Enlarged Filtrations and Indistinguishable Processes

Let \(\{W_{t}\}_{t\geq 0}\) be a Brownian motion with respect to a probability measure \(\mathbb{P}\). Take a constant \(\theta\), and consider \(X_{t} = W_{t} +\theta t\), \(0 \leq t <\infty\), which is called a Brownian motion with drift. Our goal is to find a probability measure \(\mathbb{Q}\) for which X t , 0 ≤ t ≤ T, is a Brownian motion for some fixed T. We require an additional condition that \(\mathbb{Q}\) is equivalent to \(\mathbb{P}\), i.e., \(\mathbb{P}(A) = 0\) if and only if \(\mathbb{Q}(A) = 0\). In other words, an event occurs with positive \(\mathbb{P}\)-probability if and only if it happens with positive \(\mathbb{Q}\)-probability. Such a condition is important in financial applications since we have to deal with the same set of asset price movements even when we switch to a new probability measure. Igor Girsanov proved the existence of such a measure \(\mathbb{Q}\). We will find first a necessary condition for the existence of an equivalent probability measure \(\mathbb{Q}\) for which a Brownian motion with drift is a Brownian motion. Such a necessary condition will turn out to be crucial in defining \(\mathbb{Q}\).

We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Levy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integro-differential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Levy process posseses only negative jumps. This is, however, the topic of a companion paper [7].

In the years following the publication of Black and Scholes (J Political Econ, 81(3), 637–654, 1973), numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump-diffusion models, and models based on Levy processes. These all have their own shortcomings, and evidence suggests that none is up to the task of satisfactorily pricing and hedging extremely long-dated claims. Since they all fall within the ambit of risk-neutral valuation, it is natural to speculate that the deficiencies of these models are (at least in part) attributable to the constraints imposed by the risk-neutral approach itself. To investigate this idea, we present a simple two-parameter model for a diversified equity accumulation index. Although our model does not admit an equivalent risk-neutral probability measure, it nevertheless fulfils a minimal no-arbitrage condition for an economically viable financial market. Furthermore, we demonstrate that contingent claims can be priced and hedged, without the need for an equivalent change of probability measure. Convenient formulae for the prices and hedge ratios of a number of standard European claims are derived, and a series of hedge experiments for extremely long-dated claims on the S&P 500 total return index are conducted. Our model serves also as a convenient medium for illustrating and clarifying several points on asset price bubbles and the economics of arbitrage.

A Visual Classification of Local Martingales

We introduce a suitable backward stochastic differential equation (BSDE) to represent the value of an optimal control problem with partial observation for a controlled stochastic equation driven by Brownian motion. Our model is general enough to include cases with latent factors in mathematical finance. By a standard reformulation based on the reference probability method, it also includes the classical model where the observation process is affected by a Brownian motion (even in presence of a correlated noise), a case where a BSDE representation of the value was not available so far. This approach based on BSDEs allows for greater generality beyond the Markovian case, in particular our model may include path-dependence in the coefficients (both with respect to the state and the control), and does not require any nondegeneracy condition on the controlled equation. We use a randomization method, previously adopted only for cases of full observation, and consisting in a first step, in replacing the control by an exogenous process independent of the driving noise and in formulating an auxiliary (“randomized”) control problem where optimization is performed over changes of equivalent probability measures affecting the characteristics of the exogenous process. Our first main result is to prove the equivalence between the original partially observed control problem and the randomized problem. In a second step, we prove that the latter can be associated by duality to a BSDE, which then characterizes the value of the original problem as well.

Delta and Gamma Hedging of Mortality and Interest Rate Risk

We study mean-field control (MFC) problems with common noise using the control randomisation framework, where we substitute the control process with an independent Poisson point process, controlling its intensity instead. To address the challenges posed by the mean-field interactions in this randomisation approach, we reformulate the admissible control as L0-valued processes adapted only to the common noise. We then construct the randomised control problem from this reformulated control process, and show its equivalence to the original MFC problem. Thanks to this equivalence, we can represent the value function as the minimal solution to a backward stochastic differential equation (BSDE) with constrained jumps. Finally, using this probabilistic representation, we derive a randomised dynamic programming principle (DPP) for the value function, expressed as a supremum over equivalent probability measures.

