−ν(w_x, w_x)

≤ 1

4|α_x|∞(w, w).

(28)

The initial condition w(x,0) = 0 implies w≡0.

The existence of the solution u(x, t) of the (25) can be proved in dierent ways, local existence via linear iteration and global existence or using the Cole−Hopf transformation.

(36)

2 STOCHASTIC SYSTEM AND BURGERS EQUATION 36 Local existence via iteration

The global existence of the solution of Burgers equation is shown to be an extension of the local solution obtained via iteration.

Given the following problem







uⁿ⁺¹_t =uⁿuⁿ⁺¹_x +νuⁿ⁺¹_xx

uⁿ⁺¹(x,0) =f(x) n= 0,1,2, ... (29) where u⁰(x, t) = f(x), the sequence (uⁿ)_n ∈ N will be shown to converge and its limit will solve the (25). Remembering that H^k(R) is the Sobolev space W^k,2 which admits an inner product dened ashu, vi_Hk =Pk

i=0hDⁱu, Dⁱvi_L2, whereDⁱis the weak derivative, then the following Theorem proves the local existence.

Theorem 2.1. (Local existence) Let T₁ =T₁(kfk_H2)>0 s.t.

kuⁿ(·, t)k_H2 ≤2kfk_H2 in 0≤t ≤T₁.

For any ν ≥0, Burgers equation has a C^∞ solution u(x, t) dened for 0≤t≤T₁. Proof.

Consider the sequence uⁿ dened by the iteration (29) and abbreviate

v =uⁿ⁺¹−uⁿ, w=uⁿ−uⁿ⁻¹. Then







v_t=uⁿv_x+uⁿ_xw+νv_xx v(x,0) = 0

implies that

1 2

d

dtkvk² ≤(v, uⁿvx) + (v, uⁿ_xw)

=−1

2(v, uⁿ_xv) + (v, uⁿ_xw)

≤const kvk²+kwk²

, 0≤t≤T1

(37)

2 STOCHASTIC SYSTEM AND BURGERS EQUATION 37

Therefore by Gronwall's Lemma uⁿ⁺¹(·, t)−uⁿ(·, t)

2 ≤K Z t

0

uⁿ(·, τ)−uⁿ⁻¹(·, τ)

2dτ,

where K is a constant independent of n.

The convergence of the sequence uⁿ(·, t) to a function u(·, t) ∈ L₂ is given by the following Lemma

Lemma 2.2. Letη^k(t), k = 0,1, ...be a sequence of non-negative continuous functions which satisfy the inequalities

η^k+1(t)≤a+b Z t

0

η^k(τ)dτ,0≤t ≤T with constant a, b≥0, then

η^k(t)≤a

k−1

X

=0

bt

! +b^kt^k k! max

0≤τ≤tη⁰(τ)

for0≤t≤T andk = 0,1, .... In particular, the sequenceη^k(t),0≤t≤T is uniformly bounded. If a= 0, then the sequence converges uniformly to zero.

We know from the theory on the PDE that max

δ^p+q

δ^pxδ^qtu^h(·, t)

: 0≤t≤T

≤c(p, q, T)

with constant independent of h.

Therefore it follows that for u ∈ C^∞, the convergence uⁿ −→ u holds pointwise and also holds for all derivatives. Then the equation (29) implies that u(x, t) solvers the Burgers equation in the[0, T₁] interval.

Global existence

Let us assume ν > 0 arbitrary but xed, then a proof of the existence of a global solutionu(x, t)∈C^∞ dened for 0≤t≤ ∞ is given by the following Theorem.

Theorem 2.2. Let f ∈ C^∞ and ν > 0. The 1-periodic Cauchy problem (25) has a unique solution u∈C^∞ dened for 0≤t <∞.

(38)

2 STOCHASTIC SYSTEM AND BURGERS EQUATION 38 Proof. The proof of this Theorem is based on the following result:

Theorem 2.3. Letf =f(x) denote1-periodic C^∞ initial data, and let u∈C^∞ denote a solution of the Burgers equation dened for 0 ≤ t ≤ T. There is a constant K, depending on kfk_H2 and , but independent from T, with

ku(·, t)k_H2 ≤K for 0≤t≤T. (30) Assuming this result is proven, we can show all-time existence for the problem (25) as follows: According to the Local Existence Theorem 2.1, there is a time T₁ >0 with a C^∞-solution u dened for 0≤t ≤T₁. By (30) we have

ku(·, t)k_H2 ≤K.

The functionx7→u(x, T₁)can be chosen as the new initial data and the Local Existence Theorem 2.1 can be applied again.

The solution starting with initial data u(·, T₁) exists in a time interval 0≤ t≤T₂, T₂ depending only on K and . Clearly, putting the two solutions together, we have a C^∞-solution u of (25) dened for 0 ≤ t ≤ T₁ +T₂. The important point is that the prior estimate (30) implies that

ku(·, T₁+T₂)k_H2 ≤K

with the same constant K as before. Thus, using the Local Existence Theorem, we can again extend the solution for a time interval of length T₂, etc.

For more details see [31].

The result can be extended to a 1-periodic Cauchy problem with f ∈L^∞.

An alternative way to prove the existence of the Burgers equation with initial condition u(x,0) = u⁰(x), u∈C^∞ is via the Cole-Hopf transformation.

