Transactions of the American Mathematical Society, Series B
For a real-valued one dimensional diffusive strict local martingale, we provide a set of smooth f... more For a real-valued one dimensional diffusive strict local martingale, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution under a local 1 2 \frac 12 -Hölder condition. Under the weaker Engelbert-Schmidt conditions, we provide a set in which the Cauchy problem has a unique weak solution. We exemplify our results using quadratic normal volatility models and the two dimensional Bessel process.
An integral representation result for strictly positive subharmonic functions of a one-dimensiona... more An integral representation result for strictly positive subharmonic functions of a one-dimensional regular diffusion is established. More precisely, any such function can be written as a linear combination of an increasing and a decreasing subharmonic function that solve an integral equation g(x) = a + v(x, y)µ A (dy) + κs(x), where a > 0, κ ∈ R, s is a scale function of the diffusion, µ A is a Radon measure, and v is a kernel that is explicitly determined by the scale function. This integral equation in turn allows one construct a pair (g, A) such that g is a subharmonic function, A is a continuous additive functional with Revuz measure µ A and g(X) exp(−A) is a local martingale. The changes of measures associated with such pairs are studied and shown to modify the long term behaviour of the original diffusion process to exhibit transience. Theory is illustrated via examples that in particular contain a sequence of measure transformations that render the diffusion irregular in the limit by breaking the state space into distinct regions with soft and hard borders. Finally, the theory is applied to find an "explicit" solution to an optimal stopping problem with random discounting.
1. To formulate the model of the market precisely, let (Ω,F , (Ft)t∈[0,1],P) be a filtered probab... more 1. To formulate the model of the market precisely, let (Ω,F , (Ft)t∈[0,1],P) be a filtered probability space satisfying the usual conditions of right continuity and P-completeness. Assume that on this probability space there exist a continuous random variable V ∈ F0 and a standard Brownian motion B, independent of V . We consider a market in which a single risky asset is traded. The value of this asset, V , will be a public knowledge at some future time t = 1. For simplicity of exposition we assume that the risk free interest rate is 0. There are three types of agents that interact in this market: i) Liquidity traders, whose demands are random, price inelastic and do not reveal any information about the value of V . In particular we assume that their cumulative demand at time t is given by Zt = σBt. ii) A single insider who knows V at time t = 0 and is risk neutral. We will denote insider’s cumulative demand at time t by Xt. The filtration of the insider, FI , is generated by observ...
0 σ(s,Xs)dWs where W is a r-dimensional Brownian motion, b is a d × 1 drift vector and σ is a d ×... more 0 σ(s,Xs)dWs where W is a r-dimensional Brownian motion, b is a d × 1 drift vector and σ is a d × r dispersion matrix. Definition 5.1. A weak solution of (5.1) is a triple (X,W ), (Ω,F , P ), (Ft), where i) (Ω,F , P ) is a probability space, (Ft) is a filtration of sub-σ-fields of F satisfying usual conditions; ii) X = (Xt) is adapted to (Ft) and W is (Ft)-Brownian motion such that (5.1) is satisfied and P [ ∫ t 0 { |b(s,Xs)|+ σ ij(s,Xs) } ds < ∞] = 1 for every t > 0. The probability measure μ on B(R) defined by μ(Λ) = P (X0 ∈ Λ) is called the initial distribution of the solution. Definition 5.2. We say that uniqueness in the sense of probability law holds for (5.1) if, for any two weak solutions (X,W ), (Ω,F , P ), (Ft) and (X̂, Ŵ ), (Ω̂, F̂ , P̂ ), (F̂t) with the same initial distribution have the same law, i.e. for any t1, . . . , tn and n ≥ 1, P (Xt1 ∈ E1, . . . , Xtn ∈ En) = P̂ (X̂t1 ∈ E1, . . . , X̂tn ∈ En), where Ei ∈ B(R). Exercise 5.1. Let X be a weak solution of (5.1...
