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In this article we will propose a novel, self-financing, dynamic and path dependent portfolio trading strategy which is based on the well known principle "sell high – buy low". Trading strategies are important also for the hedging problem selling/buying an option. The main problem of the writer of an option is how to invest the amount that she has received selling the option therefore the proposed trading strategy play an important role here. We will see that the hedging problem reduces to an optimization one and therefore the portfolio optimization and the hedging problem are closely related. We will also propose a deterministic portfolio selection method (i.e., without making any assumption-guess about the assets) and a notion of a deterministic fair price of an option. [ABSTRACT FROM AUTHOR]

There is no method of predicting the price of an option other than hedging strategies such as the binomial hedging strategy, the Black–Scholes hedging strategy and others. We will study these two basic hedging strategies in terms of their feasibility, and we will see that the Black–Scholes hedging strategy is not feasible because this strategy demands instantaneously rebuilding the replicating portfolio. Consequently, the real world prices of the options are not relevant at all with the Black–Scholes hedging strategy! We will suitably redefine the binomial hedging strategy so that it will be practically useful and present other feasible and generally more effective hedging strategies with some of them practically useful for options with no tradable underlying assets. Finally, we will mention some open questions related to the above. [ABSTRACT FROM AUTHOR]

We are interested in the numerical approximation of solutions of nonlinear stochastic differential equations, that appear in financial mathematics. Here, we study the Aït-Sahalia model. We propose an explicit numerical scheme where we actually approximate the Lamperti transformation of the original stochastic differential equation and then transform back. The proposed method is domain preserving and is proven to converge strongly to the solution process with order at least 1 with no extra restrictions on the step-size Δ. Numerical experiments verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this note, we describe a new approach to the option pricing problem by introducing the notion of the safe (and acceptable) price for the writer of an option, in contrast to the fair price used in the Black–Scholes model. Our starting point is that the option pricing problem is closely related with the hedging problem by practical techniques. Recalling that the Black–Scholes model does not give us the price of the option but the initial value of a replicating portfolio, we observe easily that this has a serious disadvantage because it assumes the building of this replicating portfolio continuously in time, and this is a disadvantage of any model that assumes such a construction. Here we study the problem from the practical point of view concerning mainly the over-the-counter market. This approach is not affected by the number of the underlying assets and is particularly useful for incomplete markets. In the usual Black–Scholes or binomial approach or some other approaches, one assumes that one can invest or borrow at the same risk-free rate r > 0 , which is not true in general. Even if this is the case, one can immediately observe that this risk-free rate is not a universal constant but is different among different people or institutions. So the fair price of an option is not so much fair! Moreover, the two sides are not, in general, equivalent against the risk; therefore, the notion of a fair price has no meaning at all. We also define a variant of the usual binomial model, by estimating safe upward and downward rates u , d , trying to give a cheaper safe or acceptable price for the option. [ABSTRACT FROM AUTHOR]

We derive a nontrivial solution for a Neumann noncoercive hemivariational inequality using the critical point theory for locally Lipschitz functionals. We use the Mountain-Pass theorem due to Chang (1981).

We study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of ℒ 2 {\mathcal{L}^{2}} -convergence of the truncated SD method and showed that it can be arbitrarily close to 1 2 {\frac{1}{2}} ; see [I. S. Stamatiou and N. Halidias, Convergence rates of the semi-discrete method for stochastic differential equations, Theory Stoch. Process. 24 2019, 2, 89–100]. We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE. Numerical simulations support our theoretical findings. [ABSTRACT FROM AUTHOR]

In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A1, A2, . . . , Ak. [ABSTRACT FROM AUTHOR]

We investigate the existence of multiple solutions to quasilinear elliptic problems containing Laplace like operators ( -Laplacians). We are interested in Neumann boundary value problems and our main tool is Brézis-Nirenberg's local linking theorem.