Cole−Hopf transformation

The Cole−Hopf transformation was developed in [32] for the Burgers equation and then applied at dierent partial dierential equations as Seventh-Order KdV Equation

(39)

2 STOCHASTIC SYSTEM AND BURGERS EQUATION 39 in [33]. This method maps the solution of the viscid Burgers equation (23) to the heat equation

ϕ_t=νϕ_xx f or ϕ∈C^∞. The Cole−Hopf transformation is dened by

u(x, t) = −2νϕ_x

ϕ, (31)

where partial derivatives are:

u_t= 2ν(ϕ_tϕ_x−ϕϕ_xt)

ϕ² , uu_x = 4ν²ϕ_x(ϕϕ_xx−ϕ²_x) ϕ³

and

uxx =−2ν²(2ϕ³_x−3ϕϕ_xxϕ_x+ϕ²ϕ_xxx)

ϕ³ .

Substituting this expression into (23)

2ν(−ϕϕ_xt+ϕ_x(ϕ_t−νϕ_xx) +νϕϕ_xxx)

ϕ² = 0⇐⇒ −ϕϕxt+ϕx(ϕt−νϕxx) +νϕϕxxx = 0

ϕ_x(ϕ_t−νϕ_xx) = ϕ(ϕ_xt−νϕ_xxx) = ϕ(ϕ_t−νϕ_xx)_x.

Therefore, if ϕ is a solution of the heat equation ϕt−νϕxx = 0, x ∈ R, then u(x, t) given by the transformation (31) solves the viscid Burgers equation (23).

Writing the (31) as

u=−2ν(logϕ)x

then

ϕ(x, t) =e⁻^R⁰^x^u(y,t)^2ν ^dy. (32)

(40)

Thus the initial condition must be transformed by (34) into ϕ(x,0) =e⁻^R⁰^x^f(y)^2ν ^dy.

Hence the problem (25) has been transformed into







ϕt−νϕxx = 0 x∈R, t≥0, ν >0

ϕ(x,0) = ϕ₀(x) =e⁻^R⁰^x^f(y)^2ν ^dy x∈R. (33) in which the function ϕ(x, t), satisfying the heat equation, is dened as

ϕ(x, t) = C s√

πνt Z

R

f(x)e⁻^(x−s)2^4νt ds. (34)

Then the solutionu(x, t) can be determined from (31).

2.1.2 Inviscid Burgers equation The inviscid Burgers equation

u_t+uu_x = 0 (35)

can be treated as a limit of (23) as ν −→ 0 which has a unique solution, u ∈ C^∞, existing for all time t≥0.

However, the smoothness of the solution u breaks down in general, at a certain time T_b, at which one or several shocks form. This can be proved by the method of characteristics.

Solution of characteristic

Lemma 2.3. Suppose u(x, t) to be a smooth solution of

(41)







ut+uux = 0

u(x,0) =f(x) (36)

Then u(x₀+tf(x₀), t) =f(x₀), x₀ ∈R in any t-interval 0≤t≤T.

Proof. Let(x(t), t), f or0≤t≤T, dened as a characteristic line of the equation (35) for the specic solution u, if

dx

dt(t) = u(x(t), t) (37)

for 0 ≤ t ≤ T and x(0) = x₀. When x(t) and u(x, t) ∈ C¹ are solutions of (37) and (35) respectively, then

d

dt[u(x(t), t)] =u_t(x(t), t) + dx

dt(t)u_x(x(t), t)

=u_t(x(t), t) +u(x(t), t)u_x(x(t), t)

= 0

i.e. u(x(t), t) is constant along the characteristic curve x(t). Therefore u(x(t), t) =u(x(0),0) = f(x₀)

which from the system (37), leads us to conclude that the characteristic curves are straight lines determined by initial data

x(t) =x₀+tf(x₀), t >0 (38) which implies u(x₀, tf(x₀), t) = f(x₀).

Let assume that there are 2 characteristics that arise from initial conditions x1 and x2 = x1 + ∆x with ∆x = x2−x1. According to (38), these characteristics will cross when

x₁+tf(x₁) = x₂+tf(x₂).

(42)

2 STOCHASTIC SYSTEM AND BURGERS EQUATION 42 Solving for t leads to

t=− x₁−x₂

f(x1)−f(x2) =− 1

f⁰(ξ) f or some ξ.

Thus,Tb the breaking time, the time at which the smoothness of the solutionu breaks down, is dened as follow







∞ if f⁰(ξ)≥0, f or some ξ

−_inf(f¹0(ξ)) otherwise. (39)

Iff⁰(ξ)≥0 ∀ξ, then x₁ < x₂ impliesf(x₁)< f(x₂);hence

(x₁+tf(x₁), t), (x₂+tf(x₂), t), t≥0 (40) never cross, therefore T_b =∞ and a smooth solution exists∀t ≥0. On the other hand in f⁰(ξ) < 0 for some ξ then there are values x₁ < x₂ with f(x₁) > f(x₂) and the characteristics (40) cross.

The solution steepens up with time. The time T_b is nite. We cannot extend the solutionu beyond T_b as a smooth solution.

Lemma 2.4. The C^∞ function u = u(x, t) dened by u(x, t) = f(x0(x, t)), x ∈ R, 0≤t≤Tb is the unique classical solution of







u_t+uu_x = 0

u(x,0) =f(x). (41)

Proof.x₀(x,0) =xand thusu(x,0) = f(x).To show thatu_t+uu_x = 0,we dierentiate

x₀+f(x₀) =x w.r.t. xand t, and nd that

x₀_x+t(f(x₀))_x = 1 ∧ x₀_t +f(x₀) +t(f(x₀))_t= 0

Stochastic particle models: mean reversion and burgers dynamics. An application to commodity markets

Acknowledgements

Contents

1 Stochastic models for price evolution

1.1 Introduction

1.2 Mean Reverting processes

1.3 An individual based model

2 Stochastic system and Burgers equation

2.1 Burgers equation