Let X be a linear diffusion taking values in (`, r) and consider the standard Euler scheme to com... more Let X be a linear diffusion taking values in (`, r) and consider the standard Euler scheme to compute an approximation to E[g(XT )1[T<ζ]] for a given function g and a deterministic T , where ζ = inf{t ≥ 0 : Xt / ∈ (`, r)}. It is well-known since [16] that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to 1/ √ N with N being the number of discretisatons. We introduce a drift-implicit Euler method to bring the convergence rate back to 1/N , i.e. the optimal rate in the absence of killing, using the theory of recurrent transformations developed in [6]. Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.
We propose a static equilibrium model for limit order book where N ≥ 1 profit-maximizing investor... more We propose a static equilibrium model for limit order book where N ≥ 1 profit-maximizing investors receive an information signal regarding the liquidation value of the asset and execute via a competitive dealer with random initial inventory. While the dealer's initial position plays a role similar to noise traders in Kyle [16], he trades against a competitive limit order book populated by liquidity suppliers as in Glosten [12]. We show that an equilibrium exists for bounded signal distributions, obtain closed form solutions for Bernoulli-type signals and propose a straightforward iterative algorithm to compute the equilibrium order book for the general case. We obtain the exact analytic asymptotics for the market impact of large trades and show that the functional form depends on the tail distribution of the private signal of the insiders. In particular, the impact follows a power law if the signal has fat tails while the law is logarithmic in case of lighter tails. Moreover, the tail distribution of the trade volume in equilibrium obeys a power law in our model. We find that the liquidity suppliers charge a minimum bid-ask spread that is independent of the amount of 'noise' trading but increasing in the degree of informational advantage of insiders in equilibrium. The model also predicts that the order book flattens as the amount of noise trading increases converging to a model with proportional transactions costs. In case of a monopolistic insider we show that the last slice traded against the limit order book is priced at the liquidation value of the asset. However, competition among the insiders leads to aggressive trading causing the aggregate profit to vanish in the limiting case N → ∞. The numerical results also show that the spread increases with the number of insiders keeping the other parameters fixed. Finally, an equilibrium may not exist if the liquidation value is unbounded. We conjecture that existence of equilibrium requires a sufficient amount of competition among insiders if the signal distribution exhibit fat tails.
We study the solutions of the inverse problem g(z) = f (y)P T (z, dy) for a given g, where (P t (... more We study the solutions of the inverse problem g(z) = f (y)P T (z, dy) for a given g, where (P t (•, •)) t≥0 is the transition function of a given symmetric Markov process, X, and T is a fixed deterministic time, which is linked to the solutions of the ill-posed Cauchy problem u t + Au = 0, u(0, •) = g, where A is the generator of X. A necessary and sufficient condition ensuring square integrable solutions is given. Moreover, a family of regularisations for above problems is suggested. We show in particular that these inverse problems have a solution when X is replaced by ξX + (1 − ξ)J, where ξ is a Bernoulli random variable, whose probability of success can be chosen arbitrarily close to 1, and J is a suitably constructed jump process.
Let X be a regular one-dimensional transient diffusion and L y be its local time at y. The stocha... more Let X be a regular one-dimensional transient diffusion and L y be its local time at y. The stochastic differential equation (SDE) whose solution corresponds to the process X conditioned on [L y ∞ = a] for a given a ≥ 0 is constructed and a new path decomposition result for transient diffusions is given. In the course of the construction Bessel-type motions as well as their SDE representations are studied. Moreover, the Engelbert-Schmidt theory for the weak solutions of one dimensional SDEs is extended to the case when the initial condition is an entrance boundary for the diffusion. This extension was necessary for the construction of the Bessel-type motion which played an essential part in the SDE representation of X conditioned on [L y ∞ = a].
International Biodeterioration & Biodegradation, 2017
Perchlorate is a naturally occurring and manufactured chemical anion and can be present in water ... more Perchlorate is a naturally occurring and manufactured chemical anion and can be present in water sources together with nitrate. This study aims at (1) determining the nitrate and perchlorate contamination in a semi-arid plain (Harran Plain) and (2) evaluating the performance of a heterotrophicautotrophic sequential denitrification process for nitrate and perchlorate removal from the groundwater of this plain. The nitrate in the groundwater samples varied between 4.07 and 83.22 mg l À1 NO 3-N. Perchlorate was added to groundwater samples externally and its concentration was increased from 100 to 1500 mg l À1. The total nitrogen concentrations in the sequential system effluent throughout the study were always below 0.5 mg l À1. C/N ratio was 2.44 which was slightly lower than the theoretical level of 2.47. Therefore the average NO 3 À-N in the heterotrophic reactor effluent was 19 ± 3.7 mg l À1 corresponding to an efficiency of 75% reduction. The remaining nitrate and nitrite were almost completely reduced in the autotrophic process. The system's perchlorate removal efficiency was above 98%, except during the last period (82%), at which influent perchlorate was 1500 mg l À1. The maximum perchlorate reduction rate throughout the study was around 15 mg/(L.d). Both perchlorate and nitrate reduction were partial in the heterotrophic reactor, but completed in the following autotrophic process.