Asymptotic optimality of the generalized [formula omitted] rule under model uncertainty

Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform

We use the Cox process (or a doubly stochastic Poisson process) to model the claim arrival process for catastrophic events. The shot noise process is used for the claim intensity function within the Cox process. The Cox process with shot noise intensity is examined by piecewise deterministic Markov process theory. We apply the model to price stop-loss catastrophe reinsurance contract and catastrophe insurance derivatives. The asymptotic distribution of the claim intensity is used to derive pricing formulae for stop-loss reinsurance contract for catastrophic events and catastrophe insurance derivatives. We assume that there is an absence of arbitrage opportunities in the market to obtain the gross premium for stop-loss reinsurance contract and arbitrage-free prices for insurance derivatives. This can be achieved by using an equivalent martingale probability measure in the pricing models. The Esscher transform is used for this purpose.

We seek to define statistical solutions of hyperbolic systems of conservation laws as time-parametrized probability measures on $p$-integrable functions. To do so, we prove the equivalence between probability measures on $L^p$ spaces and infinite families of \textit{correlation measures}. Each member of this family, termed a \textit{correlation marginal}, is a Young measure on a finite-dimensional tensor product domain and provides information about multi-point correlations of the underlying integrable functions. We also prove that any probability measure on a $L^p$ space is uniquely determined by certain moments (correlation functions) of the equivalent correlation measure. We utilize this equivalence to define statistical solutions of multi-dimensional conservation laws in terms of an infinite set of equations, each evolving a moment of the correlation marginal. These evolution equations can be interpreted as augmenting entropy measure-valued solutions, with additional information about the evolution of all possible multi-point correlation functions. Our concept of statistical solutions can accommodate uncertain initial data as well as possibly non-atomic solutions even for atomic initial data. For multi-dimensional scalar conservation laws we impose additional entropy conditions and prove that the resulting \textit{entropy statistical solutions} exist, are unique and are stable with respect to the $1$-Wasserstein metric on probability measures on $L^1$.

Market Imperfections, Equilibrium and Arbitrage

This paper represents a model for risk management in a firm which exercises control of its risk as well as potential profit by choosing different business activities among those available to it. Furthermore, the firm has an option of investing its reserve in a financial market consisting of a risk-free asset (bond) and a risky asset (stock). The example we consider is that of a large corporation such as an insurance company, whose liquid assets in the absence of control and investments fluctuate as a Brownian motion with a constant positive drift and a constant diffusion coefficient. We interpret the diffusion coefficient as risk exposure, while drift is associated with potential profit. At each moment of time there is an option to reduce risk exposure, simultaneously reducing the potential profit, like using proportional reinsurance with another carrier for an insurance company. The company invests its reserve in a financial market, which is described by a classical Black–Scholes model. The management of the company also controls the dividend pay-outs to shareholders. The objective is to find a policy, consisting of investment strategy, risk control and dividend distribution scheme, which maximizes the expected total discounted dividends paid out until the time of bankruptcy. We apply the theory of controlled diffusions to solve the problem and show that there is a level u 1>0 such that the optimal action is to distribute all the reserve in excess of u 1 as dividends. Furthermore, there exists a constant x 0, with x 0<u 1, such that the risk exposure monotonically increases on (0, x 0) from zero to the maximum possible. The optimal choice of investments depends on the market price of risk , where r 0, r 1 denotes the mean rate of return of bond and stock respectively and σP denotes the volatility of the stock price. We get the following results. (1) m p≤0: invest everything in bond. (2) m p is large: invest everything in stock. (3) m p is small: there exists x 0<x 1<u 1, such that the optimal fraction of the reserve invested in stock is constant when the current reserve x is less than x 0 and it is an increasing function of x on [x 0, x 1] with all the reserve to be invested in stock whenever the reserve level exceeds x 1.

We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions B1t and B2t, the main result is that the range of possible values for is the same for Markovian pairs and all pairs of Brownian motions, that is with φ being the cumulative distribution function of a standard Gaussian random variable.

Let b be a Borel measurable IRd ‐ valued function, defined on some Borel subset of IRd. Consider the d‐ dimensional SDEmagnified imagewith singular drift b. A local solution (up to σ) is a tuple (X, W, Q,σ) where X is a stochastic process, W is a Brownian motion under the probability measure Q, and σ is a strictly optional time (i.e., stopping time) such that the above equation is satisfied for all t &lt; σ. Such a local solution was constructed by the author in an earlier paper under very mild conditions on b. In this paper we give criteria for the global existence of the solution, i. e., for Q(σ = ∞) = 1.