In this paper we prove two theorems for noncoercive elliptic boundary value problems using the critical point theory of Chang and the subdifferentiable of Clarke. The first result is for a Dirichlet noncoercive problem and the second one is for Neumann elliptic problem with nonlinear multivalued boundary conditions. We use the mountain-pass and the saddle-point theorems to obtain nontrivial solutions for these problems.

Our aim in this note is to extend the semi-discrete technique by combine it with the split step method.We apply our new method to the Ait-Sahalia model and propose an explicit and positivity preserving numerical scheme. [ABSTRACT FROM AUTHOR]

We are interested in the numerical solution of stochastic differential equations with non-negative solutions. Our goal is to construct explicit numerical schemes that preserve positivity, even for super-linear stochastic differential equations. It is well known that the usual Euler scheme diverges on super-linear problems and the tamed Euler method does not preserve positivity. In that direction, we use the semi-discrete method that the first author has proposed in two previous papers. We propose a new numerical scheme for a class of stochastic differential equations which are super-linear with non-negative solution. The Heston 3/2-model appearing in financial mathematics belongs to this class of stochastic differential equations. For this model we prove, through numerical experiments, the 'optimal' order of strong convergence at least 1/2 of the semi-discrete method. [ABSTRACT FROM AUTHOR]

We study the two-factor CIR model and using the main idea of two of our previous papers, we propose explicit numerical schemes that preserves positivity and converges strongly in the mean square sense to the true solution. Our results apply for the multi-factor and the one-factor cases as well. [ABSTRACT FROM AUTHOR]

In this paper we generalize an explicit numerical scheme for the CIR process that we have proposed before. The advantage of the new proposed scheme is that preserves positivity and is well posed for a (little bit) broader set of parameters among the positivity preserving schemes. The order of convergence is at least logarithmic in general and for a smaller set of parameters is at least 1/4. [ABSTRACT FROM AUTHOR]

In this note we work on the construction of positive preserving numerical schemes for a class of multidimensional stochastic differential equations. We use the semi discrete idea that we have proposed before proposing now a numerical scheme that preserves positivity on some multidimensional stochastic differential equations converging strongly in the mean square sense to the true solution. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new point of view in numerical approximation of stochastic differential equations. By using Ito–Taylor expansions, we expand only a part of the stochastic differential equation. Thus, in each step, we have again a stochastic differential equation which we solve explicitly or by using another method or a finer mesh. We call our approach as a semi-discrete approximation. We give two applications of this approach. Using the semi-discrete approach, we can produce numerical schemes which preserves monotonicity so in our first application, we prove that the semi-discrete Euler scheme converge in the mean square sense even when the drift coefficient is only continuous, using monotonicity arguments. In our second application, we study the square root process which appears in financial mathematics. We observe that a semi-discrete scheme behaves well producing non-negative values. [ABSTRACT FROM AUTHOR]

An existence result is proved for backwards stochastic differential equations (BSDEs) with a generator f(t, x, z) which is possibly discontinuous in the x variable. For this comparison results are first established for BSDEs with the generator satisfying a generalized Lipschitz condition in its x variable. [ABSTRACT FROM AUTHOR]

In this paper we consider stochastic equations in Banach spaces. Our first result is a comparison theorem. As an application we prove an existence theorem in the case when the drift coefficient is nonsmooth. The present studies extend some results both for deterministic and stochastic equations in infinite dimensional case. [ABSTRACT FROM AUTHOR]

The purpose of this article is to consider a stochastic integral equation driven by semimartingale with discontinuous and increasing drift part. We discuss the existence of strong solutions using lower and upper solutions method and a fixed point theorem for ordered topological space. Finally we present some applications in finance. [ABSTRACT FROM AUTHOR]

In this paper we give two existence theorems for a class of elliptic problems in an Orlicz-Sobolev space setting concerning both the sublinear and the superlinear case with Neumann boundary conditions. We use the classical critical point theory with the Cerami (PS)-condition. [ABSTRACT FROM AUTHOR]

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals. [Copyright &y& Elsevier]