Stochastics An International Journal of Probability and Stochastic Processes, 2015
For a squared Bessel process, X, the Laplace transforms of joint laws of (U, Ry 0 X p s ds) are s... more For a squared Bessel process, X, the Laplace transforms of joint laws of (U, Ry 0 X p s ds) are studied where Ry is the first hitting time of y by X and U is a random variable measurable with respect to the history of X until Ry. A subset of these results are then used to solve the associated small ball problems for Ry 0 X p s ds and determine a Chung's law of iterated logarithm. Ry 0 X p s ds is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The findings are then used to price a class of exotic derivatives on interest rates and determine the asymptotics for the prices of some put options that are only slightly in-the-money.
We discuss the pricing of defaultable assets in an incomplete information model where the default... more We discuss the pricing of defaultable assets in an incomplete information model where the default time is given by a first hitting time of an unobservable process. We show that in a fairly general Markov setting, the indicator function of the default has an absolutely continuous compensator. Given this compensator we then discuss the optional projection of a class of semimartingales onto the filtration generated by the observation process and the default indicator process. Available formulas for the pricing of defaultable assets are analyzed in this setting and some alternative formulas are suggested.
Let X be a Markov process taking values in E with continuous paths and transition function (P s,t... more Let X be a Markov process taking values in E with continuous paths and transition function (P s,t). Given a measure µ on (E, E), a Markov bridge starting at (s, ε x) and ending at (T * , µ) for T * < ∞ has the law of the original process starting at x at time s and conditioned to have law µ at time T *. We will consider two types of conditioning: a) weak conditioning when µ is absolutely continuous with respect to P s,t (x, •) and b) strong conditioning when µ = ε z for some z ∈ E. The main result of this paper is the representation of a Markov bridge as a solution to a stochastic differential equation (SDE) driven by a Brownian motion in a diffusion setting. Under mild conditions on the transition density of the underlying diffusion process we establish the existence and uniqueness of weak and strong solutions of this SDE.
Abstract We solve explicitly a Bayesian sequential estimation problem for the drift parameter μ o... more Abstract We solve explicitly a Bayesian sequential estimation problem for the drift parameter μ of a fractional Brownian motion under the assumptions that a prior density of μ is Gaussian and that a penalty function is quadratic or Dirac-delta. The optimal stopping time for this case is deterministic.
ABSTRACT We study a stochastic model for a market with two tradeable assets where the price of th... more ABSTRACT We study a stochastic model for a market with two tradeable assets where the price of the first asset is implied by the value of the second one and the state of a partially ‘hidden’ control process. We derive a closed expression for the value of the first asset, as a function of the price for the second and the most recent observation of the control process. We show how the model can be applied to EU markets for carbon emissions. The 5th Actuarial and Financial Mathematics Day took place on 9 February 2007.
Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own fi... more Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X 1 = Z 1. We call X a dynamic bridge, because its terminal value Z 1 is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration F X and the filtration F X,Z jointly generated by X and Z. Our construction is heavily based on parabolic PDE's and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading, that can be viewed as a non-Gaussian generalization of Back and Pedersen's [3], where insider's additional information evolves over time.
The mainstream in finance tackles portfolio selection based on a plug-in approach without conside... more The mainstream in finance tackles portfolio selection based on a plug-in approach without consideration of the main objective of the inferential situation. We propose minimum expected loss (MELO) estimators for portfolio selection that explicitly consider the trading rule of interest. The asymptotic properties of our MELO proposal are similar to the plug-in approach. Nevertheless, simulation exercises show that our proposal exhibits better finite sample properties when compared to the competing alternatives, especially when the tangency portfolio is taken as the asset allocation strategy. We have also developed a graphical user interface to help practitioners to use our MELO proposal.
This document is the author's final manuscript accepted version of the journal article, incorpora... more This document is the author's final manuscript accepted version of the journal article, incorporating any revisions agreed during the peer review process. Some differences between this version and the published version may remain. You are advised to consult the publisher's version if you wish to cite from it.
International Journal of Theoretical and Applied Finance, 2009
We propose a model for trading in emission allowances in the EU Emission Trading Scheme (ETS). Ex... more We propose a model for trading in emission allowances in the EU Emission Trading Scheme (ETS). Exploiting an arbitrage relationship we derive the spot prices of carbon allowances given a forward contract whose price is exogenous to the model. The modeling is done under the assumption of no banking of carbon allowances (which is valid during the Phase I of Kyoto protocol), however, we also discuss how the model can be extended when banking of permits is available. We employ results from filtering theory to derive the spot prices of permits and suggest hedging formulas using a local risk minimisation approach. We also consider the effect of intermediate announcements regarding the net position of the ETS zone on the prices and show that the jumps in the prices can be attributed to information release on the net position of the zone. We also provide a brief numerical simulation for the price processes of carbon allowances using our model to show the resemblance to the actual data.
We study an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm ... more We study an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm in the framework of Back [2]. The market consists of a risk-neutral informed agent, noise traders and a market maker who sets the price using the total order. When the insider does not trade, the default time possesses a default intensity in market's view as in reduced-form credit risk models. However, we show that, in the equilibrium, the modelling becomes structural in the sense that the default time becomes the first time that some continuous observation process falls below a certain barrier. Interestingly, the firm value is still not observable. We also establish the no expected trade theorem that the insider's trades are inconspicuous.
Transactions of the American Mathematical Society, Series B
For a real-valued one dimensional diffusive strict local martingale, we provide a set of smooth f... more For a real-valued one dimensional diffusive strict local martingale, we provide a set of smooth functions in which the Cauchy problem has a unique classical solution under a local 1 2 \frac 12 -Hölder condition. Under the weaker Engelbert-Schmidt conditions, we provide a set in which the Cauchy problem has a unique weak solution. We exemplify our results using quadratic normal volatility models and the two dimensional Bessel process.
An integral representation result for strictly positive subharmonic functions of a one-dimensiona... more An integral representation result for strictly positive subharmonic functions of a one-dimensional regular diffusion is established. More precisely, any such function can be written as a linear combination of an increasing and a decreasing subharmonic function that solve an integral equation g(x) = a + v(x, y)µ A (dy) + κs(x), where a > 0, κ ∈ R, s is a scale function of the diffusion, µ A is a Radon measure, and v is a kernel that is explicitly determined by the scale function. This integral equation in turn allows one construct a pair (g, A) such that g is a subharmonic function, A is a continuous additive functional with Revuz measure µ A and g(X) exp(−A) is a local martingale. The changes of measures associated with such pairs are studied and shown to modify the long term behaviour of the original diffusion process to exhibit transience. Theory is illustrated via examples that in particular contain a sequence of measure transformations that render the diffusion irregular in the limit by breaking the state space into distinct regions with soft and hard borders. Finally, the theory is applied to find an "explicit" solution to an optimal stopping problem with random discounting.
1. To formulate the model of the market precisely, let (Ω,F , (Ft)t∈[0,1],P) be a filtered probab... more 1. To formulate the model of the market precisely, let (Ω,F , (Ft)t∈[0,1],P) be a filtered probability space satisfying the usual conditions of right continuity and P-completeness. Assume that on this probability space there exist a continuous random variable V ∈ F0 and a standard Brownian motion B, independent of V . We consider a market in which a single risky asset is traded. The value of this asset, V , will be a public knowledge at some future time t = 1. For simplicity of exposition we assume that the risk free interest rate is 0. There are three types of agents that interact in this market: i) Liquidity traders, whose demands are random, price inelastic and do not reveal any information about the value of V . In particular we assume that their cumulative demand at time t is given by Zt = σBt. ii) A single insider who knows V at time t = 0 and is risk neutral. We will denote insider’s cumulative demand at time t by Xt. The filtration of the insider, FI , is generated by observ...
0 σ(s,Xs)dWs where W is a r-dimensional Brownian motion, b is a d × 1 drift vector and σ is a d ×... more 0 σ(s,Xs)dWs where W is a r-dimensional Brownian motion, b is a d × 1 drift vector and σ is a d × r dispersion matrix. Definition 5.1. A weak solution of (5.1) is a triple (X,W ), (Ω,F , P ), (Ft), where i) (Ω,F , P ) is a probability space, (Ft) is a filtration of sub-σ-fields of F satisfying usual conditions; ii) X = (Xt) is adapted to (Ft) and W is (Ft)-Brownian motion such that (5.1) is satisfied and P [ ∫ t 0 { |b(s,Xs)|+ σ ij(s,Xs) } ds < ∞] = 1 for every t > 0. The probability measure μ on B(R) defined by μ(Λ) = P (X0 ∈ Λ) is called the initial distribution of the solution. Definition 5.2. We say that uniqueness in the sense of probability law holds for (5.1) if, for any two weak solutions (X,W ), (Ω,F , P ), (Ft) and (X̂, Ŵ ), (Ω̂, F̂ , P̂ ), (F̂t) with the same initial distribution have the same law, i.e. for any t1, . . . , tn and n ≥ 1, P (Xt1 ∈ E1, . . . , Xtn ∈ En) = P̂ (X̂t1 ∈ E1, . . . , X̂tn ∈ En), where Ei ∈ B(R). Exercise 5.1. Let X be a weak solution of (5.1...
Let X be a linear diffusion taking values in (`, r) and consider the standard Euler scheme to com... more Let X be a linear diffusion taking values in (`, r) and consider the standard Euler scheme to compute an approximation to E[g(XT )1[T<ζ]] for a given function g and a deterministic T , where ζ = inf{t ≥ 0 : Xt / ∈ (`, r)}. It is well-known since [16] that the presence of killing introduces a loss of accuracy and reduces the weak convergence rate to 1/ √ N with N being the number of discretisatons. We introduce a drift-implicit Euler method to bring the convergence rate back to 1/N , i.e. the optimal rate in the absence of killing, using the theory of recurrent transformations developed in [6]. Although the current setup assumes a one-dimensional setting, multidimensional extension is within reach as soon as a systematic treatment of recurrent transformations is available in higher dimensions.
We propose a static equilibrium model for limit order book where N ≥ 1 profit-maximizing investor... more We propose a static equilibrium model for limit order book where N ≥ 1 profit-maximizing investors receive an information signal regarding the liquidation value of the asset and execute via a competitive dealer with random initial inventory. While the dealer's initial position plays a role similar to noise traders in Kyle [16], he trades against a competitive limit order book populated by liquidity suppliers as in Glosten [12]. We show that an equilibrium exists for bounded signal distributions, obtain closed form solutions for Bernoulli-type signals and propose a straightforward iterative algorithm to compute the equilibrium order book for the general case. We obtain the exact analytic asymptotics for the market impact of large trades and show that the functional form depends on the tail distribution of the private signal of the insiders. In particular, the impact follows a power law if the signal has fat tails while the law is logarithmic in case of lighter tails. Moreover, the tail distribution of the trade volume in equilibrium obeys a power law in our model. We find that the liquidity suppliers charge a minimum bid-ask spread that is independent of the amount of 'noise' trading but increasing in the degree of informational advantage of insiders in equilibrium. The model also predicts that the order book flattens as the amount of noise trading increases converging to a model with proportional transactions costs. In case of a monopolistic insider we show that the last slice traded against the limit order book is priced at the liquidation value of the asset. However, competition among the insiders leads to aggressive trading causing the aggregate profit to vanish in the limiting case N → ∞. The numerical results also show that the spread increases with the number of insiders keeping the other parameters fixed. Finally, an equilibrium may not exist if the liquidation value is unbounded. We conjecture that existence of equilibrium requires a sufficient amount of competition among insiders if the signal distribution exhibit fat tails.
We study the solutions of the inverse problem g(z) = f (y)P T (z, dy) for a given g, where (P t (... more We study the solutions of the inverse problem g(z) = f (y)P T (z, dy) for a given g, where (P t (•, •)) t≥0 is the transition function of a given symmetric Markov process, X, and T is a fixed deterministic time, which is linked to the solutions of the ill-posed Cauchy problem u t + Au = 0, u(0, •) = g, where A is the generator of X. A necessary and sufficient condition ensuring square integrable solutions is given. Moreover, a family of regularisations for above problems is suggested. We show in particular that these inverse problems have a solution when X is replaced by ξX + (1 − ξ)J, where ξ is a Bernoulli random variable, whose probability of success can be chosen arbitrarily close to 1, and J is a suitably constructed jump process.
Let X be a regular one-dimensional transient diffusion and L y be its local time at y. The stocha... more Let X be a regular one-dimensional transient diffusion and L y be its local time at y. The stochastic differential equation (SDE) whose solution corresponds to the process X conditioned on [L y ∞ = a] for a given a ≥ 0 is constructed and a new path decomposition result for transient diffusions is given. In the course of the construction Bessel-type motions as well as their SDE representations are studied. Moreover, the Engelbert-Schmidt theory for the weak solutions of one dimensional SDEs is extended to the case when the initial condition is an entrance boundary for the diffusion. This extension was necessary for the construction of the Bessel-type motion which played an essential part in the SDE representation of X conditioned on [L y ∞ = a].
International Biodeterioration & Biodegradation, 2017
Perchlorate is a naturally occurring and manufactured chemical anion and can be present in water ... more Perchlorate is a naturally occurring and manufactured chemical anion and can be present in water sources together with nitrate. This study aims at (1) determining the nitrate and perchlorate contamination in a semi-arid plain (Harran Plain) and (2) evaluating the performance of a heterotrophicautotrophic sequential denitrification process for nitrate and perchlorate removal from the groundwater of this plain. The nitrate in the groundwater samples varied between 4.07 and 83.22 mg l À1 NO 3-N. Perchlorate was added to groundwater samples externally and its concentration was increased from 100 to 1500 mg l À1. The total nitrogen concentrations in the sequential system effluent throughout the study were always below 0.5 mg l À1. C/N ratio was 2.44 which was slightly lower than the theoretical level of 2.47. Therefore the average NO 3 À-N in the heterotrophic reactor effluent was 19 ± 3.7 mg l À1 corresponding to an efficiency of 75% reduction. The remaining nitrate and nitrite were almost completely reduced in the autotrophic process. The system's perchlorate removal efficiency was above 98%, except during the last period (82%), at which influent perchlorate was 1500 mg l À1. The maximum perchlorate reduction rate throughout the study was around 15 mg/(L.d). Both perchlorate and nitrate reduction were partial in the heterotrophic reactor, but completed in the following autotrophic process.
Stochastics An International Journal of Probability and Stochastic Processes, 2015
For a squared Bessel process, X, the Laplace transforms of joint laws of (U, Ry 0 X p s ds) are s... more For a squared Bessel process, X, the Laplace transforms of joint laws of (U, Ry 0 X p s ds) are studied where Ry is the first hitting time of y by X and U is a random variable measurable with respect to the history of X until Ry. A subset of these results are then used to solve the associated small ball problems for Ry 0 X p s ds and determine a Chung's law of iterated logarithm. Ry 0 X p s ds is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The findings are then used to price a class of exotic derivatives on interest rates and determine the asymptotics for the prices of some put options that are only slightly in-the-money.
We discuss the pricing of defaultable assets in an incomplete information model where the default... more We discuss the pricing of defaultable assets in an incomplete information model where the default time is given by a first hitting time of an unobservable process. We show that in a fairly general Markov setting, the indicator function of the default has an absolutely continuous compensator. Given this compensator we then discuss the optional projection of a class of semimartingales onto the filtration generated by the observation process and the default indicator process. Available formulas for the pricing of defaultable assets are analyzed in this setting and some alternative formulas are suggested.
Let X be a Markov process taking values in E with continuous paths and transition function (P s,t... more Let X be a Markov process taking values in E with continuous paths and transition function (P s,t). Given a measure µ on (E, E), a Markov bridge starting at (s, ε x) and ending at (T * , µ) for T * < ∞ has the law of the original process starting at x at time s and conditioned to have law µ at time T *. We will consider two types of conditioning: a) weak conditioning when µ is absolutely continuous with respect to P s,t (x, •) and b) strong conditioning when µ = ε z for some z ∈ E. The main result of this paper is the representation of a Markov bridge as a solution to a stochastic differential equation (SDE) driven by a Brownian motion in a diffusion setting. Under mild conditions on the transition density of the underlying diffusion process we establish the existence and uniqueness of weak and strong solutions of this SDE.
Abstract We solve explicitly a Bayesian sequential estimation problem for the drift parameter μ o... more Abstract We solve explicitly a Bayesian sequential estimation problem for the drift parameter μ of a fractional Brownian motion under the assumptions that a prior density of μ is Gaussian and that a penalty function is quadratic or Dirac-delta. The optimal stopping time for this case is deterministic.
ABSTRACT We study a stochastic model for a market with two tradeable assets where the price of th... more ABSTRACT We study a stochastic model for a market with two tradeable assets where the price of the first asset is implied by the value of the second one and the state of a partially ‘hidden’ control process. We derive a closed expression for the value of the first asset, as a function of the price for the second and the most recent observation of the control process. We show how the model can be applied to EU markets for carbon emissions. The 5th Actuarial and Financial Mathematics Day took place on 9 February 2007.
Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own fi... more Given a Markovian Brownian martingale Z, we build a process X which is a martingale in its own filtration and satisfies X 1 = Z 1. We call X a dynamic bridge, because its terminal value Z 1 is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration F X and the filtration F X,Z jointly generated by X and Z. Our construction is heavily based on parabolic PDE's and filtering techniques. As an application, we explicitly solve an equilibrium model with insider trading, that can be viewed as a non-Gaussian generalization of Back and Pedersen's [3], where insider's additional information evolves over time.
The mainstream in finance tackles portfolio selection based on a plug-in approach without conside... more The mainstream in finance tackles portfolio selection based on a plug-in approach without consideration of the main objective of the inferential situation. We propose minimum expected loss (MELO) estimators for portfolio selection that explicitly consider the trading rule of interest. The asymptotic properties of our MELO proposal are similar to the plug-in approach. Nevertheless, simulation exercises show that our proposal exhibits better finite sample properties when compared to the competing alternatives, especially when the tangency portfolio is taken as the asset allocation strategy. We have also developed a graphical user interface to help practitioners to use our MELO proposal.
This document is the author's final manuscript accepted version of the journal article, incorpora... more This document is the author's final manuscript accepted version of the journal article, incorporating any revisions agreed during the peer review process. Some differences between this version and the published version may remain. You are advised to consult the publisher's version if you wish to cite from it.
International Journal of Theoretical and Applied Finance, 2009
We propose a model for trading in emission allowances in the EU Emission Trading Scheme (ETS). Ex... more We propose a model for trading in emission allowances in the EU Emission Trading Scheme (ETS). Exploiting an arbitrage relationship we derive the spot prices of carbon allowances given a forward contract whose price is exogenous to the model. The modeling is done under the assumption of no banking of carbon allowances (which is valid during the Phase I of Kyoto protocol), however, we also discuss how the model can be extended when banking of permits is available. We employ results from filtering theory to derive the spot prices of permits and suggest hedging formulas using a local risk minimisation approach. We also consider the effect of intermediate announcements regarding the net position of the ETS zone on the prices and show that the jumps in the prices can be attributed to information release on the net position of the zone. We also provide a brief numerical simulation for the price processes of carbon allowances using our model to show the resemblance to the actual data.
We study an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm ... more We study an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm in the framework of Back [2]. The market consists of a risk-neutral informed agent, noise traders and a market maker who sets the price using the total order. When the insider does not trade, the default time possesses a default intensity in market's view as in reduced-form credit risk models. However, we show that, in the equilibrium, the modelling becomes structural in the sense that the default time becomes the first time that some continuous observation process falls below a certain barrier. Interestingly, the firm value is still not observable. We also establish the no expected trade theorem that the insider's trades are inconspicuous.
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